ijf^a 
 
 f 
 
 
IN MEMORIAM 
 FLORIAN CAJORl 
 
/-u 
 
 it^try 
 
TREATISE ON ALGEBRA, 
 
 EMBRACING, 
 
 BESIDES THE ELEMENTARY PRINCIPLES, ALL THE 
 HIGHER PARTS USUALLY TAUGHT IN 
 
 COLLEGES 
 
 COXTAINING 
 
 MOREOVER, THE NEW METHOD OF CUBIC AND HIGHER EQUATIONS 
 
 AS WELL AS THE DEVELOPMENT AND APPLICATION 
 
 OF THE MORE RECENTLY DISCOVERED 
 
 THEOREIM OF STURM. 
 
 BY GEORGE R. PERKINS, A. M., 
 
 PROFESSOR OP MATHEMATICS IX NEW-VORK STATE NORMAL SCHOOL. AUTHOR 
 " ELEMENTART ARITHMETIC,"' '" HIGHKR ARITHMETIC," "' ELEMENTS 
 OF ALGEBRA," ETC., ETC. 
 
 SECOND EDITION, REVISED, ENLARGED AND IJIPROVED. 
 
 NEW YORK: 
 PUBLISHED BY D. APPLETON AND CO. 
 
 UTICA: 
 
 HAWLEY, FULLER AND CO. 
 
 1850. 
 
FIRST EDITION. 
 
 Entered, according to Act of Congress, in the year 1841, by 
 
 GEORGE R. PERKINS, 
 in the Clerk's Office of the Northern District of New- York. 
 
 SECOND EDITION. 
 Entered, according to Act of Congress, in the year 1847, by 
 
 GEORGE R. PERKINS, 
 in the Clerk's Office of the Northern District of New-York. 
 
 C. VAN BENTHUYSEN AND CO. 
 fiTEREOTYPERS. 
 

 PREFACE 
 
 TO THE FIRST EDITION. 
 
 In presenting: this volume to the Public, I would not claim to 
 have unfolded many new principles of Algkgra; I only claim to 
 have judiciously combined and arraii^^ed principles already known. 
 By commencing this work with the most elementary parts, and 
 gradually ascendinjj to the more comftlicated, I have designed to 
 adapt it to the wants of students of every grade. 
 
 While I acknowledge, that, in general, the principles have long 
 been known, I think I am justifiable in claiming some of the methods 
 of operation as original. 
 
 This work will be found to contain, for the first time, I believe 
 in any American school book, a demonstration and application of 
 Stubm's Theohkm; by the aid of which, we may at once deter- 
 mine the number of real roots, of any Algebraic Equation, with 
 much more ease, than could be done by any previously discovered 
 methods. 
 
 The method of finding the numerical values of the roots of 
 cubic and higlier equations, as fully explained under the last chapter, 
 will, no doubt, be new to many, and interesting to all lovers of this 
 science. It is particularly interesting on account of the ease witli 
 which it resolves itself into the method of extracting any root of a 
 number, as explained in my IIioiikk Akitiimetic. 
 
 It would be extrcmelv diflficult to point out the exact sources 
 from which I hiive drawn for this work, and even could I do so, these 
 principles have been so long in use, we could not with safety say 
 when, and with whom, they each originated. While I acknowledge 
 the aid of many works on this science, I would give by far the 
 greatest share of credit, to the eighth edition of Bourdon's most 
 excellent treatise on Algebra. 
 
 rtica, July, 1842. GEORGE R. PERKINS. 
 
 iv'tmn-9i'<a 
 
PREFACE 
 
 TO THE SECOND EDITION. 
 
 TiiK ]>ri'sei.t Edition luis been very carefully Revised and 
 cinsideiably Enlarg'cd. One entire Chapter on tiic subject of Con- 
 •ii.xuED FkactU/Xs, which are treated in quite a general manner 
 has been added. The subject of Recurring Series has been 
 re-written, and n-.uch ^^n1pli^led, and many other chans^es, which 
 we deemed to be ini'provemenls, have been introduced. 
 
 Ilavin;^ almost daily made lue of tl.is w<^'rk in my Classes, since 
 its Publication, and always having- had in view tho ciianges which it 
 would be desirable to make, in order to improve the work, we feel 
 that we are now ])rej arcil to present thi* pr^ sent edition as quite an 
 improvement upon ihe first. It is bclitved il will be found to con- 
 tain a i)rclty lull ami C( nijilele dcvelopmrnt of all ihe \arioussul) 
 jects of Algebra, usually taught in our Colleges. 
 
 As wehavealrcady ])rej);ire(l a smalhr work, especially dcsignod 
 tor i)rimary schools, it has been our aim to adapt this Treatise to 
 the wants of the more advanced Schools and Colleges. 
 
 Utica, Januarti, 1S17. GEORGE R. PE.'vKI.'.'S. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 DEFINITIONS AND PRELIMINAKY RUI.Ei>. 
 
 Definitions, 9 
 
 Addition, 15 
 
 Subtraction, 21 
 
 Multiplication, 24 
 
 Division 30 
 
 CHAPTER 11. 
 
 ALGEBRAIC FRACTIONS. 
 
 Reduction of monomial fractions. 38 
 
 Greatest common measure , 40 
 
 Reduction of polynomial fractions, 46 
 
 Redaction of a mixed quantity to the form of a fraction, 48 
 
 Reduction of a fraction to an entire or mixed quantity, 50 
 
 Reduction of fractions to a common denominator, 52 
 
 Addition of fractions, 53 
 
 Subtraction of fractions, 54 
 
 Multiplication of fractions, 56 
 
 Division of fractions, 57 
 
 CHAPTER III. 
 
 SIMPLE EQUATIONS. 
 
 Equations defined, 59 
 
 Axioms used in solving equations CO 
 
 Clearing equations of fractions, 60 
 
 Transposinij the terms of an equation C,2 
 
 General rule for solving simple equations 64 
 
 QuPitions involving simple equations 67 
 
 ECiUATIONS OF TWO OR MORE UNKNOWN QUANTniES. 
 
 Elimination by addition and subtraction, 79 
 
 Elimination by comparison, 84 
 
 Elimination by substitution, 87 
 
 Questions involving two or more unknown quantities, 91 
 
 General solution of literal equations, 106 
 
 Checker-board process, 113 
 
n 
 
 CHAPTER IV. 
 
 INVOLUTION. 
 
 To involve a monomial, 122 
 
 To involve a polynomial, 124 
 
 EVOLUTION. 
 
 To extract a root of a monomial, 127 
 
 General rule for extracting any root of a polynomial, 131 
 
 Enunciation of the square of any polynomial, 133 
 
 Rule for extracting the square root of a polynomial, 134 
 
 New rule for extracting the cube root of a polynomial, 136 
 
 SURD QUANTITIES. 
 
 To reduce a quantity to the form of a surd, 141 
 
 To reduce surds to a common index, 142 
 
 To reduce surds to their simplest form, 143 
 
 Addition and subtraction of surds 145 
 
 Multiplication and division of surds, 146 
 
 Extraction of the square root of a binomial surd, 147 
 
 To find multipliers which will cause surds to become rational,.... 152 
 
 IMAGINARY QUANTITIES. 
 
 Defined, 159 
 
 Multiplication of imaginaries, 161 
 
 Division of imaginaries, 162 
 
 Interpretation of the symbols --, — , -, io6 
 
 CHAPTER V. 
 
 QUADRATIC EQUATIONS. 
 
 Incomplete quadratic equations, 168 
 
 Complete quadratic equations, 172 
 
 First rule for quadratic equations, 174 
 
 Second rule for quadratic equations, 180 
 
 Equations of several unknown quantities involving quadratic equa- 
 tions, 181 
 
 Questions involving quadratic equations 196 
 
 Properties of the roots of quadratic equations, 202 
 
 Examples giving imaginary roots, 214 
 
 CHAPTER VI. 
 
 RATIO AND PROGRESSION. 
 
 Ratio defined, 217 
 
 Table giving formulas for the 20 cases of arithmetical progression, 221 
 
 Geometrical ratio, 224 
 
 Table of all the formulas of geometrical progression 230 
 
 Harmonical proportion, 234 
 
CONTENTS, VU 
 
 CHAPTER VII. 
 
 Method of indeterminate coefficients, 237 
 
 Binomial Theorem demonstrated, 243 
 
 Application of the binomial theorem, 248 
 
 Multinomial Theorem demonstrated, 253 
 
 Examples under the multinomial theorem, 257 
 
 Reversion of series, 258 
 
 Differential method of series, 261 
 
 Summation of infinite series, 267 
 
 Recurring series, 272 
 
 A geometrical series may be also a recurring series, 277 
 
 CHAPTER VIII. 
 
 COMTINUED FRACTIONS. 
 
 Defined, 283 
 
 General rule for converting a continued fraction into its approxima- 
 tive values, 286 
 
 To convert a common fraction into a continued one, 287 
 
 Rule for converting the common kind of continued fractions into its 
 
 approximative ralues, 294 
 
 The square roots of surds found by continued fractions, 295 
 
 CHAPTER IX. 
 
 LOGARITHMS. 
 
 Defined, 309 
 
 Logarithmic formula found 311 
 
 Numerical calculation of logarithms, 313 
 
 Exponential Theorem 320 
 
 Application of logarithms, 323 
 
 Exponential equations resolved by logarithms, 325 
 
 Compound interest and annuities by logarithms 326 
 
 CHAPTER X. 
 
 OEWERAL PROPERTIES OF EqUATIONS. 
 
 The root of an equation defined 336 
 
 The number of positive and negative roots, 339 
 
 The relation between the roots and coefficients, 341 
 
 Impossible roots occur in pairs, 342 
 
 The sum of the 7nth powers of the roots, 34-1 
 
 To cause the second term to vanish, 345 
 
 Method of finding the derived polynomials, 348 
 
 Equations having equal roots, 352 
 
 Recurring equations, 355 
 
Vlll CONTENTS. 
 
 Binomial equations, 359 
 
 General solution of an equation of the third degree, 365 
 
 General solution of an equation of the fourth degree. 372 
 
 Sturm's Theorem defined, 375 
 
 Demonstration of Sturm's theorem, 376 
 
 Application of Sturm's theorem, 380 
 
 General method of elimination among equations above the first 
 degree, 385 
 
 CHAPTER XI. 
 
 Numerical solution of cubic equations, 392 
 
 Numerical solution of equations above the third degree 413 
 
 I 
 
TREATISE ON AL&EBRA. 
 
 CHAPTER I. 
 
 DEFINITIONS AND PRELIMINARY RULES. 
 
 DEFINITIONS. 
 
 {Article 1.) Algebra is that branch of Mathematics, in 
 which the calculations are performed by means of letters and 
 signs or symlols. 
 
 (2.) In Algebra, quantities, whether given or required, 
 are usually represented by letters. The first letters of the 
 alphabet are, for the most part, used to represent known 
 quantities ; and the final letters are used for the unknown 
 quantities. 
 
 (3.) The symbol =, is called the sign oi Equality ; and 
 denotes that the quantities between which it is placed, are 
 equal or equivalent to each other. Thus $1 = 100 cents, 
 which is read, one dollar equals one hundred cents. Again, 
 a = 6, which is read, a equals h. 
 
 (4.) The symbol -f? is called plus ; and denotes that the 
 quantities between which it is placed, are to be added 
 together. Thus, a -|- 6 = c, which is read, a and h added, 
 equals c. Again, a-\-h-\-c=d-\-x^ which is read, o, 
 h and c added, equals d added to x. 
 2 
 
10 DEFINITIONS. 
 
 (5.) The symbol — , is called minus; and denotes that 
 the quantity \vhich is placed at the right of it is to be sub- 
 tracted from the quantity on the left. Thus, a — 6 = c, 
 which is read, a diminished by b equals c. 
 
 (6.) The symbol X, is called the sign oi inultiplication ; 
 and denotes that the quantities between which it is placed 
 are to be multiplied together. Thus, a X b = c, which is 
 read, a multiplied by b, equals c. Multiplication is also 
 represented by placing a dot between the factors, or terms 
 to be multiplied. Thus, a . bis the same as a X b. Another 
 method, which is used as frequently as either of the above, 
 is to unite the quantities in the form of a word. Thus, a6c 
 is the same as a X 6 X c, or a . 6 . c. 
 
 (7.) The symbol -r, is called the sign of divisioji ; and 
 denotes that the quantity on the left of it is to be divided 
 by the quantity on the right. Thus, a -^ b = c^ which is 
 read, a divided by b equals c. Division is also indicated by 
 placing the divisor under the dividend, with a horizontal 
 
 X 
 
 line between them like a vulgar fraction. Thus, - is the 
 
 fa Jy 
 
 same as x -r y. 
 
 (8.) When quantities are enclosed in a parenthesis, brace, 
 or bracket, they are to be treated as a simple quantity. Thus, 
 (a 4- 6) -7- c, indicates that the sum of a and b is to be divided 
 by c. Again, (x — y) -^z=[x — y]^z = \x — y]^Zj 
 each of which expressions is read, y subtracted from x and 
 the remainder divided by z. The same thing may also be 
 expressed by a bar or vinculum. Thus, x — y -r- z, which 
 is read the same as the last three expressions. 
 
 (9.) The symbol >, is called the sign of tne^wa/zYy; and 
 is used to express that the quantities between which it is 
 placed are unequal. Thus, 6 > a indicates that b is greater 
 than a ; and 2> < c denotes that b is less than c. 
 
UEFINlTIOiVS. 11 
 
 (10.) When a quantity is added to ilscli several liiues, as 
 c -\- c -'r c -{- c, we need write it but onre, by placing before 
 it a number to show how many times it has been taken. 
 Thus, f -{- c -f- c -}- c = 4c. The number which is thus 
 placed before the quantity is called the coefficitnt of the 
 quantity. In the above example, 4 is the coefficient of c. 
 A coefficient nniy consist, itself, of a letter. Thus, n is the 
 coefficient of x in the ex])ression nx; so also may x be 
 regarded as the coefficient of n in the same expression. 
 
 (11.) The continued product of a quantity into itself is, 
 usually, denoted by writing the quantity once, and placing 
 a number over the quantity, a little to the right. Thus, 
 o X (t X a is the same as u^. The number thus placed over 
 the quantity, is called the exponmt of the quantity. Thus, 
 5 is the exponent of a in the expression a^, and denotes 
 that a is to be multiplied into itself, as a factor, five times. 
 
 (12.) When a quantity is multiplied continually into 
 itself, the result is called a power of the quantity. Thus, a^ 
 is the sixth power of a, and a^ is the third power of a, the 
 exponent always indicating the degree of the power. 
 
 W^hen a quantity is written without any exponent, it is 
 understood that its exponent is a unit. 
 
 Thus, a is the same as a*, and (x + v) X m is the same 
 as (x-f-y)' X in\ 
 
 (13.) The symbol v/, is called the radical sign; and' 
 denotes that a root of the quantity, over which it is placed,, 
 is to be extracted. Thus, 
 
 V ar, or simply -/ a:, denotes the square root of x. 
 
 V X denotes the cube root of x. 
 
 V X denotes the fourth root of x. 
 
 The number placed over the radical is called the index 
 of the root. Thus, 2, 3, and 4 are, respectively, the indices 
 of the square root, cube root, and fourth root. 
 
12 DEFINITIONS. 
 
 (14.) A root of a quanlUy may also be represented by 
 means of a fractional exponent. Thus, the square root of a 
 
 is a"; the cube root of a is a'; th • fourth root of a is a* ; 
 and so on for other roots. 
 
 By the same notation, a^ is the cube root of the square 
 of a, or the square of the cube root of a. For the same 
 
 3 
 
 reason a^ is the fifth root of the third power of «, or the 
 third power of the fifth root of a. 
 
 (15.) The reciprocal of a quantity is a unit divided by 
 
 that quantity. Thus, - is the reciprocal of a; also x is the 
 reciprocal of 3. 
 
 (16.) The symbol .-., is equivalent to the phrase, there- 
 fore, or consequently. 
 
 (17.) When algebraic quantities are written without any 
 sign prefixed, the sign plus is understood, and the quantities 
 are said to be positive or affirmative ; and those having the 
 sign minus prefixed are called negative quantities. Thus, 
 7. = -f- 0, /) = + 6, are each positive quantities ; whilst 
 — a, — h, are negative quantities. When the sign — , is 
 prefixed to an isolated term, as — a, — 6, it is not to be 
 considered as a symbol of operation, but as a symbol of 
 condition, merely showing that a and & are in a state or 
 condition directly opposite to that denoted by -|- « and -{- h. 
 Thus, if the degrees of the thermometer above zero are 
 called +, then those below must be called — . 
 
 (18.) An algebraic expression composed of two or more 
 terms connected by -f- or — , is called a polyyiomial . A 
 polynomial composed of but two terms, is called a hhio- 
 mial ; one composed of three terms, is called a trinomial. 
 
DEFINITIONS. 13 
 
 Thus, 
 
 3a + 46, ) 
 
 7a:' — 3]/, > are binomials. 
 3o^ — x"', ) 
 3a= + 46 — x, ) 
 4771 — y + a, > are trinomials. 
 5g — a; + 3/, 5 
 (19.) Each of the literal factors which compose any terra, 
 is called a dimension of this term : the degree of a term is 
 the number of the dimensions or factors. Thus, 
 7a, ) are terms of one dimension, or of 
 56, ^ the first degree. 
 5gx, } are terms of two dimensions, or 
 5a:]/, ^ of the second degree. 
 la'P = laabbh, ? are terms of five dimensions, or 
 3x^ = oxxxxx, ) of the fifth degree. 
 
 (20.) A polynomial is said to be homogeneous, when all 
 its terms are of the same degree. Thus, 
 
 So — 5a; -|- 2y, ^ are homogeneous polynomials of 
 
 b — y -\- m,j \ the first degree. 
 4ff* + 2x^ — jy, ? are homogeneous polynomials of 
 1am — c" + a^, \ the second degree. 
 5a"b^ — 6o^ — 4z'*y, ) are homogeneous polynomials 
 3a^6 —b^ -f-4a3/)2, J of the fifth degree. 
 
 (21.) Any combination of letters,by the aid of algebraic 
 signs, is called an algebraic expression. Thus, 
 
 _ ? is an algebraic expression, denoting seven times 
 '^ the quantity x. 
 
 ) is an algebraic expression, denoting that 
 3 </ b -\- a"\ the quantity a is to be squared, and then 
 
 ) added to three times the square root of h. 
 ( 4- na ^ '^ an algebraic expression, denoting that 
 ^ ' ' ^ the sum of a and b is to be squared. 
 
 The algebraic expression (a -f 6)(" — b) =^ a" — /•', is 
 read, the sum of a and b, multiplied by the dilurence of a 
 and b. equals the difference of the squares of a and b. 
 
14 DEFINITIONS. 
 
 (22.) We will give some identical algtbralc expressions, 
 which may serve to exercise the student in reading algebraic 
 formulas. 
 
 (a-\-b)'^=a^-^2ab+b'. (1) 
 
 (a — 6)* = a2 — 2a6 + R (2) 
 
 {a + by -^ {a — bY = 2a' + 2b-. (3) 
 
 i{a + b)-Ha-b) = b. (5) 
 
 Expression (1) is read, " the square of the sum of a and 
 b is equal to the square of o, plus twice the product of a 
 and 6, plus the square of 6." 
 
 Expression (2) is read, " the square of a diminished by b 
 is equal to the square of a, minus twice the product of a 
 and b, plus the square of 6." 
 
 The student should read the remaining expressions for 
 himself, and should also form other expressions, which he 
 may in like manner translate into common language. He 
 should also substitute particular values for a and 6, in the 
 above expressions, and see if the results on both sides oi 
 the equations are identical. 
 
 Thus, the above expressions become, when 
 a :^ 2, and 6=1. 
 (2 + 1^ = 4+4 + 1=9. (1) 
 
 (2_1)2==4 — 4 + 1 = 1. (2) 
 
 (2 + 1)3+ (2—1) 2=. 8 + 2 = 10. (3) 
 
 H2+l) + i(2-l) = 2. (4) 
 
 U2+l)-H2-l) = l. (5) 
 
 If a = J , and 6=1, they will become 
 
 (i + i)''=i + i + ^ = H. (1) 
 
 U— ^r = i — H-f=3V. (2) 
 
 + ir + (5-i)^=i+f = lf. (3) 
 
 i(i + *)+KJ — 0=1. (4) 
 
 Hh+^)-i{h-h) = h _ _ (5) 
 
 In this way the student should be exercised, until he be- 
 comes familiar with the nature of algebraic expressions. 
 
ADDITION. 16 
 
 ADDITION. 
 
 (23.) Addition, in Algebra, is finding the simplest ex- 
 pression for several algebraic quantities, connected by + 
 
 Suppose we wish to find the sum of 
 
 We first seek the sum of the positive quantities, by pla- 
 cing them under each other as in arithmetical addition, thus, 
 
 + 3a'^6 
 
 -f- 14a^6 = sum of the positive terms. 
 
 Proceeding in the same way with the negative terms, we 
 find 
 
 — lOa'b 
 
 — 5a^b 
 
 — 2a^h 
 
 — 17a^6 = sum of negative terms. 
 
 Therefore the total sum is -|- 14a'6 — 17a'6= — 3a"h. 
 
 We could proceed in a similar way for expressions of a 
 like kind. 
 
16 ADDITION. 
 
 CASE I. 
 
 (24.) When the quantities are alike but have unlike signs, 
 we have this 
 
 RULE. 
 
 I. Place the different terms under each other, add the 
 coefficients of the positive quantifies into one sum, and the 
 coefficients of the negative quantities into another. 
 
 II. Subtract the less from the greater. 
 
 III. Prefix the sign of the greater sum to the remainder, 
 and. annex the common letters. 
 
 EXAMPLES. 
 
 1. What is the sum of 
 
 2ahx — labx — 2ahx -\- 12ahx -\- abx — 3abx ? 
 2abx 
 12abx 
 abx 
 
 Idabx = sum of positire terms. 
 
 — labx 
 
 — 2abx 
 
 — 3abx 
 
 I2abx = sum of negative terms. 
 
 Therefore 15abx — 12abx = Zabx = sum total. 
 
 2. What is the sum of 
 
 7amn — 3ainn -\- 2amn + 5amn — lOamn % 
 
 Ans. amn. 
 
ADDlTIOy. 17 
 
 3. What is the sum of 
 
 ■idaxy — 31axy — lOaxij -\- lOOaxy — laxij -\-Aaxi/ ? 
 
 Ans. 99axy. 
 
 4. What is the sum of 
 
 3 Vox -\- 7 ^/ax — 5 y/ax — 3 v/o.r+10 ^ax — 4 y/ax ? 
 
 Ans. S-yax. 
 
 5. What is the sum of 
 
 i 11 
 
 ia^b — oab- +41av/6 — lab' + lOob'? 
 
 Ans. 43a6^ 
 
 6. What is the sum of 
 
 13ah' +GJvh — 20Jb^ + lOah' 1 
 
 Ans. 9ah\ 
 
 7. What is the sum of 
 
 Ua^b- Vx + aW'x' — 20a-^b^x' + Ga'b" \/x ? 
 
 Ans. — 2a'^b~\/x. 
 
 CASE II. 
 
 (25.) When both quantities and signs are unlike, or some 
 like and others unlike. 
 
 RULE. 
 
 I. Find, the, sum of the like terms as in Case I. 
 
 II. Then write the su7ns one after another, loith their 
 •proper signs. 
 
 EXAMPLES. 
 
 1. What is the sum of 
 
 Sax — 2ah + 4xy — 2ax + 3xy -\-lab — 2xy -{- 6ax 7 
 3 
 
18 
 
 3ax 
 -2ax 
 Gax 
 
 '7ux = sum of the terms containing 
 —■2ah 
 
 5ab = snm. of the terms containino; ab. 
 
 4:xy 
 3xy 
 2xy 
 
 oxy = sum of the terms containing xy. 
 
 Therefore lax + 5ab -\- bxy = total sum. 
 
 2. What is the sum of 
 
 2d^x — Zax^ + 2a5 — la^x + 4.ax^ — 8a 6 — 6a=x 
 -h 10aa:= -}- \2ab ? 
 
 2a" X — 3ax- -|- 2fl6 
 
 — 7a-x + dnx" — Sah 
 
 — 6a"a:+ lOax'^ + 12a6 
 
 Ans. — \\a~x-\-\\ax'^ -\- &ab 
 
 3. 4. 
 
 SaSft*— 7a6*+5axy 4awi— Sa;^"— 6a6 
 
 ■Id^b^ — 2aft'* — axy — lam-\- 4a7n,* + 'i^ 
 
 8a"5^ + ab* — laxy — 8am — lOam^ — 6a6 
 
 a-h^ — IQab* + 3axy am + am^ + 20a6 
 
 5a268_18a6* —10am— 8am2 + 9a6 
 
ly 
 
 5. • 6. 
 
 1 s/y — 4(a + m) 4a* •+• ban 
 
 3v/y — 2(a-j-??i) — 3«- — Ian 
 
 s/y + 7(rt -}- '^0 2fl- — 3a7i 
 
 5 s'y + (''' + //' ) 5«- + lOovi 
 
 16 ^/// + 2(a + m) 8a^ 4- 5a/i 
 
 7. 8. 
 
 a{a-^h)-\- 3v^fl— X 3xy + 2m^/a+ 3 
 
 ~4'.(«-f?,)— lOv^a- 
 
 .r y 
 
 -|- Im \/ « — 
 
 7 
 
 I3x"^y 
 
 — m-^/a — 
 
 10 
 
 -2.^;v. 
 
 + 3/71 \/a + 
 
 2 
 
 ox ?/- 
 
 — 4w\/a — 
 
 1 
 
 21x*y 
 
 — 7w v/a — 
 
 13 
 
 -7fl(a-t-6) — 4v'a — x 
 
 9. 
 
 i i , 1 ^ 10. 
 
 x^m^+ xjrn^ 26+ 8x + 6y 
 
 -Wx'.in-— 6x"'m' 36— Sx — 4y 
 
 14x^;;i^+ 5x^m^ —76— 3x- y 
 
 1 ^ J- -i 86 + 10x + 3.y 
 
 — 7x"*m" — 12x"77i'' oj, 
 
 9o — X — y 
 
 _33:V^_12x^m' 15/> + 6x + 3y 
 
 1 1 . What is the sura of "^ag -j- 6am — 9xy + 3a6 — xy 
 -f 4«o- + 10am — 7xy — 6a6 + 5xy + 4a o- — l3om ? 
 
 Ans. Wag -\- Sam — 12xy — 3a6. 
 
20 ADDITION. 
 
 12. What is the sum of 'ia'^x — 5a^y -{-lam — Sa"x 
 — lOa^y — .iam + da^y — lirx — 13am + Ca'y — Wa'x 
 -f- am — a^y -\- a'^^y — Ga'^x 1 
 
 Ans. — 23n^x — 9am. 
 
 13. What isthesumof — 2>xy -\-bn -\- 3ax — \Oam — 6xy 
 -f In — -iax -\- Sam — 2xy + lO?^ + Gam — 4ax ? 
 
 Ans. — 11 X y -\- 22n — 5fla: -f- 4cw. 
 
 14. What is the sum of 4a- + 5a-6V^ — 9a^ + 6a^6'-c- 
 4- lOa'x + Ta^o: + Scr— I3a^6^c^ + oa" — 3a^x + Sa^bh' 1 
 
 Ans. 8a* + a-&V + 14a3x. 
 
 15. What is the sum of ^a — 'ixy — V^ — n -\- %xy 
 + 5 v^a 4- 7n — 7xy + 9 v'o — 7 Vni + 16;? — 5 v/a ? 
 
 Ans. lOv/fl— 4x7/ — 8 Vin +22n. 
 
 16. What is the sum of 6a v^6 -\-bx^y — 7x + 5a h 
 -h 6x — 3a*6^ — 4x2 y^ _|_ 2a\/6 — lOx + 8a*6* ? 
 
 Ans. l8«*6'^H-x^y*— llx. 
 
 17. What is the sum of 3 \^a-\-h — 5 ^/x-\-bxry^-\-l ^ a-{-h 
 4-3x^^3 — 7 s/x-f y/ fl-f 6 — 8x'2/" + 2 y/x 4- 10 VT+l ? 
 
 Ans. 21v/a4-6 — lOv/x. 
 
 18. What is the sum of ba^h-c — 4a6ac^ 4- 2a26-c* — 7ax 
 4- Qa^Pc — baVc^ — 13ax — 70^6=02 4- 3a6'^c' — Sa^J'c ? 
 
 Ans. 3a'62j, „ ^^52^3 _5a-.'62(.2 _ gOox. 
 
SIBTRACTIOX. 21 
 
 SUBTRACTION. 
 
 (26.) SuBTRACTiox, ill Algebra, is the finding the sim- 
 plest expression for the ditference of two algebraic expres- 
 sions. 
 
 If we subtract the positive quantity b from a, we obvi- 
 ously obtain 
 
 a — b, 
 which is the same as the addition of a and — b. 
 
 Again, if we wish to subtract b — c from a, we obtain 
 by subtracting 6 from a, a — 6, but we have subtracted too 
 much by the quantity c, therefore adding c, we get 
 
 a — 6 + c, 
 which is the same as the addition of a and — b -{- c. 
 
 From this, we see that subtracting a quantity is the same 
 as adding it after the signs are changed. 
 
 Hence, for the subtraction of algebraic quantities we 
 have this 
 
 RULE. 
 
 I. Write the terms to be subtracted wider the similar 
 terms y if there are any^ oy those from which they arc to be 
 subtracted. 
 
 II. Conceive the signs of the terms of the polynomial to 
 be subtracted, to be changed, and then proceed as iji additian. 
 
29 SUBTRACTION. 
 
 EXAMPLES. 
 
 1. 2. 
 
 From lac — Sab -{- d- From Samx — 4-^2/+ 5y^ 
 Take 4ac + Sab + id' Take damx+lOxy—Uf 
 
 Rem. 3ac — Uub — 3d- Ri.'m. — amx—Uxy-]-l6y^ 
 
 3. 4. 
 
 From 6a:y — 3ac + 2,71^ From4o^v/x- — oa i/y -\- x 
 
 Take 4xy — lac — 9m" Take 3<r'' ^/x + 3a Vy — ""'^ 
 
 Rem. 2a:y + 4ac + Ib/i^ Rem. a^ v^o: — 8a Vy + 8^ 
 
 5. From 3a"6c — 7oxy-|- 37ny -f- " take d'bc -\-'$>axy 
 +6a ~4:my. 
 
 Alls. 2a-6c — I5axy -\- Imy — 5a. 
 
 6. From 9>abVc — I2a^h -\-(icx — Ixy take 9a6v/c 
 
 — I3a^6 + Sxy — an -\- 2cx. 
 
 Ans. — ab^c -\- a^b + 2cx — Ibxy + an. 
 
 7. From Iba^x — 14a-y + 3a6' + Qamn take Qmg + 3a 
 
 — ba^x — 7a-y -\-?>ab^ — 4flwm — 4. 
 
 Ans. 20a^x — 7a^y + lOam^t — Q)iiig — 3a + 4. 
 
 8. From I3a^a;^y + 3a.r — lab -\- Qmg — x'^y'^ take bxy 
 + 4aV2/ — Qax + 9aA + 2mg + bxhf. 
 
 Ans. ^a*x^y + 9c3- — 16a6 + 4mo- — Gary^ — bxy. 
 
 1 1 
 
 9. From 7a-6-'-|-3a~ — 4i''x — 3ax//-f-4x- — 3x?/^take 
 
 3a6 — 17 +4a2_5a'6i — 7i^j- + 3xy^ 
 
 Ans. 12a^6^—a-'4-3//'a—3afi/4-4a;-— 6x2/3— 3a6 + 17. 
 
 10. From 4a^ix — 7axy + 3//'7 +17 — x take 4,p'^q 
 
 — 13 -|- 7a^6x + Suxy — 7x + 3/ — mg -\- n. 
 
 Aus. — 3a^i'X — 15ajy — jrq +30 +6x — 3/+ mg — n. 
 
SIBTRACTIOX. 23 
 
 1 1 . From 6am -\- x take Sam -|- y. 
 
 Ans. 3am -\-x — y. 
 
 12. From 3a'm — Qx-y'^^2xy take 4:(rm-\- 6x^7/^ -j-o^y. 
 
 Ans. — a"m — I2xh/ — 2xy. 
 
 13. From "^amx — 43 -j- x — y-f-27d take 15/i + Ig — 3 
 4-4]/ — Sd-\- lamx — x-\-p(l — rs. 
 
 Ans. — ^amx — 40-}-2a: — by-\-2>bd — Ion — Ig — jjq-j-^s. 
 
 14. From a -{- b take a — b. 
 
 Ans. 2b. 
 
 (27.) We ean express the subtraction of one polynomial 
 from another, by writing the polynomial which is to be sub- 
 tracted, after enclosing it within a parenthesis, immediately 
 after the other polynomial from which it is to be subtracted, 
 observing to place the negative sign before the parenthesis. 
 
 Thus, ab — Gxy-j- 3am — (4«/; -f- 3xy -)- am) 
 denotes, that the polynomial enclosed within the parenthesis 
 is to be subtracted from the one which precedes it ; and since, 
 by (Art. 26), to perform subtraction, we must change all the 
 signs of the terms to be subtracted, we may remove the pa- 
 renthesis provided we change the signs of the terms which it 
 encloses : and conversely, we may enclose any number of 
 terms within the parenthesis, with a negative sign before it, if 
 we observe to change the signs of the terms thus enclosed. 
 
 In this way we can transform the expression 
 a'b-\-xy — 1am — {7nx-\-6 — iSx*), 
 
 mto 
 into 
 into 
 into 
 
 a'b-\-xy — lam — 7nx — 6-j- I3x-, 
 a%-\-xy — {lam-j-mx-^G — 13a;-), 
 a^6 — ( — xy-\-1am-\-mx-\-6 — iSx"), 
 a'^b-{-xy — lam — mx — (6 — I3x'). 
 
94 MULTIPLICATION. 
 
 MULTIPLICATION. 
 
 (28.) If we wish to multiply a by 6, we must repeat a as 
 many times as there are units in &, which, by (Art. 6), is 
 done by writini^ h immediately after a, thus, a multiplied 
 by ft = ab. 
 
 Again, if we wish to multiply a by — 5, we obserA'e that 
 this is the same as to multiply — h by a, hence we must re- 
 peat — 6 as many times as there are units in a: repeating a 
 minus quantity once, twice, thrice, .or any number of times 
 can not change it to a positive quantity. Hence, — h multi- 
 plied by c, or, which is the same, a multiplied by — 6= — ah. 
 
 Finally, if we wish to multiply a — 6 by c — (/, we will 
 first multiply a — 6 by c, we thus obtain 
 a — 6 
 
 ac — 6c for a — h repeated c times. 
 This result is evidently too great by the product of a — h 
 by d, since it was required to repeat a — 6 as many times as 
 there are units in c lessd. 
 
 Then repeating a — 6 as many times as there are units in 
 rf, we have 
 
 a — 6 
 d 
 
 ad — hd for a — b repeated d times. 
 Subtracting this last result from the former, we have 
 ac — be — {ad — bd), which, by (Art. 27), becomes 
 ac — be — ad -j- bd for the product of a — b by c — d. 
 
MTLTIPLICATIOX. 25 
 
 Hence, we see thai — b, when multipiied by — d, product s 
 the product + bd 
 
 If we wished to multiply a by — 6, it would hardly be 
 correct to say, that we are to repeat a mi7ius b times ; for 
 a quantity cannot be repeated a 7ninus number of times. 
 But when we wish to multiply a by — 6, we evidently 
 wish to repeat a as many times as there are units in 6, and 
 then to give to the product the negative sign ; that is, when 
 the multiplier is negative, we must multiply as though it 
 were positive, and then give to the product a contrary sign. 
 
 Applying this principle to the case of — a multiplied by 
 
 — b. We know that — a multiplied by -|" ^ gives — ab 
 for the product ; therefore — a multiplied by — b must 
 give the same product taken with a contrary sign ; that is, 
 
 — a multiplied by — b must give + o^- 
 
 (29.) From all this, we discover, that the inoduct will 
 have the sign +, iL'hen both factors have like signs, and the 
 product will have the sign — , lohen the factors have con- 
 trary signs. 
 
 If we wish to multiply 3a-6 by 4a'6-, we observe that 
 
 3a'b = Saab 
 
 4a^b~ = 4:aaabb 
 
 Hence, the product will be 
 
 2aab X -iaaabb = 12aaaaabbb = 12a^b^. 
 
 Here we discover that the exponent of a, in the product, 
 is equal to the sum of the exponents of a in the factors ; 
 likewise the exponent of fe, in the product, is equal to the 
 sum of the exponents of b in the factors. 
 
 (30.) Hence the product of several letters of different ex- 
 ponents is equal to the product of all the letters, having for 
 exponents the sums of their respective exponents inJhe fac- 
 tors. 
 
26 MULTIPLICATION. 
 
 CASE I. 
 
 (31.) From what has been said, we have, for multiplying 
 this 
 
 RULE. 
 
 I. Multiply the coefficients^ observing to prefix the sion -J- 
 whcn both factors have like signs; and the sign -r- when they 
 have contrary signs. 
 
 II. Write the letters one after another; if the same letter 
 occur in both factors^ add the exponents for anew exponent. 
 
 (32.) The product will be the same in whatever order the 
 letters are placed, but it will be found more convenient, in 
 practice, to have a uniform order for their arrangement. 
 The order usually adopted is to place them alphabetically. 
 
 EXAMPLES. 
 
 1. Multiply llax'^y by 3ay. 
 
 Ans. 33aV2/''. 
 
 2. What is the product of 3a7n^ by Qa'^b^x ? 
 
 Ans. iSd^Wiivx. 
 
 3. What is the product of \OcH^ by ^ahdl 
 
 Ans. 'dQah'd^ 
 
 4. Multiply — 13ac^ by —40^6 V. 
 
 Ans. b2a^b'^c^. 
 
 5. Multiply a'Vc'' by a''b\ 
 
 6. Multiply — 11 xhj by ?>xyz. 
 
 Ans. a'"^"h"^'c''. 
 
 Ajis. — blx^y-z. 
 
 Mamply -ab'^cd by -axy. 3 
 
 4 ' Ans — n- 
 
 Ans. -—a-h^cdxy. 
 28 ^ 
 
MULTIPLICATION. 27 
 
 8. Multiply —-xyzhy~xYz\ j 
 
 2 " '2 
 9. Multiply Im^n^p"^ by Gmn'^p^ 
 
 Ans. xV-^ 
 
 4 -^ 
 
 CASE II. 
 
 (33.) Polynomials may be multiplied together by the fol- 
 lowing 
 
 RULE. 
 
 I. Multiply all the terms of the multiplicand succes- 
 sively by each term of the multiplier, and observe the same 
 rules for the signs and exponents as in Case I. 
 
 II. When there arise several partial products alike, they 
 must he placed under each other, and then added together 
 in the total product. 
 
 (34.) The total product will be the same in whatever 
 order we multiply by "the terms of the multiplier, but for 
 the sake of order and uniformity, we begin with the left- 
 hand term. 
 
 EXAMPLES. 
 
 1 . What is the product of Sa' — 6a.T -f- y by 3a — m ? 
 
 OPERATION. 
 
 3a2 — dax -|- y 
 3a — m 
 
 Ans. 9a' — iSa^ x-\-Zay — 3a2 m -{- Qamx — my . 
 
i MULTIPLICATION. 
 
 2. What is the product of Cx^— Sy'-f-a by a;«— 2y»— a ? 
 
 OPERATION. 
 
 6x2— 32/3 + a 
 X- — 2y^ — a 
 
 6x*— 3xy+ 0x2 
 
 — 12x-y^ — 6ax--\-6y^ — 2ay ' 
 
 -{-2ay^ — a^ 
 
 Ans. 6x^— 15xy — 5ax-+6/-f ay^—a\ 
 
 3. What is the product of b~m — Say by 6x — 3 ? 
 
 Ans. 66~mx — ISaxy — db"m -\- 9ay. 
 
 4. What is the product of 7/— 2m— 9 by 3/— 11m ? 
 
 Ans. 21/'— 83/m— 27/4-22;?i-+99m. 
 
 5. Multiply 2a-\-ol+3i—5e by 3a + 106 + 15/ 
 
 . { 6a=2+35a6+9ac— 15ae+50/>-+306r 
 ^"s. I _50ic+30rr/+75//+45c/— 75e/. 
 
 6. Multiply a + 6 + c+d by a — J- c — (/. 
 
 Ans. a~ — b' — 2bc — 26d — c' — 2c(Z — d^ 
 
 7. Multiply a^-\-a*-{- a- by a- — 1. 
 
 Ans. a — (I-. 
 
 8. Multiply a^ -f- flz + z- by a-—az-{- z". 
 
 Ans. o^ + a"z^ + c''. 
 
 9. Multiply a + 6 by a + 6. 
 
 Ans. a= + 2fl6+6^ 
 
 10. Multiply a — 5 by a — b. 
 
 Ans. a' — 2ah-\-b'\ 
 
 11. Multiply a + 6 by a — b. 
 
 Ans. a* — 6*. 
 
 (35.) The last three examples, when translated into com- 
 mon language, give three distinct and important theorems, 
 which we will proceed to illustrate. 
 
 Example 9 is the same as 
 
 (a + 5) X (a + ft) = (a + bf = a^ + 2ab + b- ; 
 which, when translated, gives 
 
MULTIPLICATION, 29 
 
 THEOREM T. 
 
 The square of the sum of two quantities is the same as the 
 square of the first ^ plus twice the product of bothj plus the 
 square of the secofid. 
 
 EXAMPLES. 
 
 1. (x+y) X {x-\-y)={x-j-yy=x Jr2xy-\-y\ 
 
 2. (2x-f-fl) X (2x-j-a)=(2.r-)-a)-=4z-+4ax-(-a*. 
 
 3. (5m+3) X (5/7i-}-3)=: (5/n4-3)==25/?i-+3077i+9. 
 
 Example 10 is the same as 
 
 {a—b) X {a—b)=^{a—by-:=^a'--2ab+b^ ; 
 which, when translated, gives 
 
 THEOREM II. 
 
 The square of the dijference of two quantities is equal to 
 the square of the first, minus twice the product of both, plus 
 the square of the second. 
 
 EXAMPLES. 
 
 1. {x — y)x{x — y)={x — y)-^x^—2xy + y\ 
 
 2. (3a — 6) X (3a —6 ) =r (3a — &)= =9a' — Gab + b^. 
 
 3. (5a — x)X(5a— a;) = (5a — x)' = 25a' — lOax+i'^ 
 Example 11 is the same as 
 
 {a + b)X{a~b) = a'—h-', 
 which, when translated, gives 
 
 THEOREM III. 
 
 The sum of two quantities multiplied by their difference, 
 is equal to the square of the greater, minus the square of 
 the less. 
 
 EXAMPLES. 
 
 1. {x + y)x{x-y)=ar' — y\ 
 
 2. (3a + 6)x(3a — 6) = 9a2— 6-. 
 
 3. (7m -f- y) X (Jm —y)= 49wr — f. 
 
30 
 
 DIVISION. 
 
 (36.) We know by the principles of Arithmetic, that, if, 
 in Division, we multiply the divisor into the quotient, tli'' 
 product will be the dividend. 
 
 Therefore, referring to what has been said under Multi- 
 plication (Art. 29), we infer that when the dividend has the 
 sign -f-j the divisor and quotient must have the same sign ; 
 but when the dividend has the sign — , then the divisor and 
 quotient must have contrary signs. 
 
 (37.) Hence, when the dividend and divisor have like sigiia, 
 the quotient will have the sign +; and .when the dividend 
 and divisor have contrary signs, the quotient loill have the 
 the sign — . 
 
 We have also seen under Multiplication (Art. 30), that 
 the product of several letters of different exponents is equal 
 to the product of all the letters with the sum of their re- 
 spective exponents for new exponents. 
 
 (38.) Hence, to divide any power of a letter by a different 
 power of the same letter, it is obvious that the quotieid will 
 be a power of the same letter, having for exponent the ex- 
 cess of the exponent in the dividend above that of the divisor. 
 (39.) If we divide continually the expression 
 a^ = aaaaa by a, we shall find 
 a^ -7- a=:a^~^ = a* = aaaa ; 
 a'^ -r- a=a'^ — ^ = a^ = aaa ; 
 a* -T- a = a^ ~ ' = a'^ := aa ; 
 a^ -r- a = a^~^ = a^ = a; 
 a* -7- a = fl' " ' = a" = 1 ; 
 
 i 
 
DIVISION. 31 
 
 '=-=reciprocal of a; 
 a 
 
 a~'-ra=a"~'~'=a~-= -=— ^reciprocal of a*; 
 a a a~ 
 
 a—'-r-a=a~'~^=a~^-^ =— =reciprocal of a^ ; 
 
 aaa a" 
 
 a-^-x-a=a~^~^^a~^= =— =reciprocal of a* ; 
 
 aaaa a* 
 
 &c. &c. 
 
 (40.) From the above scheme, we see, that whenever the 
 exponent of a quantity becomes 0, its value is reduced to 1 . 
 
 (41.) That whenever it is negative, it is the reciprocal of 
 what it would he were it poaitive. 
 
 (42.) Hence, changing the sign of the exponent of a 
 quantity is the same as taking its reciprocal. 
 
 CASE I. 
 
 (43.) From what has been said, we have, for dividing one 
 monomial by another, this 
 
 RULE. 
 
 L Divide the coefficient of the dividend by that of the 
 divisor, observing to prefix to the quotient the sign -\- when 
 the signs of the dividend arid divisor are alike, and fhc 
 iign — when they are contrary. 
 
 II. Subtract the exponents of the letters in the divisor 
 from the exponents of the corresponding letters of the divi- 
 dend J if letters occur in the divisor ichich do not in the 
 dividend, they may (Art. 42) be written in the quotient by 
 changing the signs of their exponents. 
 
 (44.) It must be recollected here, and in all cases here- 
 after, that when the exponent of a letter is not written, 1 is 
 
32 DIVISION- 
 
 always understood (Art. 12)j and when the exponent is 0, 
 the value of the power is 1. (Art. 40.) 
 
 EXAMPLES. 
 
 1. What is the quotient of 14a^a;° divided by lax^y ? 
 Dividing the coefficients we find 2, to which if we annex 
 
 the letters after subtracting the exponents, we have 
 
 the X has disappeared, since its exponent became 0, and its 
 value therefore was 1, by (Art. 40.) And since the y oc- 
 curred in the divisor and not in the dividend, it was written 
 in the quotient with the sign of the exponent changed. 
 (Art. 42.) 
 
 2. What is the quotient of Sof.-'^Pc divided by bahcl 
 
 Ans. lah"^. 
 
 3. What is the quotient of — 44;n7ix* divided by 22 ahcx ? 
 
 Ans. — 2a-^'b~^c~^mnx. 
 
 4. Divide — Ix'^y by lOx^y. 
 
 5. Divide Za^rri^n^ by — Qamn. 
 
 6. Divide 35a;^2/cMjy — Txyz^ 
 
 7. Divide cd}' by — 13cd*^ 
 
 8. Divide — 3a"'6" by — 4a''6 c\ 
 
 9. Divide —llaH^'m-^ by -^a^H^ 
 
 Ans. 
 
 10 
 
 Ans. — 
 
 1 . ■» ', 
 
 Ans. 
 
 —bxy-'\ 
 
 Ans. 
 
 -Id-K 
 13 
 
 A 3 „ 
 
 Ans. -a" 
 4 
 
 '~'ir-^c-\ 
 
 'm-\ 
 
 
 Ans 
 
 4 
 
DIVISION. 33 
 
 10. Divlile 13j-'?/-^ by — 26x2/. 
 
 Ans. a—' J/-®. 
 
 2 '^ 
 
 (45.) To divide one polynomial by another, we shall imi- 
 tate the arithmetical method of long division. And in the 
 arrangement of the work we shall follow the French method 
 of placing the divisor at the right of the dividend. Thus, 
 to divide 
 
 a^ 4- a'^x -\- ub -\- bx by a-\-x, 
 
 we proceed as follows : 
 
 OPERATION. 
 
 Dividend = a" -(- a'x -\- ab -\~ bxla -j- x = divisor 
 
 a^ -f- a'^x I 
 
 a^ -\- b = quotient 
 
 ah + bx 
 ab -\-hx 
 
 
 Having placed the divisor at the right of the dividend, we 
 seek how many times its left-hand term is contained in the 
 left-hand term of the dividend, which we find to be a'-*, which 
 we place directly under the divisor, and then multiply the 
 divisor by it, and subtract the product from the dividend ; 
 then bringing down the remaining terms, we again seek how 
 many times the left-hand term of the divisor is contained 
 in the left-hand term of this remainder, which we find to 
 be b ; we then multiply the divisor by b, and again subtract- 
 ing there remains nothing ; so that n'^-\-h is the complete 
 quotient. 
 
 That the operation may be the most simple, it will be 
 necessary to arrange both dividend and divisor according to 
 the powers of some particular letter, commencing with the 
 highest power. 
 
34 
 
 CASE II. 
 
 (46.) To divide one polynomial by another, we have this 
 
 RULE. 
 
 I. Arrange the dividend and divisor with reference to a 
 certain letter; then divide the first term on the left of the 
 dividend by the first term on the left of the divisor, the re- 
 sult is the first term of the quotient; multiply the divisor 
 by this term, and subtract the product from the dividend. 
 
 II. Then divide the first term of the remainder by the first 
 term of the divisor, which gives the secon ' term of the quo- 
 tient ; multiply the divisor by this second term, and subtract 
 the product from the result after the first operation. Con- 
 tinue this process until we obtain for remainder; or lohen 
 the division does not terminate,which is frequently the case^ 
 we can carry on the above process as far as we choose, and 
 then place the last remainder over the divisor , forming a 
 fraction, which must be added to the quotient. 
 
 EXAMPLES. 
 
 1. What is the quotient of 2a'b -\- b^ -\- 2ab^ -\- a' di- 
 vided by a" -\- b^ ■]- ab 1 
 
 Arranging the terms according to the powers of a, and 
 operating agreeably to the above rule, we have 
 
 Dividend = a' + 2a''b + 2a6^ + b 
 
 OPERATION. 
 
 a^ -|- ab -{-b^ = divisor 
 
 a -\-b = quotient 
 
 „2^+ ab^ + b^ 
 a''b+ a6' + 6' 
 
 
 
DIVISION. 35 
 
 2. What is the quotient of a=i— Sa^-f-Safe— 6«— 46-f22 
 iliviiledby 6 — 3? 
 
 OPERATION. 
 
 10 
 
 6—3 
 
 a-b- 
 a-b- 
 
 -3a' 
 -Zir 
 
 +'2ab- 
 
 _6g— .li-l-92 b 
 
 a 
 -6a 
 -6a 
 
 —3 
 
 ^_|_2fl— 4 + 
 
 
 
 2ab- 
 2ab- 
 
 
 
 —46+22 
 — i6-fl2 
 
 10= 
 
 ^remainder. 
 
 3. Divide x^—x*-{-x'—x''-\-2x—l by x^ -f-x— 1. 
 
 OPERATION. 
 
 :j:^^x*+x'—x'i-2x—l 
 x^-\-x'—x* 
 
 — x^-{-x^ — x'^ 
 
 x*—x'+2x 
 x'+x'—x^ 
 
 — x'+2x— 1 
 — x^ — x^4-x 
 
 x"-\-x — 1 
 x^-f-x— 1 
 
 X- + X — 1 
 
 :4_a;3_^a.2_3._|.i 
 
 4. What is the quotient of x' — Sax^-j-Sa'x — a^ divided 
 ? 
 
36 
 
 OPERATION. 
 
 X' — 3ox- + 3a'x — o3'x ■ 
 
 X3 QTT 
 
 \x^ — 2ax -\- a'^ 
 
 — 2ax" -f- 3a'x 
 
 — 2ax^ 4- 2a''x 
 
 a^x — a" 
 a^x — a^ 
 
 
 
 5. Divide Uaf— 21hf-\- lcf+ Sag — %g -j- 3cg by 
 7/+ 3 a-. Ans.-2a — 36 + c. 
 
 6. Divide 4x3 + 4x2 — 29x + 21 ^^ 2x — 3. 
 
 Ans. 2x--|-5x — 7. 
 
 7. Divide 119c2 — 200cf/-|-408ce— 113cA — 39c/=^ 
 -f- 72de + 31 dh — 96e/i + 20h' by 17c + 3d — 4/i. 
 
 Ans. 7r — 13(? + 24c — 5/i. 
 
 8. Divide 72x* —ISx'y — lOx^.v* + Hxy^ -|- Sy^ by 
 Gx'^ — 4x1/ — J/?. 
 
 Ans. 12x2 — 5xy — 3y2. 
 
 9. Divide 36aH — e3ab^-}-20b^ by 12o6 — 5/r. 
 
 Ans. 3a— 46. 
 
 10. Divide a^ — b'- by a — b. 
 
 Ans, a -{-b. 
 
 11. Divide a^ — 6^ by a — 6. 
 
 Ans. a^ + a"b-i-ab"-\-bK 
 
 (47.) The following examples cannot be accurately per- 
 formed, there being still a remainder, however far the divi- 
 sion be carried. 
 
12. Dividing 1 by 1 — 6, we have in succession 
 b 
 
 1^(1-6) = 1 + 
 
 1 — 6 
 
 b' 
 
 ='^'+l-b 
 = 1 + 6 + 6-^+ ^ 
 
 = 1 + 6 + 6= +6 
 
 1 — & 
 b' 
 
 1 — 6 
 
 &c. &c. 
 
 ,3. 1->(1 + 6) = 1--A-^ 
 
 
 l_6-{_52_63 
 
 1 + 6 
 
 6^ 
 
 1+6 
 &c. &c. 
 
 U. (l+a:)^(l-a:) = l+ ^"^ 
 
 = l+2x-i 
 
 l—x 
 
 2x^ 
 
 l — x 
 
 2x^ 
 
 l+2x + 2x" + 
 
 l — x 
 
 37 
 
 = 1 + 2x + 2x' + 2x3+ ^^' 
 
 1— :r 
 &c. &c. 
 
38 ALGEBRAIC FRACTIONS. 
 
 CHAPTER tt 
 
 ALGEBRAIC FRACTIONS. 
 
 (48.) In our operations upon algebraic fractions, we shall 
 follow the corresponding operations upon numerical frac- 
 tions, so far as the nature of the subject will admit. 
 
 CASE I. 
 
 To reduce a monomial fraction to its lowest term, we 
 have this 
 
 RULE. 
 
 I. Find the greatest common measure of the coefficients of 
 the numerator and denominator. (See Arithmetic.) 
 
 II. 7%en, to this greatest common measure annex the let- 
 ters which are common to both numerator and denominator^ 
 give to these letters the lowest exponent which they have, 
 whether in the numerator or denominator ; the result will 
 be the greatest common measure of both numerator and de- 
 nominator. 
 
 III. Divide . both numerator find denominator by this 
 greatest common measure^ (by Rule under Art. 43,) and the 
 resulting fraction will be in its lowest terms. 
 
ALGEBRAIC FRACTIONS. 39 
 
 ^ „ , 315a^bxy , ., , 
 
 1. Reduce '-- to its lowest terms. 
 
 The greatest common measure of 373 and 15 is 15, to 
 which annexing abxy, we have loabxy for the greatest 
 common measure of both numerator and denominator. 
 Dividing the numerator by Ibabxrj^ we find 
 yiba^bxy -i- loabxy = 25a^. 
 In the same way we find 
 
 loab'^xy^ -r- loabxy = by- ; 
 hence, we have 
 
 2>1ba^bxy^ 25a- 
 
 loab-xy^ by' 
 which, by Rule under Art. 44, becomes 
 
 — - - = 2oa-b-^y-'. 
 by' 
 
 - ,, , 42ax'^7/r'' . , 
 2. Reduce „-;: — ;^to its lowest terms. 
 3oxy^z^ 
 
 In this example, the greatest common measure of the 
 
 numerator and denominator is Ixyz^ ; hence, dividing both 
 
 numerator and denominator of our fraction by Ixyz^, we 
 
 find 
 
 4:2ax'yz' Gax'z'' , • ^ • • ., , 
 
 — — r^ ==:: —,-~r-i which IS in its lowest terms. 
 
 35xy^z^ oy' 
 
 _ „ , — iSmnx^y^ ^ . , 
 
 6. Reduce :- to its lowest terms. 
 
 12/nxY' 
 
 Ans. — — . 
 
 1 3x^ 
 4. What is the simplest form of — — — 1 
 ^ 26xy^ 
 
 2y* 
 
40 ALGEBRAIC FRACTIONS. 
 
 5. What IS the simplest form of ? 
 
 Ans. 9/>-fi«. 
 
 (49.) Fro??i iij/ia^ has been said (Art. 42), we infer that 
 we may transfer a letter from the numerator to the denomi- 
 nator, or from the denominator to the numerator, by changing 
 the sign of the exponent. 
 
 Thus, 
 
 , laxyz 7z ^ o . 7 
 1. f-r-= — ^ =7ca:-"v~ = : — r- 
 
 „ 1^ , , , 17x^ 17 
 
 49aic^ .. . • 
 ■35^ 
 all the letters to the numerator. 
 
 3. Reduce ^^ ^ to its simplest terms and then transfer 
 
 7c« 7 
 
 4. Reduce in a similar manner the fraction 
 
 lOSaSft^ 
 
 Ans. — ^^ — = -a~'b~^cdm~'^ . 
 4ia-bhn 4 
 
 GREATEST COMMON MEASURE OF POLYNOMIALS. 
 
 (50.) Before proceeding to the reduction of polynomial 
 fractions, it is necessary to show how to find the greatest 
 common measure of two polynomials, which may be effected 
 by this 
 
 RULE. 
 
 Divide one of the polynomials by the other, and the pre- 
 ceding divisor by the last remainder, till nothing remains ; 
 the last divisor will be the greatest common measure. 
 
 This rule may be demonstrated as follows : 
 
ALGEBRAIC FRACTIONS. 41 
 
 (51.) Let JV and /i be two polynomials, of Avhich .Yis 
 greater than n ; then, performing the divisions as directed 
 in the above rule, we have 
 
 OPERATION. 
 
 w) ./V ((71 = first quotient. 
 7iqi_ 
 First remainder = ri) 7i (70= second quotient. 
 
 Second remainckr = r.2) ri (93 = third quotient. 
 ^273 
 Third remainder = 
 The numerals placed at the bottom of the letters q and r, 
 are called Subscript JS'mnbers, and show the order in which 
 the quotients and remainders occur. 
 
 Letters marked like the above, are as independent as 
 though they were different letters. The reason why we use 
 them in preference to different letters, is because we can 
 the more readily discover what they are designed to repre- 
 sent. 
 
 (52.) Now', since the dividend equals the divisor multi- 
 plied by the quotient and increased by the remainder, we 
 have the following conditions : 
 
 JV=5in + ri. (1) 
 
 n = q-iVi + r.2. (2) 
 
 ri = qsTo. (3) 
 
 Substituting 73/2 for ri in (1) and (2), and they will be- 
 come 
 
 Y ^= qiTi -\- q^r^. (4) 
 
 n = q-zq^ra + To. (5) 
 
 The right-hand member of (5) is divisible by r^, and 
 therefore its left-hand member must also be divisible by r^ ; 
 that is, n is divisible by ro. 
 
 6 
 
42 ALGEBRAIC FRACTIONS. 
 
 The value of n, (5), being substituted in (4), gives 
 J\r= qiQoqsrz + ^ira + ^3^2. (6) 
 
 The right-hand member of (6) will divide by ro, and 
 therefore its left-hand member will also divide by ro ; that 
 is, JV* is divisible by r-z : hence, r-2 is a common measure of 
 M and n. It is also the greatest common measure. For 
 every common measure of JVand ?i, is also a measure of 
 JY — nqi = ri ; and every common measure of n and ri, is 
 also a measure of n — riq-i = r-2. But the greatest measure 
 of r-2 is itself. This, then, is the greatest common measure 
 of ^Yandn. 
 
 In the above case we have supposed the third remainder 
 7-3 to =0. Had the process of dividing extended still far- 
 ther, it might still be shown, that the last divisor is the great- 
 est common measure ; hence the truth of the above rule. 
 
 (53.) It is obvious, that any factor common to but one 
 of the two polynomials, may be struck out before dividing, 
 without affecting the accuracy of the work. 
 
 (54.) Also, either of the polynomials may be multiplied 
 by any factor before dividing.* 
 
 EXAMPLES. 
 
 1. What is the greatest common measure of a'* — x'*, and 
 a* -j- a^x — ax~ — x' ? 
 
 Arranging the terms according to the powers of cr, and 
 dividing according to Rule under Art. 46, we have for the 
 
 * If the above demonstration is deemed too difficult, on account of its 
 making use of some of the principles of equations, which have not yet 
 been fully explained, the student must pass it by, until he has gone 
 through with the chapter on simple equations, and then he can return 
 to it with pleasure and profit. 
 
ALGEBRAIC FRACTIONS. 43 
 
 FIRST OPERATION. 
 
 a^ -\- arx — ax^ — x^ 
 
 a* -\- a^x — crx- — ax^ 
 
 — a^x-j- u'-x" -\- ax^ — x* 
 
 — a^x — a^x~ -\- ax^ -{- x* 
 
 2a'X- — 2x^ = first remainder. 
 We must now divide o3-[-a*a; — aa:" — x'^ by 2a-x^ — 2x^; 
 but before performing the division, we will expunge from 
 2a'^x^ — 2x^ the factor 2x~ (Art. 53), which gives a^ — x^ for 
 the divisor ; hence, we have for the 
 
 SECOND OPERATION. 
 
 a3 
 
 + 
 
 a'x- 
 
 -CX2 — 
 
 -Z3 
 
 a' 
 
 
 
 -ax' 
 
 
 
 
 a^x 
 
 
 
 x' 
 
 
 
 a~x 
 
 — 
 
 x' 
 
 + x 
 
 
 There being no remainder, the process must terminate. 
 The last divisor, or greatest common measure, is therefore 
 a« — x^. 
 
 2. What is the greatest common measure of 6a' -\- llax 
 -j- 3x^ and 6a^ -f- lax — 3x' ? 
 
 In this example, we may take either of the polynomials 
 as the divisor, since they are each of the same degree. 
 
 FIRST OPERATION. 
 
 6c^ + llax + 3x2l6a= + lax — 3x^ 
 
 6a-^ + lax — 3x- 
 
 4ax + 6x^ = first remainder. 
 Before dividing 6a^ + lax — 3x' by 4flx -(- 6x^ we ex- 
 punge from 4ax + 6x' the factor 2x, and thus have 
 
44 
 
 ALGEBRAIC FRACTIONS. 
 
 SECOND OPERATION. 
 
 6a2-f7aa: — 3ar 
 6a2 -f 9ax 
 
 2a + 3x 
 
 3a — X 
 
 — 2ax — 3x^ 
 
 — 2ax — 3x" 
 
 Therefore, 2a -j- 3a: is the greatest common measure. 
 3. What is the greatest common measure of 
 
 as — a% + Zah^ — 363 and a^ — bah + ib"- ? 
 
 FIRST OPERATION. 
 
 a3- 
 a'- 
 
 - a'b-{- 
 
 — ba~h + 
 
 3a6^ 
 4a6=^ 
 
 — 3¥ 
 
 
 4a=^6 — 
 4a'6 — 
 
 ab^ 
 20a¥ 
 
 — 363 
 + 166^ 
 
 5a6 + 465 
 
 a + 46 
 
 19ah" — 1963= first remainder. 
 Before dividing «" — bah + 46* by 19a6" — 196^, we ex- 
 punge from this last polynomial the factor 196'. 
 
 SECOND OPERATION. 
 
 a2 — 5a64-462 
 
 a — 6 
 
 a — 46 
 
 _4a6 4-46" 
 _4a5_|_4fc2 
 
 
 Therefore, a — 6 is the greatest common measure. 
 4. We will now seek the greatest common measure of 
 these polynomials after the terms have been arranged ac 
 cording to the powers of 6, as follows : 
 
 — 36^ + 3a6= — a^b + a' and 46* — bab -\- a^ ? 
 
ALGEBRAIC FRACTIONS. 45 
 
 Before dividing, we must multiply the polynomial — 36" 
 -\-3ab' — arb -\- a^ hy 4, in order that its left-hand term 
 may be divisible by the left-hand term of the other poly- 
 nomial. (Art. 54.) 
 
 FIRST OPERATION. 
 
 — l2b^-{-12ab-— 4a%-i- 4ta'Mb- — 5ab + a^ 
 
 — Ub'^lbab-^— da'b ' — — 
 
 I— 36 — 3n 
 
 Multiplying by 4, — 3a6- — a^b -\- 4a^ 
 
 — 12a6^— 4a26 + 16a3 
 
 —12ab- -{- 15a^b — oa^ 
 
 — 19a*6 -f- 19a^ = first remainder. 
 Before dividing W — 5ab-\-a~ by — IQa^fi -f 19a\ we 
 expunge from this last polynomial, the factor 19a-, and then 
 dividing, we have for the 
 
 SECOND OPERATION. 
 
 — 6-fa 
 
 W — 5ab + a- 
 462 — 4a6 
 
 — 46 + 
 
 — ab-\-a^ 
 
 — ab-\-a^ 
 
 
 Therefore, — b -\- a, or a — 6, is the greatest common 
 measure, same as before. 
 
 4. What is the greatest common measure of the two poly- 
 nomials 5 ^^'^^ + l^"'^ + ^°'^' + ^"'^' - ^"^'' 
 nomiais ^ ^20^6'^ + 38a'^6^ + 16«6^ — 106^ ? 
 
 Ans. 3a" + 2ab — 6^. 
 
 5. What is the greatest common measure of 
 
 x^ — b'^x and ar^ + 26x + 6^ ? 
 
 Ans. X -\- b. 
 
46 ALGEBRAIC FRACTIONS. 
 
 6. What is the greatest common measure of 
 
 a- — (ii — 2b' and a^ — Sub + 2b~ 1 
 
 Ans. a — 2b. 
 
 In this example it is immaterial which polynomial we 
 consider as the divisor, since they are of the same degree. 
 
 7. What is the greatest common divisor of 
 
 C a;6 _|_ 42,5 __ 3^,4 _ 16^3 ^ 1 1^"- _f- i2x — 9, 
 
 I %x^ + 20a;'' — \2x^ — 48x2 ^ 22a: + 12 / 
 
 Ans. x^ -f~ ^' — ox -\-'i. 
 
 8. What is the greatest common divisor of 
 
 { 20z« — 12x^ + 16x^ — \bx^ + 14x= — lox 4- 4, 
 J 15a;*— 9x3-f-47x=^ — 21x +28? 
 
 Ans. 5x=^— 3x + 4. 
 
 CASE II. 
 
 (55.) To reduce a polynomial fraction, that is, a fraction 
 of which the numerator or denominator, or both, are poly- 
 nomials, to its lowest terms, we have this 
 
 RULE. 
 
 Divide both numerator and denominator by their greatest 
 common measure., found by Rule under Art. 50. 
 
 EXAMPLES. 
 
 , r • 36x° — 18x^ — 27x^ + 9x3 . .^ 
 
 1. Reduce the fraction —^^, -3-— — 77—^— to its 
 
 27x"'y — 18x*y'' — \ix^y^ 
 
 simplest form. 
 
 We see, by a mere glance of the eye, that the numerator 
 
 and denominator can both be divided by 9x^, by which di- 
 
 4x3 — 2x'^ — 3x+l 
 
 vision the fraction becomes 
 
 Szaya _ 2x3/2 
 
ALGEBRAIC FRACTIONS. 47 
 
 We must now seek the greatest common measure of 
 
 4x3 _ 2x2 — 3a: + 1 ^nd Sary^ _ ^xy- — if. 
 Dividing the second of these by y" (Art. 53), and multi- 
 plying the first by 3 (Art. 54), we have the 
 
 
 
 FIRST OPERATION. 
 
 
 12x3- 
 
 -6x= — 
 
 ?x + 
 
 33x2 — 2x — l 
 
 i 
 
 
 12x2- 
 
 -8x-— 
 
 4x 
 
 '4x + 2 
 
 Multiplying by 3, 
 
 2x' — 
 
 5x + 
 
 3 
 
 
 
 6x2 — 
 
 15x + 
 
 9 
 
 
 
 6x"- — 
 
 4x — 
 
 2 
 
 — llx-f-11 = first remainder. 
 We must now repeat the operation upon Sx^ — 2x — 1 
 and — llx + 11. Dividing the second of these by 11 (Art. 
 53), we have for the 
 
 SECOND OPERATION. 
 
 3x2- 
 
 -2x- 
 
 -1 
 
 -x+1 
 
 3x2- 
 
 -3x 
 
 
 — 3x— 1 
 
 
 X — 
 
 -1 
 
 
 
 X — 
 
 -1 
 
 
 
 
 Hence, the greatest common measure of the numerator 
 
 4^3 2x- 3x-f-l 
 
 denominator of the fraction ^ ^ ^ — -, — ~r- is — x -\-\ 
 
 3x2y2 — 2xy* — 1/2 
 
 or X — 1. Dividing both numerator and denominator, of 
 
 4x8 ^ 2x 1 
 
 the above fraction, by x — 1, it becomes — - — „ for 
 
 3xy2 -f- 2/2 
 
 the reduced value of the given fraction. 
 
 2. Reduce — -, — ^ to its lowest terms. 
 
 x^-[-2xT/ + y^ 
 
48 ALGEBRAIC FRACTIONS. 
 
 In this example the greatest common measure of the nu- 
 merator and denominator is found to be x -{- y. Hence, 
 
 the fraction reduced becomes 
 
 x^ — ry 
 
 3. Reduce ^^— — to its simplest form. 
 
 m^ — mrn — mn* -\-n^ 
 
 Ans. ' — . 
 
 m — n 
 
 4. Reduce — — — to its simplest form. 
 
 Ans. — —-. 
 a — 
 
 _ ^ . exy-\~Sx-{- 9y+12^ . . , , 
 
 5. Reduce — —^-^ — ' ^^—- to its simplest iorm. 
 
 lOxy — Sx + ldy—12 ^ 
 
 Ans ?^- 
 *5y— 4' 
 
 ^ „ , 6x^ — 4x^ — 11x3 — 3x' — 3.T—l . . 
 
 b. Reduce — ; ~ ;;-- — to its sun- 
 
 4X-' -I- 2x^ — I8x^ 4- 3x — 5 
 
 plest form. 
 
 Ans. 
 
 2xH-5 
 
 CASE III. 
 
 (56.) To reduce a mixed quantity to the form of a frac- 
 tion. 
 
 RULE. 
 
 Multiply the entire part hy the denominator of the frac- 
 tion^ to which product add the numerator^ and under the 
 result place the given denominator. 
 
ALGEBRAIC FRACTIONS. 49 
 
 1. Reduce llx -| ^^^ to the form of a fraction. 
 
 7x 
 
 In this example the entire part is llx, which multiplied 
 
 by the denominator 7x, gives llx", to Avhich adding the 
 
 numerator x-\-y, we have 77x- -\- x-\-y for the numerator 
 
 of the fraction sought, under which placing the denomi- 
 
 11 x^ —I— X -\- y 
 
 nator Ix. we finally obtain C i for the reduced 
 
 Ix 
 
 X I 1/ 
 
 form of llx -| !-^. 
 
 IX 
 
 bx ~\~ x^ 
 
 2. Reduce x to the the form of a fraction. 
 
 . mx — bx — x^ 
 
 Ans. , 
 
 m 
 
 3. Reduce y -{- 3x — to the form of a fraction. 
 
 S -\- a 
 
 Ans 3v+9x + ay-f-3a a: — 6 
 3 -]-a 
 
 a" 62 
 
 4. Reduce x to the form of a fraction. 
 
 X 
 
 Ans. — . 
 
 X 
 6 3^ 
 
 5. Reduce 3a^ — 6-1 to the form of a fraction. 
 
 1 — y 
 
 A 21a^ — 36 — 3aV + 6y — r' 
 1 — y 
 
 6. Reduce 9 A to the form of a fraction. 
 
 a — x^ 
 
 9a — 6x=^ — 8c* 
 Ans. . 
 
50 
 
 ALGEBHAIC FRACTIOXS. 
 
 CASE IV. 
 
 (57.) To reduce a fraction to an entire or mixed quantity. 
 
 RULE. 
 
 Divide the numerator by the deno/iiinator, the quotient 
 will be the entire part ; if there is a remainder^ place it over 
 the denominator for the fractional part. 
 
 9« — 6; 
 
 EXAMPLES. 
 
 to a mixed quantity. 
 
 1. Redi 
 
 a — X' 
 
 Dividing- the numerator by the denominator, we find this 
 
 FIRST OPERATION. 
 
 9fl— 6x- — Sc^la — x- 
 
 9« — 9^' 
 
 19 = integral part. 
 
 3a:" — Sc'* = numerator of fractional part. 
 
 ^2" • 
 
 Therefore the quantity sought is 9 + 
 
 a — X 
 
 We will now change the order of the terms of the nume- 
 rator and denominator, by placing the x^ first ; we thus find 
 this 
 
 SECXDND OPERATION. 
 X^ + a 
 
 — 6x9-h9a — 8c* 
 — 6x2 + 6a 
 
 3a 
 
 6 = integral part. 
 Sc* = numerator of fractional part 
 3a — 8c^ 
 
 Therefore the quantity sought is 6 -j . 
 
ALGEBRAIC FRACTIONS. 51 
 
 These two results are equivalent, but under illfTerent 
 forms. 
 
 2. Reduce to an entire quantity. 
 
 X 
 
 Ans a — X. 
 
 3. Reduce — to a mixed quantity, 
 
 Ans.2x-?^±^^. 
 3a: +1 
 
 4. Reduce — to an entire quantity. 
 
 in — y 
 
 Ans. )iL^ H- my + ?/^ 
 
 . p , 20a* — 10(7 -f G ^ . , 
 
 0. Reduce to a mixed quantity. 
 
 5fl 
 
 Ans. 4f/ — 2H . 
 
 5a 
 
 _ „ , 9v'— lSy + 8aV. • , 
 
 6. Reduce —^ — ' -^ to a mixed quantity. 
 
 yy 
 
 Ans. .V- — 2+-^. 
 
 „ r. , ! t//'' 21/? ^ . , 
 
 /. Reduct to a nnxed quantity. 
 
 Ans. 2m^ — — . 
 m 
 
52 ALGEBRAIC FRACTIONS. 
 
 CASE V. 
 
 (58.) To reduce fractions to a common denominator. 
 
 RULE. 
 
 ' Multiply successively each numerntor into all the denomi- 
 nators^ except its own ,, for new numerators^ and all the de- 
 nominators together for a common denominator. 
 
 EXAMPLES. 
 
 1. Reduce-, -, — to equivalent fractions having: a cora- 
 
 mon denominator. 
 
 a X 2 X 7a = 14a^ = new numerator of first fraction. 
 h XxX 7a = 7a6a:=new numerator of second fraction, 
 f X .T X 2 =2ca: = new numerator of third fraction. 
 
 and X X 2 X 7a = 14ca;=common denominator. 
 
 ,_, . 14a^ lahx 2cx . • i ^ r 
 
 rhereiore, : : are the equivalent trac- 
 
 ' 14ax ' Uax ' Uax ^ 
 
 lions sought. 
 
 2. Reduce — , — , and y. to fractions havino; a common 
 
 2a' 3x' ^' ^ 
 
 denominator. 
 
 . 9mx 4a6 6axy 
 Ans. —- : - — : -— ^. 
 box OCX OCX 
 
 1 x' a^ + x^ 
 
 3. "Reduce -, — , to equivalent fractions having a 
 
 ^ o a -J- X 
 
 common denominator. 
 
 3a-{- 3x 2ax2-f-2x3_ 6a«-f6x' 
 6a -(- 6x ' 6a -|- 6x ' 6a -^ 6x 
 
ALGEBRAIC FRACTIONS. 53 
 
 4. Reduce —, — , — to fractions having a com- 
 
 3b DC a 
 
 mon denominator. 
 
 5cdx iSbdx" 15aV)c — l5bcx- 
 
 Ans. 
 
 Idbcd ' lobcd ' lobcd 
 
 X X 1 a- + 2 
 
 5. Reduce-, — - — , — - — to fractions having a com- 
 
 mon denominator. 
 
 . 12a: 8x — 8 6a:-+12 
 ^"'•"2^' "24"' "^4~- 
 
 CASE VI. 
 
 (59.) To add fractional quantities. 
 
 RULE. 
 
 Reduce the fractions to a common denominator ; then add 
 the numerators, and jilace their sum over the common de- 
 nominator. 
 
 EXAMPLES. 
 
 1. What is the sum of ;^, I, ^? 
 3a 3 7 
 
 These fractions, when reduced to a common denominator, 
 
 . 21x 21a 9ay ,,..!.• . u 
 
 become — — , — — , — -- ; adding their numerators, we have 
 63o 63a D3a 
 
 21a; 4- 21a -{- day ; placing this over the common denomi- 
 nator, we find 
 
 X I 1 . y 21x-j-21a + 9oy 7x-|-7a-|-3ay 
 
 3^ "^ 3 "^ 7 63^^ ~ 21^ 
 
54 ALGEBRAIC FRACTIONS. 
 
 2x 8x 
 
 2. What is the sum of 3x -f- — and x — ? 
 
 5 9 
 
 Ans. 3x 4- ^^ . 
 45 
 
 3. What is the sum of— , — , 
 
 2x Ix 2x + 1 
 3 ' 4 ' ~5~ 
 
 ^ , 49x4-12 
 
 Ans. 2x A -!- — . 
 
 ^ 60 
 
 4. What IS the sum ot — , — , — ? 
 4 5 6 
 
 , 45x + 48x4-50x „ , 23x 
 
 An>;. 1 ! =2x+— — . 
 
 60 60 
 
 5. What IS the sum of — ^ , — — r 
 
 Ans. 
 
 6. What IS the sura of ■ , :: •' 
 
 4 4 
 
 Ans. "lili'. 
 
 CASE VII. 
 
 (60.) To subtract one fraction from another. 
 
 K, U L E . 
 
 Reduce the fractions to a common denominator^ then 
 auhtract the numerator of the subtrahend from the nume- 
 rator of the minuend, and place the difference over the 
 common denominator. 
 
ALGEBRAIC FRACTIONS. 55 
 
 EXAMPLES. 
 
 ^ 3x4- (I 2x — a 
 
 1. from — subtract — - — . 
 
 4 3 
 
 These fractions, when reduced to a common denominator, 
 
 , 9x-\-3a Sx — 4a 
 
 become -^•— and — -- — . bubtraclmc: the numera- 
 
 tois we have 9a; + 3a — (8a: — 4o) = j--}-7a, placing this 
 over the common denominator 12, we find 
 
 3x-|-a 2x — a--f-7a 
 
 4 3~'~~I2~ ■ 
 
 2. From^r-^' subtract !^-t_^ 
 5 4 
 
 19m — y 
 
 Ans. 
 
 20 
 
 3. From 3y-j-- subtract i/ — ^^^ ". 
 
 c 
 Ans. 2y -j- 
 
 4 From — '—^ subtract -. 
 
 2 2 
 
 Ans. y 
 
 en. a;- + 2a-y -^ y" . i^ — 2xy -4- y" 
 
 5 From ! £_LjL subtract ■ ^ ' ■ . 
 
 4x2/ 4a-y 
 
 Ans. 1 
 
 6 From subtract a 
 
 3a; + 2i/ 
 
 Ans. 2 — a ' — - . 
 
56 ALGEBRAIC FRACTIONS. 
 
 CASE VIII. 
 
 (61.) To multiply fractional quantities together. 
 
 RULE. 
 
 If any of the quantities to he multiplied are mixed, they 
 musty by Case III, he reduced to a fractional form ; then 
 multiply together all the numerators for a numerator, and 
 all the denominators together for a denominator. 
 
 EXAMPLES. 
 
 1. Multiply .-p by -:i^. 
 
 The product of the numerators will be 
 
 (x + c) X{x -\- h) = X' -{- ax -\- hx -^ ah ; 
 
 and the product of the denominators is 2 X 3 = 6. 
 
 _-. X A- a x-\-h x'^-\-ax-\-hx-\- ah 
 
 Hence, -t x ^ = -IL^^-. 
 
 2. Multiply ?lf±^yii±'. 
 
 '^ •' be '' b-\-c 
 
 x* — h* 
 
 Ans. 
 
 hH + bc' 
 
 « 3 J. 4_l_3; 3 
 
 3. What is the continued product of — -- , — -— , and - 1 
 
 r 7 ' 2 ' 7 
 
 36 — 3x — 3x2 
 
 Ans. _ . 
 
 98 
 
 4. What is the product of ^^, ^=^ ? 
 
 , Ans. : — . 
 
ALGEBRAIC P'RACTIONS, 57 
 
 5. What is the continued product of — , ' , - , and 
 
 ?7l X ' c 
 
 r 1 ■ . 2b/ix — 2bdx 
 
 ^ Ans. . 
 
 cmrx — cmx 
 
 6, What is the product of 1/ + by ^ , ? 
 
 Ans. ^r-^y + r-l, 
 
 2V- 
 
 CASE IX. 
 
 (62.) To divide one fraction by another. 
 
 RULE. 
 
 If there are any mixed quantities^ reduce them to a frac- 
 tional form ^ by Case III. ; then invert the divisor, and mul- 
 tiply as in Case VIIL 
 
 EXAMPLES. 
 
 ^ ^. .. 3x + 7, 43:— 1 
 
 1. Divide — ' — by . 
 
 4 • o 
 
 If we invert the divisor, and then multiply, we have 
 
 Zx + I 5 15^ + 35- . . ^ 
 
 — X = — r for the quotient. 
 
 4 4a; — 1 16x — 4 
 
 2. Divide 5^''- by ^-!±l:. 
 
 . x'u — y3 
 x' + xy- 
 
58 ALGEBRAIC FRACTIONS. 
 
 3. Divide -„^T- by ^• 
 
 3m' "^ 5m 
 
 4. Divide ^^y?^^. 
 8cd ^ 4d 
 
 . 3oa 
 
 Ans. 
 
 12mY 
 
 Qc'^y ' 
 
 5. Wiiat is the quotient of divided by - ? 
 
 Ans. 
 
 a:— 1 
 
 6. What is the quotient of — - — divded by — — ? 
 
 Ans. . 
 
 x—l 
 
SIMPLE EQUATIONS. 59 
 
 CHAPTER III. 
 
 SIMPLE EQUATIONS. 
 
 (63.) Jin equation is an expression of two equal quanti- 
 ties with tlie sign of equality placed between them. 
 
 The terms or quantities on the left-hand side of the sign 
 of equality constitute the first member of the equation, 
 those on the right constitute the second member. 
 
 Thus, x + 2 = a, (1) 
 
 1-1 = ^ (2) 
 
 3a; _^ 7 = c, (3) 
 
 are equations ; the first is read, " x increased by 2 equals o." 
 
 The second is read, " one-half of x diminished by 1 equals 
 6." 
 
 The third is read, " three times x increased by 7 equals 
 c." 
 
 (64.) Nearly all the operations of algebra are carried on 
 by the aid of equations. The relations of a question or 
 problem are first to be expressed by an equation, containing 
 known quantities as well as the unknown quantity. After- 
 wards we must make such transformations upon this equa- 
 tion as to bring the unknown quantity by itself on one side 
 of the equation, by which means it becomes known. 
 
60 SIMPLE EQUATIONS. 
 
 (65.) An equation of the first degree^ or a simple equa- 
 tion^ is one, in which the unknown has no power above the 
 first degree. 
 
 (66.) A quadratic equation^ is an equation of the second 
 degree, that is, the unknown quantity is involved to the 
 second power, and to no greater power. 
 
 (67.) An equation of the third, fourth, &c., degree, is 
 one which contains the unknown quantity to the third, 
 fourth, &c., degree ; but to no superior degree. 
 
 And in general, an equation which involves the mth power 
 of the unknown quantity, is called an equation of the mth 
 degree. 
 
 (68.) The following axioms will enable us to make many 
 transformations upon the terms of an equation without de- 
 stroying their equality. 
 
 AXIOMS. 
 
 I. If equal qua^itities he added to both members of an 
 equation, the equality of the members will not be destroyed. 
 
 II. If equal quantities be subtracted from both members 
 of an equation, the equality of the members will not be 
 destroyed. 
 
 III. If both members of an equation be multiplied by 
 the same quantity, the equality will not be destroyed. 
 
 IV. If both members of an equation be divided by the 
 same quantity, the equality will not be destroyed. 
 
 CLEARING EQUATIONS OF FRACTIONS. 
 
 (69.) When some of the terms of an equation are frac 
 tional, it is necessary to so transform it, as to cause the de 
 nominators to disappear, which process is called clearing of 
 fractions. 
 
SIMPLE EQUATIONS. 
 
 61 
 
 Let it be required to clear of fractions, the following equa- 
 tion. 
 
 Now, by Axiom III, we can multiply all the terms of this 
 equation by any number we please, without destroying the 
 equality. If we multiply by a multiple of all the denomi- 
 intors, it is evident they will disappear. 
 
 Tf we choose the least multiple of the denominators as a 
 multiplier, it is plain that the labor of multiplying will be 
 the least possible. 
 
 Thus, in the above example, multiplying all the terms 
 of both sides of the equation by G, which is the least multi- 
 ple of 2, 3, and 6, we have 
 
 3a: + 2x + a; == 6a; + 6. (2) 
 
 This equation is now free of fractions. 
 
 (70.) Hence, to clear an equation of fractions, we deduce, 
 from what has been said, this 
 
 RULE. 
 
 Multiply all the terms of the equation by any multiple oj 
 (heir denominators. If we choose the least common multi- 
 pie of the denominators, for our multiplier, the terms of 
 the fraction, when cleared, will be in their simplest form. 
 
 EXAMPLES. 
 
 1. Clear of fractions the equation ^ = — — . 
 
 ^ 5 2 7 
 
 In this example the least common multiple of the deno- 
 minators 5, 2, and 7, is 70 ; hence, multiplying all the terms 
 of our equation by 70, we find 
 
 14a: — 14a = 35x + 356 — 10, 
 for the equation when cleared of fractions. 
 
62 SIMPLE EQUATIONS. 
 
 « r^^ x- r • ^ ^ , X 4- U X /; . X 
 
 2. Clear ot fractions — 1 ]— -— =x-\- — 
 
 o 4 ^ lb 
 
 Ans. 2z — 4 4- 4x + 4a — 8x -f- 86 = 16x -j- x. 
 
 (71.) We must observe that when a fraction has the sign 
 — , it requires its value to be subtracted, so that, if it is writ- 
 ten without the denominator, all the signs of the numerator 
 must be changed. 
 
 ^ J x-l-1 T 3 c 
 
 3. Clear of fractions — \- -^ "— j— = « + '' — ;:- 
 
 Ans. 42x-42+28a:+28— 21a:+63 =S4fl+846— 12c. 
 
 4. Clear the equation - + - + - + ^ + " =25 1 of fractions. 
 
 2 3 4 5 6 
 
 Ans. 303: + 20x + 15a: + 12x -}- lOx = 15060. 
 
 5. Clear the equation 1 \- - == g o{ fractions. 
 
 Ans. a- dm -\-bdx + cmx = dgmx. 
 
 ^rx, , • X , X — 3 a- — 5m. 
 
 6. Clear the equation ; — -— r ~ = — ot 
 
 ^ u'—b- a-\-b a — b a 
 
 fractions. 
 
 -f- Sab — a"x — abx -|-5a=' 
 
 . { ax-\-a-x — abx — 3a 
 ^^^- I -\r5ab = a=m — b-7n 
 
 m. 
 
 TRANSPOSITION OF THE TERMS OF AN EQUATION. 
 
 (72.) The next thing to be attended to, after clearing the 
 equation of fractions, is to transform it so that all the terms 
 containing the unknown quantity may constitute one mem- 
 ber of the equation. 
 
 If we take the equation 
 
 we have, when cleared of fractions, 
 
SIMPLE EQUATIONS. 63 
 
 6a — 3x + 2bx = 48a:. (2) 
 
 If we add to both members of this equation 3a; — 2bx 
 (Axiom L), it becomes 
 
 6a — 3x -I- 2bx + 3a; — 26a; = 48a; -j- 3x — 26a;. (3) 
 
 All the terms of the left-hand member cancel each other 
 except 6a. 
 
 Therefore we have 
 
 6a = 48r + 3x — 26a;, (4) 
 
 in which all the terms of the right-hand member contain x. 
 
 If we compare equation (4) with (2), we shall discover, 
 that the terms — 3a; + 26x, which are on the left side of 
 equation (2), are on the right side of equation (4), with 
 their signs changed. 
 
 Hence, we conclude that the terms of an equation may 
 change sides, provided they change signs at the same time. 
 
 (73.) To transpose a term from one side of an equation 
 to the other, we must observe this 
 
 RULE. 
 
 Jiny term may be transposed from one side of an equation 
 to the other J by changing its sign. 
 
 EXAMPLES. 
 
 X -4- 6 5x 
 
 1. Clear the equation -— - — f- 26 = j + 2 of fractions, 
 
 and transpose the terms so that all those containing r may 
 constitute the left-hand member. 
 
 First, clearing the above equation of fractions, by Rule 
 under Art. 70, we have 
 
 2x-f 12-fl04 = 5x-f8. 
 
64 SIMPLE EQUATIONS. 
 
 Secondly, transposing 2a; from the left member to the 
 right member, and 8 from the right member to the left, we 
 have 12 -j- 104 — 8 = 5a: — 2a; for the result required. 
 
 2. Clear the equation II1_=: 7^ of fractions, and 
 
 . "-^ 
 
 transpose the terms. 
 
 Ans. 3x — 2a; = 45 -|- 2a. 
 
 3. Clear of fractions, the equation — -3i-[-7. = q-|- - 
 
 and transpose the terms. 
 
 Ans. 14a; + 3x — 2x = 36 + 60. 
 
 4. Clear of fractions and transpose the terms of the 
 
 X 2-\-x c 
 
 equation -^ . 
 
 ^ a — ba-\-b a-^—b- 
 
 Ans. ax-\-bx — ax-\-bx= c -\-2a — 2b. 
 
 (74.) We are now prepared to find the value of the un- 
 known quantity. If we take the last example, it may be 
 written thus, 
 
 (a -[-6 — a -\-b) x:=c -\-2a — 2b ; 
 or uniting the like terms within the parenthesis, it becomes 
 
 2bx = c -\-2a — 2b. 
 Dividing both sides of this equation by 26, (Axiom IV.), 
 
 c , c-l-2a — 26 
 
 we nncl x = : 
 
 26 ' 
 
 hence, the value of x is now known, since it is equal to the 
 
 c + 2a— 26 
 expression — ■ — — . 
 
 (75.) From what has been done, we discover that an 
 equation of the first degree may be resolved by the follow- 
 ing general 
 
SIMPLE EQUATIONS. 65 
 
 K U L E . 
 
 I. If any of the terms of the equation are fractional, t'i> 
 equation mtist be cleared of fractions, by Rule under ^irt. TO. 
 
 II. The terms must then he xo transposed, that all Ihose 
 containing the unknown quantity may constitute one side c.r 
 vicinbcr of the equation, by Rule under Art. 73. 
 
 III. Then divide the algebraic sum of those terms on that 
 side of the equation which are independent of the unknown 
 quantity, by the algebraic sum of the coefficients of the terms 
 containing the unknown quantity, the quotient will be the 
 value of the unknown quantity. 
 
 EXAMPLES. 
 
 X X 
 
 1. What is the value of a: in the equation 1- - =x — 10 1 
 
 3 4 
 
 This, cleared of fractions, becomes 
 
 4a;-|-3x=12a:— 120. 
 When the terms are transposed and united, we have 
 
 120 = 5x. 
 Dividing by 5, we get 24 = x. 
 
 2. What is the value of x in the equation 
 
 2a: + 1 a; 4- 3 , 
 
 X = — '■ — ' 
 
 Ans. ar = 13. 
 
 - _. 21— 3a; 4a: + 6 5a: + 1 ^ . 
 
 3. Given -^ — =6 ! — to find x. 
 
 3 9 4 
 
 Ans. a-^= 3. 
 
 4. Find x from the equation 3ox -j — — 3 = bx — a. 
 
 6 — 3fl 
 
66 SIMPLE EQUATIONS. 
 
 ^ ^. a; — 2 3.r , 15x n^ . r i 
 
 5. Given 1- — — = 3 / , to find x. 
 
 4 2 2 
 
 Ans. a; = 6. 
 
 ^ . , , ... 3tx 26x . ,. 
 
 6. Find X so as to satisty the condition — 4 =/• 
 
 •^ a m 
 
 rt/m + 4am 
 
 Ans. 0:=:; — -r- 
 
 3cw — 2a6 
 
 „ „. , ^ , . Snx — b 36 . , c 
 
 7. r ind x trom the equation — 7r='* — " — 7i- 
 
 7 2 2 
 
 56 4-96— 7c 
 Ans. a; = — ■• 
 
 167J. 
 
 S. Given ^-^ + '^-+— = 3x- — 12, to find x. 
 
 Ans. a: =6. 
 
 ^^. 3a: — 5, 4a; — 2 .T^ri 
 
 9. Given a; — \- — - — = a: + 1, to find x. 
 
 ^ Ans. a; = 6. 
 
 10. Given ^4^ — 3x + ?^=^ + 3 = a-, to find x. 
 
 3 5 
 
 Ans. a; = 2. 
 
 , ^. 3x — 2 , 3a; + 2 ^ ^ c ^ 
 
 11. Given -— 1 -^=x— l,to finda:. 
 
 12. Given -— -+x=ll, to find x. 
 
 O I 
 
 Ans. 
 
 Ans. a;^9/j. 
 
 „ ^. (a-{-b)x , X a; + 1 ^ „ , 
 
 13. Given ^ ^ / + -. r^ = — ^, to find x. 
 
 a — 6 'a'» — b^ a + b 
 
 Ans. x = 
 
 a^ + 2ab + b^ — a-\-b-\-l 
 
SIMPLE EQUATIONS. 67 
 
 QUI^IONS, THE SOLUTION OK WHICH KEQUIRE EQUAT!C)NS 
 OF THE FIRST DEGREE. 
 
 (76.) In the solution of questions, by the aid of algel)ra 
 the most difficult part is to obtain the proper equation which 
 shall include all the necessary relations of the question. 
 When once this equation of condition is properly found, 
 the value of the unknown quantity is readily obtained by 
 the Rule under Art. 75. 
 
 Suppose Ave wish to solve, by algebra, the lollowir 
 (;uestion. 
 
 1. What number is that, whose half increased by i 
 third part and one more shall equal itself ? 
 
 If we suppose x to be the number sought, its half will bt 
 
 -, which increased by its third part, becomes z -[- 55 ^nd 
 
 'bis increased by one, becomes -+--}- 1, which by the 
 
 question must equal itself. 
 
 Therefore, we have - -|- -f- 1 ^ x for the equation of 
 
 ••ondition. 
 
 Solving this, by Rule under Art. 75, we have x = 6, 
 
 VERIFICATION. 
 
 ? = iof6=3, 
 
 ?=iof6 = 2, 
 1 =1, 
 
 Therefore, - -}- ^+ 1 == 6, which sliows that 6 is 
 
 truly the number sought. 
 
CS SIMPLE EQUATIONS, 
 
 Again, let us cmleavor to solve tills question : 
 
 2. What number is that whose third part exceeSi its 
 
 fourth part by 5 / 
 
 Suppose X to be the number, then will its third part 
 
 X . X 
 
 — : its fourth part =-. 
 
 Therefore, the excess of its third part over its fourth part 
 
 XX. 
 
 is expressed by , which, by the question, must equal 5 . 
 
 X X 
 
 Hence, we have the following equation = 5, 
 
 3 4 
 
 this solvci!, gives x = 60 ; the third part of which is 20, 
 
 and its fourth part is 15, so that its third part exceeds its 
 
 r.iurth part by 5, hence, this is the correct number sought. 
 
 (77.) The method of forming an equation from the con- 
 rlitions of a question, is of such a nature as not to admit of 
 any simple rule, but must be in a measure left to the inge- 
 nuity of the student. 
 
 It will hoM'ever be of assistance to pay attention to the 
 followins 
 
 RULE 
 
 letter^ we must indicate, by algebraic symbols, the same 
 operation, as it would be necessary to perform upon the true 
 number, in order to verify the conditions of the question. 
 
 3. Out of a cask of wine Avhich had leaked aAvay a third 
 part, 21 gallons were afterwards drawn, and the cask was 
 then found to be half full : how much did it hold ? 
 
 Suppose X to be the number of gallons which the cask 
 held. 
 
SIMPLE EQUATIONS. 69 
 
 Then, the part leaked away must be -. 
 
 <j 
 
 And tlic part leaked away, together with the quantity 
 
 X 
 
 drawn off, must be - -j- 21. 
 o 
 
 Now, by the question, the cask is still half full ; so that 
 
 what has leaked out, together with what has been drawn off 
 
 must be -. 
 
 Hence, we have this equation, - =— -|- 21, 
 
 which, cleared of fractions, becomes, 3x = 2a--|- 126 ; 
 transposing and uniting terms, we have a:= 126. 
 
 4. There are two numbers which are to each other as 6 to 
 5, and whose ditference is 40. What are the numbers. 
 
 Suppose the numbers to be denoted by 6a; and 5x, which 
 are obviously as 6 to 5 for all values of x. Now, by the 
 question, the diflference of these numbers is 40. Therefore, 
 we have 6rc — 5a: = 40 ; that is, x = 40. 
 
 Hence, 6a-= 6x40 =240 ?, , . ' 
 
 ^ .^ ^^^ / the numbers sought. 
 5^ = 5X40=200^ ° 
 
 5. A farmer had two flocks of sheep, each containing the 
 same number. Having sold from one of these 39, and 
 from the other 93, he finds twice as many remaining in the 
 one as in the other. How many did each flock originally 
 contain 1 
 
 Suppose the number in each flock to be denoted by x. 
 
 Then the flock from which he sold 39 will have remain- 
 ing x— 39. 
 
 And the one from which he sold 93 will have remaining 
 X — 93. 
 
 Hence, by the the question, we have 
 
 2x(x — 93)=x — 39, or 2x— 186=x — 39j 
 transposing and uniting terms, x = 147. 
 
70 SIMPLE EQUATIONS. 
 
 6. Divide the number 36 into three such parts, that | of 
 the first, -^ of the second, ] of the third, shall be equal to 
 each other. 
 
 If we denote the Ihree parts by 2.r, 3x, 4a:, it is plain that 
 ■ of the first, i of the second, ^ of the third, will be equal 
 for all values of x. 
 
 Now, by the question, the sum of these three parts must 
 equal 36. 
 
 Therefore, 2x + 3x + 4x = 36 ; 
 
 uniting terms, we have 9x = 36 ; 
 
 dividing by 9, and we obtain x=^^. 
 
 Consequently, 2a; =2X4= 8^ 
 
 3a: =r 3 X 4 = 12 / tlie parts sought. 
 4a; = 4x4 = 16) 
 
 7. Two pieces of cloth are of the same price by the yard, 
 but of different lengths ; the one cost $5, the other $6^. If 
 each piece had been 10 yards longer, their lengths would 
 have been as 5 to 6. What was the length of each piece '\ 
 
 Since the price per yard was the same for both pieces, 
 their lengths must have been to each other the same as the 
 number of dollars which they cost, or as 5 to 6' , or, which 
 is the same, as 10 to 13. 
 
 Therefore we will denote their lengths by lOx and 13a;. 
 
 These become, when increased by 10, 
 
 10a; + 10 and I3x + 10, 
 which, by the question, must be as 5 to 6. 
 
 Hence, 6(10a;4- I0) = 5(l3x+ 10) ; 
 
 or, expanding, 60a; -f- 60 =: 65a; -j- 50 ; 
 transposing and uniting terms, we getl0 = 5x, and a;:=2. 
 Therefore, lOx =10x2 =20 
 
 13x= 13X2=26 
 
 the lengths sought. 
 
SIMPLE EQUATIONS. 71 
 
 8. Twelve oxen have in 4 weeks eaten all the grass which 
 grew on 3^ acres of land, in such a manner that they not 
 only ate all the grass which at first was there, but also that 
 which grew during the time they were grazing. In like 
 manner, have 21 oxen, in 9 weeks, eaten all the grass upon 
 10 acres of land. How many oxen can, in this way, graze 
 for 18 weeks upon 24 acres of land 1 
 
 Let X = the growth in acres of one acre of grass for one 
 week ; then will the growth of 3i acres for 4 weeks equal 
 
 3iX4Xa:=-^; 
 o 
 
 also, the growth of 10 acres for 9 weeks will equal 
 
 10x9xa: = 90x. 
 Therefore, the whole quantity of grass eaten in the first case, 
 equals 
 
 40x_10+40.r 
 ^^+-3"-"~3"~- 
 The quantity eaten in the second case equals 10 -|- 90x. 
 Hence the quantity which one ox eat in one week equals 
 10+ 40a: 1 1 5 + 20a: . , ^ 
 
 Again, in the second condition the quantity which one ox 
 eat in one week equals 
 
 Now, by the question, an ox in the first case ate the same 
 as an ox in the second case ; therefore we have 
 
 5-f 20x _10 + 90a: . 
 
 72 '~ 189 ■ ^ ^ 
 
 This, solved, gives x= — of an acre. 
 
 This value of x substituted in either member of (1) gives 
 
72 . SIMPLE EQUATIONS. 
 
 — for the fractional part of an acre eaten by one ex in one 
 
 5^ 
 
 54 
 
 week ; therefore, the quantity which 1 ox eats in 18 weeks is 
 
 5 5 
 
 — X 18 = — acres. 
 
 54 3 
 
 Now, the 24 acres increasinf^ IS weeks, at the rate of — 
 
 of an acre for each acre for each Avetk, will amount to CO 
 
 acres. 
 
 5 
 Hence, 60 -H - = 36 oxen for the answer. 
 
 o 
 
 9. Divide the number 237 into two such parts, that the 
 one may be contained in the other 1:| times. What are 
 these parts? 
 
 Ans. 1051 and 131 §. 
 
 10. The sum of $1200 is to be divided between two per- 
 sons, A and B, so that A's share may be to B's share as 
 2 to 7. How much does each receive ? 
 
 Ans. A$266|, B$933),. 
 
 The above question, when generalized, becomes like the 
 following question. 
 
 11. Divide a number a into two such parts, that the first 
 
 part may be to the second as m to n. What are the parts ? 
 
 . ma na 
 
 Ans. -, . 
 
 m -\- 11 ?ii -J- 71 
 
 12. Divide the number 46 into two unequal parts, so that 
 when the greater is divided by 7, and the less by 3, the quo- 
 tients together may amount to 10. What are these parts ? 
 
 Ans. 28 and 18. 
 
 13. In a company of 266 persons, consisting of officers, 
 merchants, and students, there were four times as many mer- 
 chants, and twice as many officers as students. How many 
 were there of each class 1 
 
 Ans. 38 students, 152 merchants, and 76 officers. 
 
SIMPLE EQUATIONS. 1?, 
 
 14. Divide the number a into three such parts, thai thi- 
 second may be m times, and the third n times as great as 
 the first. What are these parts ? 
 
 1 -|- Di -f- « 1 -}- m + n 1 -\-m-\- n 
 
 15. A fiekl of 864 square rods is to be divided among 
 three farmers, A, B, C, so that A's part shall be to B's as 
 5 to 11, and C may receive as much as A and B together. 
 How much docs each receive ? 
 
 Ans. A 135, B 297, C 432 square rods. 
 
 16. Divide the number a into three such parts, that the 
 first shall be to the second as m to ??, and the second part 
 to the third as p to q. What are these parts ? 
 
 . mpa npa nqa 
 
 II I p -\- lip -{- nn VI p -\- np -j-w?' mp -\- np -\- nq' 
 
 17. Divide $1520 among three persons. A, B, C, so that 
 B may receive $100 more than A ; and C §270 more than 
 B. How much does each receive? 
 
 Ans. A $350, B $450, C $720. 
 
 IS. A certain sum of money is to be divided amongst 
 three persons. A, B, C, as follows : A shall receive $3000 
 less than the half of it, B $1000 less than the third part, 
 and C is to receive $800 more than the fourth part of tlie 
 whole sura. What is the sum to be divided 7 and what 
 does each receive '? 
 
 Ans. The whole sum is $38400. A receives $16200, 
 B $11800, C $10400. 
 
 19. A mason, 12 journeymen, and 4 assistants, receive 
 together $72 wages for a certain time. The mason receives 
 $1 dailv; each journeyman \ 
 
74 SIMPLE EQUATIONS. 
 
 cents. How many days must they have worked for this 
 money ? 
 
 Ans. 9 days. 
 
 20. A courier who had started from a certain place 10 
 
 days ago, is pursued by another from the same place, and 
 
 by the same way. The first goes 4 miles every day, the 
 
 other 9. How many days wull the second need to overtake 
 
 the first ? 
 
 Ans. 8 days. 
 
 21. A courier left this place n days ago, and makes a 
 miles daily. He is pursued by another making h miles daily. 
 How many days will the second require to overtake the first 1 
 
 Ans. days. 
 
 h — « 
 
 22. But, in what time will the second courier overtake 
 
 the first, when it is supposed that the second starts 12 days 
 
 later than the first, and his speed is to that of the first as 8 
 
 is to 3 ? 
 
 Ans. 7} days. 
 
 23. Two bodies start from the same place, one after the 
 other, in a straight line ; the second starts n seconds later 
 than the first, and its speed is to that of the first as q is to 
 p. In what time will these two bodies be together ? 
 
 Ans. -- — seconds after the setting out of the second. 
 <1 — V 
 
 24. Two bodies move in opposite directions ; one runs 
 c feet in each second, the other C feet. The two places 
 from which they start at the same time, are distant d feet 
 from one another. When will they meet 1 
 
 Ans. y^r— seconds. 
 C-j-c 
 
SIMPLE EQUATIONS. 75 
 
 25. But, 111 what time will these two bodies come to- 
 gether, when that which goes C feet each second, runs af- 
 ter the other ? 
 
 Ans. — — — seconds. 
 C — c 
 
 Is the problem, as here stated, always possible ? And 
 what is required for it to be possible ? What does the ex- 
 pression — signify, when C=cl What does it denote 
 
 when C<:cl 
 
 26. At 12 o'clock, both hands of a clock are together. 
 When, and how often, will these hands be together in the 
 next 12 hours ? • 
 
 '^The hands will meet 11 times ; these ren- 
 j centers will be at 5y\ minutes past 1, lOyy 
 " "^" ^minutes after 2, 16yV alter 3, and so on, in 
 vcach successive hour o/p minutes later. 
 
 27. Two bodies move after one another in the circumfe- 
 r nee of a circle which measures p feet. At first they arc 
 distant from each other by an arc measuring d feet ; the first 
 moves c feet, the second C feet in a second. When will 
 these two bodies be together for the first time, second time, 
 luid so on, supposing that they do not disturb each other's 
 motion 1 
 
 -'^"^- c=-o' & ^^-^",&c., seconds. 
 
 28. But when will they meet, when the first begins to 
 move t seconds sooner than the second ? 
 
 29. But if the first starts t seconds later than the second, 
 when will they meet 1 
 
 Ans. i=^, e±±^, ?£±i^',&c., seconds. 
 C — c C — c C — c 
 
76 SIMPLE EQUATIONS. 
 
 80. But if the first, instead of preceding the second, runs 
 against it, and starts from the same place t seconds sooner, 
 when do they meet ? 
 
 . d — ct r)-\-d — ct 2p-\-d — ct . , 
 
 -^"^- C + l'-C^TT' ^— ,&-,secon,ls. 
 
 31. A cistern can be filled by three pipes j by the first in 
 1} hours, by the second in 3^ hours, and by the third in 5 
 hours. In what time will this cistern be filled when all 
 three pipes are open at once 1 
 
 Ans. 48 minutes. 
 
 32. In order to make the foregoing problem m.ore general, 
 let th« time which the first pipe alone takes in filling the cis- 
 tern = cr, the time which the second takes in doing the same 
 = 6, and the time required by the third = c. What ex- 
 pression gives the time in which all three pipes together 
 will fill it ? 
 
 . ahc 
 
 ' ah -f ac -(- he 
 
 33. A servant received from his master $40 wages, 
 yearly, and a suit of livery. After he had served 5 months 
 he asked for his discharge, and received for this time, the 
 livery, together with $6} in money. How much did the 
 livery cost 1 
 
 Ans. $18. 
 
 34. A master hired a journeyman, and promised him 8 
 shillings for each day that he worked for him ; but if he 
 worked anywhere elscj then the journeyman must pay him 
 5 shillings daily for his board. At the expiration of 50 
 days they settle, and the journeyman receives 10 pounds 
 and 18 shillings. How many days has he worked for his 
 
 master 1 
 
 Ans. 36 days. 
 
SIMPLE EQUATIONS. 77 
 
 35. Find a number such that i thereof increased by i of 
 the same, shall be equal to l of it if increased by 35. 
 
 Ans. 84. 
 
 36. A gentleman spends l of his yearly income in board 
 
 and lodging, | of the remainder in clothes, and lays by $200 
 
 a year. What is his income .'* 
 
 Ans. $1800. 
 
 EQUATIONS OF TWO OR MORE UNKNOWN 
 QUANTITIES. 
 
 (78.) Suppose we have given the two equations 
 
 X — y = 11, 
 
 to find the value of x and y. 
 
 If we take the sum of the two equations, we shall have 
 
 2x = 30. 
 Dividing by 2 , we find 
 
 X = 15. 
 Again, subtracting the second equation from the first, we 
 have 
 
 2y = S. 
 
 Dividing by 2, we obtain 
 
 2/ = 4. 
 2. Suppose we nave given the two equations 
 
 M=8, (1) 
 
 l + fe = '' (^) 
 
 to find the value of x and y. 
 
78 SIMPLE EQUATIONS. 
 
 We will first clear these equations of fractions, by mul- 
 tiplying the first by 12 and the second by 48 (Art. 70); we 
 thus obtain 
 
 4a:+3y= 96, (3) 
 
 8a:-f3.y = 144. (4) 
 
 Now, subtracting (3) from (4) we have 
 4x = 48. 
 
 Divided by 4 we find 
 
 x= 12. 
 If we multiply (3) by 2 it becomes 
 
 8a: -f 6y = 192. (5) 
 
 Now, subtracting (4) from' (5) we find 
 
 3i/ = 48. 
 
 Dividing by 3 we find 
 
 y=16. 
 
 3. If we have given the two equations 
 
 2x — 3i/= 4, (1) 
 
 8a: — 6y = 40, (2) 
 
 to find X and y, we proceed as follows : 
 
 Dividing (2) by 2 it becomes 
 
 4x — 3y = 20. (3) 
 
 Subtracting ( 1 from (3) we find 
 
 2a:=16.-.x = 8. 
 
 Multiplying (1) by 4 we have 
 
 8a: — 122/ = 16. (4) 
 
 Subtracting (4) from (2) we get 
 
 63/ = 24 .•. y =4. 
 
SIMPLE EQUATIONS. 79 
 
 ELIMINATION BY ADDITION AND SUBTRACTION. 
 
 (79.) From Avhat has been clone, we discover that an 
 unknown quantity may be eliminated from two equations, 
 by the following 
 
 RULE. 
 
 Operate upon the two given equations^ by multiplication 
 or division, so that the coefficients of the quantity to he 
 eliminated may become the same in both equations; then add 
 or subtract the tu-o equations, as may be necessary, to cause 
 these two terms to disappear. 
 
 EXAMPLES. 
 
 4. Given, to find x and y, the two equc 
 
 itions 
 
 
 3x— y=3, 
 
 
 (1) 
 
 2/ + 2x = 7. 
 
 
 (2) 
 
 If we add the two equations, we have 
 
 
 
 5x=l0.-.x = 2. 
 
 
 (3) 
 
 Again, multiplying(3) by 2, we get 
 
 
 
 2x = 4. 
 
 
 (4) 
 
 Subtracting (4) from (2) we obtain 
 
 
 
 7/ = 3. 
 
 
 
 5. Given, to find x and y, the two equations 
 
 
 -+•-=6, 
 2^3 ' 
 
 
 (1) 
 
 3^2 
 
 
 (2) 
 
80 SIMPLE EQUATIONS. 
 
 Clearing these equations of fractions, by inultlplyiiu 
 each by 6, they become 
 
 3a: + 2y = 36, (3) 
 
 ,2x + 3y=39. (4) 
 
 Multiplying (3) by 3, and (4) by 2, they become 
 
 9it:H-6y=108, 
 
 (5) 
 
 4x + 6y= 78. 
 
 (6) 
 
 Subtracting (6) from (5) ^ve get 
 
 
 5x = 30, 
 
 
 and x=6. 
 
 (') 
 
 Multiplying (7) by 2, it becomes 
 
 
 2a:=12. 
 
 {^) 
 
 Subtracting (8) from (4), we find 
 
 
 3?/ = 27 .-.^ = 9. 
 
 
 6. Suppose we wish to find x, y, and z, from the three 
 equations 
 
 5x— 62/ + 4c = 15, (1) 
 
 7x+4y-3c = 19, (2) 
 
 2a: + y + 6z=46. (3) 
 
 We will first eliminate y: for this purpose multiply (3), 
 first by 4 and then by 6, and it will give 
 
 8a: + 4y + 242=384, (4) 
 
 12a: + 6y 4- 36r = 276. (5) 
 
 Add (1) to (5) ; and subtract (2) from (4), and we have 
 17x-f 40-r=291, (6) 
 
 x + 27c =165. (7) 
 
 We have now the two equations (6) and (7), and but 
 two unknown quantities x and c. 
 
 Multiply (7) by 17 and it will become 
 
 17x+459z = 2805. (8) 
 
SIMPLL EQUATIONS. 81 
 
 Subtracting (6) from (8) we obtain 
 
 419c = 2514. (9) 
 
 Dividing (9) by 419, we find 
 
 c = 6. (10) 
 
 Multiplying (10) by 27, we find 
 
 27z=162. ;il) 
 
 Subtracting (11) from (7) we get 
 
 x = 3. (12) 
 
 Multiplying (10) by 6, and (12) by 2, and then taking 
 their sum, we find 
 
 6c + 2.T = 42. (13) 
 
 Subtracting (13) from (3), we get 
 
 (80.) We will now repeat the solution of this last ques- 
 tion, adopting a simple and easy method of indicating the 
 successive steps in the operations. 
 
 The method which we propose to make use of, is to indi- 
 cate by algebraic signs, the same operations upon the respec- 
 tive numbers of the different equations, as we wish to have 
 performed upon the equations themselves. 
 
 Thus, 
 
 (Q\ ^(A\y,oS shows, that equation (6) is obtained by 
 ( multiplyino; equation (4) by 3. 
 
 iplying equation (4) by 
 Jshf 
 Ihy 
 
 (ll)^(6)-(3) 
 
 /,QN /r-x I /-.N ^ shows, that equation (10) is obtained 
 
 V ^ — '^ ' ^ + ^ ^ ; by adding equations (7) and (1). 
 
 shows, that equation (11) is obtained 
 by subtracting equation (3) from (G). 
 
 /,c\ /-|4^x _^<^ S shows, that equation (15) is obtained 
 
 ( by dividing equation (14) by 3. 
 
 And so on for other combinations. 
 11 
 
82 STMPI.E EQUATIONS. 
 
 This kind of notation will become familiar by a little 
 practice. 
 
 We will now resume our equations of last example. 
 
 r Sx — 6?/ + 4=rrzl5 (1)^ 
 
 Given < 7x 4- 4y — 3c = 19 (2) > , to find x, y, and z. 
 
 (2x+ y + 6z = 4.6 (3)) 
 
 8x--f 4y+ 24^ = 184 (4)=:(3)x4 
 
 12a: 4- 6y-|- 36z==276 (5)r=(3)x6 
 
 17x+ 40c = 291 (6) = (l)+(5) 
 
 a: + 27- =165 (7) = (4)— (2) 
 
 17a: 4-459r= 2805 (8) = (7)xl7 
 
 419c = 2514 (9) = (8)— (6) 
 
 c = 6 (10) = (9)^419 
 
 27z = 162 (11) = (10) X 27 
 
 a: = 3 (12)=(7)-(11) 
 
 62 = 36 (13) = (10)X6 
 
 2x = 6 (14) = (12)X2 
 
 6c-f-2a: = 42 (15) = (13) + (14) 
 
 y=4 (16) = (3)-(15) 
 
 Collecting equations (12), (16), and (10),w^e have 
 
 ra: = 3. (12) 
 
 Ans. < 1/ = 4. (16) 
 
 (z = 6. (10) 
 
 We will solve one more set of equations by this method, 
 giving all the steps at length, the better to illustrate this 
 notation. 
 
 Given* 
 
 7a: — 22 + 3w=17 
 
 (1)^ 
 
 
 4y — 2z+ f=n 
 
 (2)/ 
 
 
 5y__3a:_2«= 8 
 
 (3)> 
 
 , to find a:, y, z, u, t 
 
 4y^Su + 2t = 9 
 
 (^)\ 
 
 
 3z-f.8M = 33 
 
 (5)^ 
 
 
SIMPLE EQUATIONS. 83 
 
 8y — 42-f2^ = 22 (G) = (2) X 2 
 
 4y — 4z-\-3u = l3 (7) = (6) — (4) 
 
 21x — 6z-t-9M = 51 (8) = (1) X 3 
 
 35y— 21x— 14w = 56 (9) = (3) X 7 
 
 35y — 6: — 5w=107 (10) = (8) + (9) 
 
 140y—140z -1-105/^=455 (11) = (7) X 35 
 
 140y— 24c— 20/^=428 (12) = (10) X 4 
 
 —116c -1-125/^ = 27 (13) = (11)— (12) 
 
 348c -f 928/^ = 3828 (14) = (5) X 116 
 
 —348c -h 375?^ = 81 (15) = (13) X 3 
 
 1303/i = 3909 (16) = (14) -f (15) 
 
 « = 3 (17) = (16) -^ 1303 
 
 8u = 24 (18) = (17) X 8 
 
 3c = 9 (19) = (5) — (IS) 
 
 c = 3 (20) = (19) ^ 3 
 
 3« = 9 (21) = (17) X 3 
 
 4c = 12 (22) = (20) X 4 
 
 4c — 3/< = 3 (23) = (22) — (21) 
 
 43/ =16 (24) = (23) + (7) 
 
 y = 4 (25) = (24)-^4 
 
 8y = 32 (26) = (24) X 2 
 
 Sy — 4c = 20 (27) = (26) — (22) 
 
 2^ = 2 (28) = (6) — (27) 
 
 t = 1 (29) = (28) -^ 2 
 
 2c = 6 (30) = (20) X 2 
 
 3u — 2c = 3 (31) = (21) — (30) 
 
 7x=14 (32) = (1) — (31) 
 
 x = 2 (33) = (32)-^7 
 Collecting equations (33), (25), (20), (17), (29), we have 
 
 'x = 2. 
 
84 SIMPLE EQUATIONS. 
 
 ELIMINATION BY COMPARISON. 
 
 (81.) We may a. so eliminate one of the unknown quan- 
 tities of two equations, by the following process : 
 Take the two equations 
 
 5y — 4a: = — 22, (1) 
 
 4y + 4a:==38. (2) 
 
 If we, for a moment, consider y as a known quantity, 
 we may then, from each of these equations, find the value 
 of X by Rule under Art. 75. 
 We thus find 
 
 22 +5y 
 
 4 
 
 (3) 
 
 38-31/ 
 X ^ — ^— . (4) 
 
 Putting these two values of x equal to each other, we have 
 22+5y 38 -By 
 
 -^- - —T- ■ ^^^ 
 
 Clearing (5) of fractions, it becomes 
 
 22 + 52/ = 38 — 3y, (6) 
 
 transposing and uniting terms, we find 
 Sy = 16 .-. y = 2. 
 This value of y substituted in either of the equations (3) 
 or (4), will give 
 
 x=8. 
 The above method of eliminating may be given as in 
 the following 
 
 RULE. 
 
 I. Find, from each of the given equations, the value oj 
 one of the unJaiown qttantities, by Rule under Art. 75., on 
 the supposition that the other qaantities are knovm. 
 
SIMPLE EQUATIONS. 85 
 
 II. Then equate these different expressions of the value of 
 the tinknown, thus found ^ and we shall thus have a number 
 of equatiojis one less than were first given; and they will 
 also contain a numher of unknown quantities one less than 
 at first. 
 
 III. Operating with these new equations as was done with 
 the given equations, we can again reduce their number one; 
 and continuing this process we shall finally have but one 
 equation containing but one unknown quantity, which will 
 then become known. 
 
 EXAMPLES. 
 
 (7x + 5y + 2c= 79 (1)) 
 1 . Given I 8a; + 7y + 92 = 122 (2) \ , to find x, y, 
 t x+4y-^oz= 55 (3) !l and c. 
 
 By Rule under Art. 75, we find, by using (1), (2) and (3), 
 _19—5y — 2z 
 
 7 
 
 (4) 
 
 122^^.^ (5) 
 
 a; = 55 — 4y— 5-. (6) 
 
 Equating (4) and (6) ; and (5) and (6), we have 
 
 79 — 5y — 2z ^. , 
 
 Y =5o—4y—5z, (7) 
 
 122 — 7y — 9z „ . . ,_, 
 ^ =o5—4y — oz. (8) 
 
 When cleared of fractions, (7) and (8) become 
 
 79 _ 5y _ 2r = 385 — 28y — 35z, 
 122 — ly — 9z = 440 — 32y — 40r. 
 
 Transposing and uniting terms, we have 
 
 23y + 33r = 306, (9) 
 
 25y-j-31c=318. (10) 
 
86 SIMPLE EQUATIONS. 
 
 Equations (9) and (10) give 
 
 _306 — 33z 
 y~ 23 ' 
 
 (11) 
 
 318 — 31z 
 
 Equating (11) and (12), we have 
 
 306— 33z 318 — 31z 
 
 (13) 
 
 23 25 ' 
 
 which reduced gives 
 
 c = 3. 
 This value of z substituted in (11) gives 
 
 2/ = 9. 
 And these values of z and y, substituted in (6), give 
 
 ( ^^4-^2/+^- = 62^ 
 2. Given ? ^x + jy -f- ^2 =47 > j ^o fi"^ ^5 V) ^^^ 2; 
 l\x-^\y + lz=ZsS 
 
 These equations, when cleared of fractions, become 
 
 6.T+ 4y+ 2,z= 744, (1) 
 
 20x+15y+12z = 2820, (^) 
 
 15a: + 12y -j- 10- = 2280. (3) 
 
 Til 
 
 From (1), (2), and (3), we find 
 
 _744 — 6a: — 4i/ 
 
 " 3 
 2820 — 20a:— 15y 
 
 "12 ' 
 
 2280— 15a:— 12y 
 
 (4) 
 
 (5) 
 (6) 
 
 10 
 
 Equating (4) with (5), and (4), with (6), we have 
 
SIMPLE KQUATIONS. 
 
 87 
 
 744 — 6x — \y 2820 — 20x — 15y 
 3 12 ' 
 
 (') 
 
 744 — 6x — 4y 2280 — 15a; — \2y 
 3 10 
 
 (8) 
 
 Equations (7) and (8) when reduced become 
 
 
 4a; + y=156, 
 
 (9) 
 
 15a; + 4y = 600. 
 
 (10) 
 
 Equations (9) and (10) give 
 
 
 y=156— 4x, 
 
 (11) 
 
 600 — 15x 
 
 y- 4 • 
 
 (12) 
 
 Equating (11) and (12) , we have 
 
 
 ,_ , 600 — 15a; 
 
 lo6 — 4a; = . 
 
 4 
 
 (13) 
 
 This reduced, gives 
 
 
 a; = 24. 
 Having found z, we readily find y and z to be 
 y=60; ^ = 120. 
 
 ELIMINATION BY SUBSTITUTION. 
 
 (82.) There is still another method of elimination. 
 1. Suppose we have given the two equations 
 
 5x + 2y = 45, 
 
 (1) 
 
 4a; + 3/ = 33. 
 
 (2) 
 
 From the first we find 
 
 
 45— 5x 
 
 y= 2 • 
 
 (3) 
 
 Substituting this value of y in (2), we have 
 
 
 , 45 — 5a; „„ 
 4x -j = 33. 
 
 (4) 
 
88 SIMPLE EQUATIONS. 
 
 Equation (4), when cleared of iVaclions, becomes 
 
 8x-f-45 — 5a: = 66. (5) 
 
 This gives 
 
 a: = 7. 
 Substituting this value of a: in (3), we find 
 3/ = 5. 
 2. Again, suppose we have given, to find x, i/, and z, the 
 three equations 
 
 2x-{-4y — 3z = 22, (1) 
 
 4.x — 2y-\-5z = 18, (2) 
 
 6x + ly— z = 63. (3) 
 
 From equation (3) we obtain 
 
 c = 6a; + 7y — 63. (4) 
 
 Substituting this value of c, in (1) and (2), and they will 
 become 
 
 2x -|- 42/ — 3(6a: 4- 73/ — 63) = 22, (5) 
 
 4a: — 21/ -f- 5(6x4- 71/ — 63) =18. (6) 
 
 Equations (5) and (6) become, after expanding, transpo- 
 sing, and uniting terms, 
 
 l6a:-(-l''y=167, (7) 
 
 34a: -\- 33y = 333. (8) 
 
 Equation (7) gives 
 
 This value of a:, substituted in (8), gives 
 
 34067-17^)^^33^^333 ^^^^ 
 
 Equation (10), when solved as a simple equation of one un 
 known quantity, gives 
 
 Substituting this value of y in (9), w^e find 
 
 z = 3. 
 Using these values of x and y in (4), we obtain 
 
 1 = 4. 
 
SIMPLE EQUATIONS. 89 
 
 (83.) This method of climiniiting- may be comprehended 
 in the followintr 
 
 RULE. 
 
 Having found the value of one of the unknown quantities^ 
 from either of the given equations^ in terms of the other 
 unknown quantities, substitute it for that unknown quantity 
 in the remaining equations, and we shall thus obtaiji a new 
 system of equations one less in number than those given. 
 Operate with these new equations as with the first, and so 
 continue until we find one single equation with hut one un- 
 known quantity, ichich will then become known. 
 
 EXAMPLES. 
 
 / X — u'= 50 {l)\ 
 , r.- \'iy — x= 120 (2)( 
 1. Criven ^2- _ y ^ 120 ^3)?' , to findw, x, y, and z. 
 
 (3iy — - = 195 (4)) 
 
 From (1) we find 
 
 w = X — 50. (5) 
 
 This value of ic, substituted in (4), gives 
 
 3(j;_50)— - = 195, or 3x— r = 345. (6) 
 Equation (6) gives 
 
 z=3x — 345. (7) 
 
 This value of z, substituted in (3), gives 
 
 2(3x — 345)— 2/= 120, orGo:— y = 8l0 (8) 
 Equation (8) gives 
 
 y = 6x — SlO. (9) 
 
 This value of y, substituted in (2), gives 
 
 3(6x— 810) — 2: = 120, (10) 
 
 or 17a: = 2550. (11) 
 
 .•.a:=150. (12) 
 
 12 
 
90 SIMPLE EQUATIONS. 
 
 This value of x causes (9) to become 
 
 3/ = 90. 
 Using the value of x in (7), we find 
 
 z=105. 
 Finally, using the value of x in (5), we find 
 w = 100. 
 rx + ly = a, {\)\ 
 
 2. Given }y-{-\z = a^ (2) > to find a:, y, and z. 
 
 {z-\r\x = a, (3)) 
 
 Equation (3) gives 
 
 • ■ ■-=—■ w 
 
 This value of c, substituted in (2) , we have 
 
 I 4a — X ,_. 
 
 y+-^^=«- (5) 
 
 Clearing of fractions and uniting terms, (5) becomes 
 
 12y — x = 8a. (6) 
 
 From (6) we find 
 
 x = 12y — Sa. (7) 
 
 This value of a:, substituted in (1) , gives 
 
 122/ - 8a +1 = a. (§) 
 
 Equation (8) gives 
 
 25y=18a, (9) 
 
 Therefore, y = — r-- 
 
 ' ^ 25 
 
 This valup. of y, substituted in (7), gives 
 
 _ 16a 
 ^~ 25 * 
 Substituting for x, in (4), its value just found, we have 
 
 ^_21a 
 
 ''~25' 
 
SIMPLE EQUATIONS. 91 
 
 Hence, collecting values, we have 
 
 We may observe that if a is any multiple of 25, the above 
 values of x, y, and z will be integers. 
 
 (84.) All equations of the first degree, containing any 
 number of unknown quantities, can be solved by either ot 
 the Rules under Articles 79, 81, and 83, or by a combina- 
 tion of the same. 
 
 The student must exercise his own judgment, as to the 
 choice of the above Rules. In very many cases he will dis- 
 cover many short processes, which depend upon the parti- 
 cular equations given. 
 
 (85.) We will now solve a few equations, and shall en- 
 deavor to effect their solution in the simplest manner possi- 
 ble. 
 
 , ^. ^ 6x -I- 53/ = 128, L . 1,, 1 ^ , 
 
 1. Given \ „ ^^' > to find the values of a; and ■?/. 
 
 nx + 43/ == 88, ) ^ 
 
 Adding the two equations, and dividing the sum by 9, we 
 find 
 
 x + y = 2i. (1) 
 
 Multiplying (1) by 3, and subtracting the result from the 
 second of the given equations, we have 
 
 i/=16. (2) 
 
 Subtracting (2) from (1), we get 
 x=8. 
 
 2. Given ^ y + r = 6, (2) > to find x, y, and 2. 
 ■C: (3)) 
 
92 SIMPLE EQUATIONS. 
 
 Dividing the sum of these three equations by 2, we find 
 .+y + . = "-+A±i. (4) 
 
 From (4) subtracting, successively, (2), (3), and (1), we 
 find 
 
 b + c-h 
 
 2 ' 
 
 a — c + b 
 
 2—' r W 
 
 a-\-b-{-c 
 
 Equations (1), (2), and (3), of this last example, are 
 so related that if in (1) we change x to y, y to z, and a 
 to b, it will correspond with (2). Again, if in (2) we 
 change y to z, z to x and b to c, it will correspond with (3). 
 Also, if in (3) we change z to x, x to y, and c to a, it will 
 give (1), from which we first started. In each change we 
 have advanced the letters one place lower in the alphabeti- 
 cal scale, observing that when we wish to change the last 
 letters of the series, as z or c, we must change them respec- 
 tively to X and a, the first of the series. 
 
 Since the above changes can be made with the primitive 
 equations (1), (2), (3), without altering the conditions of 
 the question, it follows that the same changes can be made 
 in any of the equations derived from those. Thus, execu- 
 ting those changes in equations (A), we find that the first 
 is changed into the second, the second into the third, and 
 the third in turn is changed into the first. 
 
SlMn-E EQUATIONS. 93 
 
 Vx y 
 
 (1) 
 
 3. GivenJl + l_A 
 
 \y ~ 
 
 (2) 
 
 (>\-' 
 
 ^3) 
 
 If we take the sum of these three equations, we shall ob- 
 tain 
 
 9 
 
 - + ^ + ^---^' + ft + c. (4) 
 
 Now, subtracting twice (2) from (4), and we have 
 
 ^- = a-Y c — b. (5) 
 
 In a similar manner subtracting twice (3) and (1), suc- 
 cessively, from (4), and we find 
 
 ? = a-o+i; 
 
 
 (6) 
 
 2 
 
 - = — a + b + c. 
 
 
 i?) 
 
 Equations (5), (6), and (7), readily 
 
 give 
 
 
 2 ^ 
 
 
 
 2 V 
 
 
 (B) 
 
 2 \ 
 
 • 
 
 
 -a-^b + c J 
 
 
 The letters in this example will admit of the same changes 
 as those pointed out in the last example. Indeed, the only 
 difference between the two examples is, that the unknown 
 quantities in the one example are the reciprocals of those in 
 the other. Consequently the expressions for a*, y^ and c, 
 
94 SIMPLE EQUATIONS. 
 
 as given by equations (B), ought to be the reciprocals of 
 those given by equations (A), which we find to be really the 
 case. 
 
 4. Given ^ 3/+6(x+z)=7j, (2) > to find x, y, and z. 
 (z-\-c{x-^y)-'--p, (3)) 
 
 If we add and subtract ax from the left-hand member of (1), 
 and add and subtract by from the left-hand member of (2), 
 and add and subtract cz from the left-hand member of (3), 
 they will become 
 
 {l-a)xi-a{x + y-\-z)=m, (4) 
 
 {l-b)y-\-b{x-]-y-{-z) = n, (5) 
 
 {l-c)z-{-c{x + y-{.z)=p. (6) 
 
 If we divide (4) by 1 — a, and (5) by 1 — b, and (6) by 
 1 — c, they will become 
 
 x+--^(x + y + z)=-^; (7) 
 
 1 — a 1 — a 
 
 ' ^ .(a: + y4-z)=-^: (8) 
 
 Taking the sum of (7), (8), and (9), we have 
 ^ _i ^ I P 
 
 (10) 
 
 Therefore, x-j- 2/ -f- 2^ f . ^^^^ 
 
 1 + ,-^+ 
 
 I — a ' 1—6 
 
SIMPLE EQT^ATIONS. 96 
 
 This value of x-j-y + 2 substituted in (7), (8), and (9), 
 gives 
 
 ( _^ [_ ^ \ P 
 
 1 — a 1 — a).,a b c 
 
 '-, (12) 
 
 1— a ' 1—b ' 1-T 
 
 n 
 
 - . l—a~^ l—b~^ 1 — c 
 1 — a 1 — b 1 — c 
 
 ._ P 
 
 > 1— g. 1 — 6"^1 — c 
 
 (14) 
 
 1 — a ' ]—b 1 — c 
 
 This example affords a l:eautiful illustration of the law 
 of permutations which can be made with the letters which 
 enter into symmetrical equations. The primitive equations 
 (1), (2), (3); the three equations (4), (5), and (G) ; and 
 the three (7), (8), and (9) ; as well as the three (12), (13) and 
 (14), can be deduced in succession from each other, by 
 simply advancing the letters one place lower in the alpha- 
 betical scale. Equations (10) and (11), which contain all 
 the different letters, are of such a form as not to change by 
 this method of permuting. Consequently the expression 
 within the braces of (12), (13), (14), which is the right- 
 hand member of (11), must remain unchanged for the values 
 ofx, y, andc. By studying carefully the different laws 
 by which changes may be made, we have great control 
 over symmetrical algebraic expressions which we could not 
 otherwise obtain. It is not always necessary that the change 
 should be in alphabetical order, but may vary according to 
 
96 SIMPLE EQUATIONS. 
 
 any other law. The principle may be thus stated : what- 
 ever changes can be made among the letters entering into 
 the primitive equations, without altering the equations, the 
 same changes may be made on any of the derived equa- 
 tions. 
 
 This method of deducing one expression from ano- 
 ther of a similar nature, is of great* use, especially in the 
 higher parts of analysis. In order that the proper permu- 
 tations may be made with ease, and without danger of error, 
 we must adopt some simple and uniform notation for the 
 different values of the quantities which enter into our ex- 
 pressions. Indeed, by a well chosen method of notation, 
 we may frequently resolve, with ease, questions which 
 would otherwise be extremely difficult. 
 
 Perhaps we can not better impress upou the student, the 
 importance of a judicious notation, than by giving, at length, 
 the solution of the two following questions. 
 
 5. Find 7i numbers, such that the first increased by Ci 
 times the sum of all the others, shall equal 61 ; the second, 
 increased by og times the sum of the others, equals 62 ; the 
 third, increased by 03 times the sum of the rest, equals ^3 ; 
 and so on for the other numbers. 
 
 Let the n numbers sought be represented by 
 Xi, Xa, X3, ------ x„. 
 
 Then, if 
 
 S = xi+X2-|-X3 4- - - - - -\-x„, (a) 
 
 we shall have, by the conditions, the following system of 
 
 equations : 
 
SIMPLE EQUATIONS. 
 
 97 
 
 3^1 + fll(S — Xi)=6i, (1) 
 
 xo + a,{S-x.)=^h,, (2) 
 
 x; + as{S—x,)=b,, (3)( 
 
 (A) 
 
 Xn-{-an{S-x„)=b.. {n). 
 
 From (A) we readily find the following system of equa- 
 tions : 
 
 "■ xs- ^' 
 
 X3- 
 
 01— 1 
 
 (1-2 
 flo— 1 
 
 03 1 
 
 «, — r 
 
 X S— 
 X 5— 
 
 b. 
 
 Oo— i' 
 as— 1' 
 
 (1': 
 
 (2') 
 
 (3')> (A') 
 
 :r„=-i^x5- ^" 
 
 On— I 
 
 ««— 1 
 
 (^') 
 
 Taking the sum of the n equations (A') , we find 
 
 S=B' X S—B". (B) 
 
 Where, for the sake of brevity, we have put 
 
 «i — l^Go— 1^03— 1^ ^a,— !'"^^ 
 
 D"=-^+ & 
 
 ai—1 ' ao—l ' 03—1"^ ^a„— 1' ^° ^ 
 
 Returning to equation (B) , we find 
 
 D' — 1 
 This value of 5 written in the n equations (A') gives 
 
 (C) 
 
 13 
 
98 SIMPLE EQUATIONS. 
 
 
 X2 
 
 D'— 1 ^ «,— 1 
 
 0. — 1 
 
 (D) 
 
 D" 
 
 If?/ =10; and bi=bo = l.3=bi = b5=b^ = bj==h=h 
 = 6io=845693; and Oi=i, «2 = i-, 03=!) 04=ij 
 
 will the above question agree with one in the Higher Arith- 
 metic, which question is there required to be solved by rules 
 purely arithmetical. The preceding question, is also a 
 particular case of the above question. 
 
 G. Suppose n individuals, Ai, Aa, A3, . - - - A„, 
 play together on this condition, that the one who loses shall 
 give to each of the others as much as they then have. 
 First Ai loses, then As, then A3, then A4, and so on, until, 
 in turn, they have all lost ; and at the end of the^ith game 
 
 their respective shares are Oj, oo, 035 "n- How 
 
 much had each before playing ? 
 
 SOLUTION. 
 
 Let their respective shares before playing be represented by 
 
 X], X2.X3J - - - - x„. 
 
 Also, put 
 
 a:, +X0 + X3+ ----- +Xn = S. (1) 
 Since Ai loses on the first game, he must, by the question, 
 give to Aa, A3, A4, &c., as much as they now have. Hence 
 
SIMPLE EQUATIONS. 99 
 
 Ai's money will be diminished by Xo -|- X3 -|- x.| H \- x„, 
 
 which, by (1), equals 5 — Xi, so that Ai's money will be 
 
 Xi — {S — Xi)=2xi—S. 
 Therefore, at the end of the first game, they will have 
 
 Ai, A.J, A3, A„, 
 
 2 Ji — S ; 2xo ; 2x3 ; 2x„ . 
 
 Now, since A2 loses on the second game, he must give to 
 Ai, A3, A4, &c., as much as they now have. Hence Aa's 
 money will be diminished by 
 
 2x1 — 5' 4- 2x3 +2x4 + 2x„. (2) 
 
 Since they, all together, always have the same amount as 
 at first, we have 
 
 2x1 — 5 4- 2x, + 2x3 H 2xn=S; (3) 
 
 .-. 2xi — iS + 2x3 + 2x4 H 2x„ = S—2X'2. 
 
 Hence, Ao, after the second game, will have 
 
 2xo —{S — 2x,) = 4x0 — S. 
 Therefore, at the end of the second game, they will have 
 Ai, Ae, A3, A4, A„, 
 
 4xi — 2 .S; 4x2 — S; 4x3; 4x4; 4x„. 
 
 Proceeding in this way, we find that after the third game, 
 they will have 
 
 Ai, Ao, A3, A4 As, A„, 
 
 Sxi— 4 S; 8x2 — 2 S; 8x3 — S; 8x4 ; 8x5; - - - 8x„. 
 
 And in general, after the 7ith game, they will have 
 
 Ai, A2, A3, A„, 
 
 2"^^ — 2"-'S; 2"x.2 — 2"-'S; 2"Xi — 2"-^ 5; - - - 2''x„ — S. 
 
 Equating these results with the values, 
 
 fil, 02, 03} 04} ^"1 
 
100 
 
 SIMPLK EQUATIONS. 
 
 we have the n followin;^ equations 
 2"Xi — 2"-'S=ai^ 
 
 2"a:3— 2"-^S=fl3, 
 
 (1') 
 (2') 
 
 (4': 
 
 (A) 
 
 2"x„_i — 2 6'=a„_i, (("-!)') 
 From tl)c above system of equations (A), we readily find 
 
 
 
 (1") 
 
 a,.S 
 -^=2^. + 2.. 
 
 
 (2") 
 
 ^^-2« + 23' 
 
 
 (3") 
 
 
 s 
 
 2"-' 
 
 , ([^-1]") 
 
 
 
 (^") 
 
 B) 
 
 Now, since they all together had as much money when 
 they left off playing, as they had before playing, it follows, 
 that 
 
 5=ai + a2 + «3+ 04 + - - - - fln- (C) 
 
 If this value of S be substituted in the system of equations 
 (B), we shall then have the values of 
 
 Ii, To? 3-3) - - - - - Xnj 
 
 in terms of known quantities. 
 If we have the relation 
 
 fli = 02 = 03 = oi = - - - - = flu, 
 
jr. t>^-*Y\ 
 
 ^ 
 
 SIMPLE EQUATIONS 101 
 
 then (C) will give 
 
 ■ind the system of equations (B) will then become 
 
 \2 ' 2"/ 1 
 
 If, in (D), we suppose 7i = 5 and ai = 32, we shall have 
 Xi = 8l; a:2=41; X3=:21; X4=ll; Z5=6. 
 
 The above supposition causes our question to agree with 
 Ques. 13, Chap. XII, Higher Arithmetic. 
 
 /'w+x + 3/ = 13, (1)) 
 \xi + x-^z=\l, (2)( 
 7. Given ^^_|_^_^^^jg^ ^4^ ? to find w, x, y, and z, 
 
 Dividing the sum of these four equations by 3, we obtain 
 u-{-x-\-y-\-z = 22,. (5) 
 
 From (5), subtracting successively, (4), (3), (2), and (1), 
 and we find 
 
 S. Given < ^ ^ ' Mo find x and y. 
 
 I x + 5y=191,i ^ 
 
 Ans. 
 
 x= 16. 
 y=36. 
 

 102 SIMPLE EQUATIONS. 
 
 9. Given < - '^ ^ ''>tofindxand 
 
 ^ 9y— 347= 5a:-420, i 
 
 )3^23 
 
 , ,a=56. 
 
 Ans. 
 
 10. Given < ^ + 2/ 3a + x I, ^^ fl^d a: and y. 
 
 \ ax -\- 2by =dy ) 
 
 \ ^ = 5 ^• 
 
 ' ^~ 36 
 
 11. A and B possess together a fortune of $570. If A's 
 fortune were 3 times, and B's5 times as great as each really 
 is, then they would have together $2350. How much had 
 each ? 
 
 Ans. A $250, B $320. 
 
 12. Find two numbers ot the following properties : When 
 the one is multiplied by 2, the other by 5, and both products 
 added together, the sum is =31 ; on the other hand, if the 
 first be multiplied by 7, and the second by 4, and both pro- 
 ducts added together, we shall obtain 68. 
 
 Ans. The first is 8, and the second is 3. 
 
 13. A owes $1200, B $2550 ; but neither has enough to 
 pay his debts. Lend me, said A to B, j of your fortune, 
 and I shall be enabled to pay my debts. B answered, I can 
 discharge my debts, if you will lend me ^ of yours. What 
 was the fortune of each ? 
 
 Ans. A's fortune is $900, and that of B $2400. 
 
 14. There is a fraction, such, that if 1 be added to the 
 numerator, its value =J, and if 1 be added to the denomi- 
 nator, its value = i. What fraction is it ? 
 
 Ans. j^. 
 
SIMPLE EQUATIONS. 103 
 
 15. The sum of two numbers is = «, the quotient r.rishiL; 
 from the division of the second, by the first is = /;. Find 
 these numbers ? 
 
 Ans.^-p-j,an,lj-^_^-j. 
 
 16. A, Bj C, owe together $2190, and none of them can 
 alone pay this sum ; but when they unite, it can be done in 
 the following ways : first, by B's putting ?■ of his property 
 to all of A's ; secondly, by C's putting t} of his property to 
 all B's; or, by A's adding § of liis property to that of C. 
 How much did each possess ? 
 
 Ans. A $1530 ; B $1540 ; and C $1170. 
 
 17. A and B possess, together, only | of the property of 
 C ; B and C have, together, 6 times as much as A ; were 
 B $680 richer than he actually is, then he would have as 
 much as A and C together. How much has each? 
 
 Ans. A, has $200 ; B, $360 ; and C, $840. 
 
 18. Three masons. A, B, C, are to build a wall. A and 
 B, jointly, could build this wall in 12 days; B and C could 
 accomplish it in 20 days ; but C and A would do it in 15 
 days. What time would each take to do it alone in ? And 
 in what time will they finish it, if all three work together? 
 
 ^^g \ A requires 20 days, B 30, and C 60 ; 
 ^^' I all three together require 10 days. 
 
 19. Three laborers are employed in a certain work. A and 
 B would, together, complete this work in a days ; B and C re- 
 quire b days ; but C and A, only c days. What time would 
 each require, singly, to accomplish it in 1 And in what time 
 would they finish it, if they all three worked together ? 
 
 Answer, 
 
 . . 2abc , T. 2abc , 
 
 A requires -r—- davs, B, • . days, 
 
 ab -\- be — ca ' be + ca — cb 
 
 ^ 2abc , T • , 2fl6c , 
 
 days,: Jointly, ^-—^-p— days. 
 
 ca -^ ab — be '' ' ^ ab -\-bc +ca 
 
104 SIMPLE EQUATIONS. 
 
 20. A certain number consists of three digits, which are 
 in an arithmetical progression. If this number be divided 
 by the sum of its digits, (that is, without considering the value 
 they have as tens and hundreds,) the quotient is 48 ; but 
 if 198 be subtracted from it, then we obtain for the remain- 
 der a number consisting of the same digits as the one sought, 
 but in an inverted order. What number is this 1 
 
 Ans. 432. 
 
 21. A cistern containing 210 buckets, may be filled by 2 
 pipes. By an experiment, in which the first was open 4, 
 and the second 5 hqurs, 90 buckets of water were obtained. 
 By another experiment, when the first w-as open 7, and the 
 other 3^ hours, 126 buckets were obtained. How many 
 buckets does each pipe discharge in an hour. And in what 
 time will the cistern be filled, when the water flows from both 
 pipes at once 1 
 
 C The first pipe discharges 15, and the 
 Ans. < second, G buckets ; it will require 10 
 ( hours for them to fill the cistern. 
 
 22. According to Vitruvius, Hiero's crown weighed 20 
 lbs., and lost 1^ lbs., nearly, in water. Let it be assu- 
 med that it consisted of gold and silver only, and that 
 20 lbs. of gold lose 1 lb. in water, and 10 lbs. of silver, 
 in like manner, lose 1 lb. How much gold, and how 
 much silver did this crown contain. 
 
 Ans. 15 lbs. of gold, and 5 pounds of silver. 
 
 23. A person has two large pieces of iron whose weight 
 is required. It is known that | of the first piece weighs 96 
 lbs. less than ^ of the other piece ; and that |- of the other 
 piece weighs exactly as much as ^ of the first. How much 
 did each of these pieces weigh ? 
 
 Ans. The first weighs 720 lbs., the second 512 lbs. 
 
 24. Two persons, A and B, can together perform a piece 
 
SIMPLE EQUATIONS. 105 
 
 of work in 16 days. After having laboured jointly 4 days, 
 A leaves, and B by laboring 36 days more, completes it. 
 How many days Avould each separately require 1 
 
 . ^ A requires 24 days, 
 I B requires 48 days. 
 
 25 . A merchant has two kinds of wine ; if he mix a gal- 
 lons of the worst wine with b of the best, the mixture is 
 worth c dollars per gallon ; but if he mix f gallons of the 
 worst with g gallons of the best, then the mixture is worth 
 h dollars per gallon. What is the price of each kind of 
 wine per gallon '? 
 
 , Price of the worst, ^ ' ^ " ; \~ ^ ' , 
 
 ^ J ag — kf 
 
 "'■ 'Price of the best, ^^ + ^) 'f-^f^ ^) ^\ 
 
 26. Seveial. detachments of artillery divided a certain 
 number of cannon balls. The first took 72 and ^ of the 
 remainder ; the next 144 and i of the remainder ; the 
 third 216 and ^ of the remainder ; and the fourth 288 and 
 I of the remainder, and so on j when it was found that the 
 balls had been equally divided. What was the number of 
 detachments and the number of balls 1 
 
 Ans. 8 detachments, and 4608 balls 
 
 27. A person has three horses and a saddle, which of 
 itself is worth 220 dollars. If he put the saddle on the 
 back of the first horse, it will make his value equal to that 
 of the second and third ; but if he put it on the back of 
 the second horse, it will make his value double that of the 
 first and third ; and if he put it on the back of the third 
 horse, it will make his value triple that of the first and 
 second. What is the value of each horse ? 
 
 Ans. 20, 100, and 140 dollars 
 13 
 
106 SIMPLE EQUATIONS. 
 
 ELIMINATION BY INDETERMINATE MULTIPLIERS. 
 
 (86.) Suppose we wish to find x arid y from the equa- 
 tions 
 
 2a: + 33/ =13, (1) 
 
 5x + 43/ = 22. (2) 
 
 Multiplying (1) by ??i, we find 
 
 2mx + Zmy = 13m. (3) 
 
 Adding (2) and (3)^ we have 
 
 (2wi+5)xH-(37?i+4)y = 13/n + 22. (4) 
 
 Assume 3m -[-4^0 which gives 
 
 ni = -h (5) 
 
 This value of m causes equation (4) to become 
 
 13m -f 22 - ,„. 
 
 ^=-^ r-^=2. (6) 
 
 2m + 5 
 
 Again, if we had assumed 2m + 5 = 0, which would have 
 
 given 
 
 m = -f, (7) 
 
 then equation (4) would have become 
 
 , = 13m±12_3. (8) 
 
 ^ 3m + 4 ^ ^ 
 
 Now, returning to our former equations, we will subtract 
 (2) from (3) ; we thus obtain 
 
 (2m — 5) :r + (3/n — 4) y = 13m — 22. (9) 
 Assume 3m — 4=0, which gives 
 
 m = f. 
 This value of m causes (9) to become 
 13m — 22 ^ 
 
 (10) 
 (11) 
 
 2m — 5 
 
 Again, assume 2m — 5 = 0, which gives 
 
 m=^ (13) 
 
 This causes (9) to become 
 
SIMPLE EQUATIONS. 107 
 
 13m + 22^ (13) 
 
 These values of a: and y are the same as just found. 
 
 It is evident that had we multiplied (2) by m, and then 
 added, or subtracted the result from (1), we should then 
 have found, in a similar manner, the same values for x 
 and y. 
 
 (87.) We will now apply this method to the two literal 
 equations, 
 
 Z:x+Fi2/ = A, " (1) 
 
 Zoa; + Y.y = M^. (2) 
 
 In these equations the capital letters are supposed to be 
 known, and their subscript numerals indicate the equation 
 to which they belong. Thus, 
 
 Xo is the coefficient of a: in the second equation. 
 
 Yi is the coefficient of y in the first equation. 
 
 ^2 is the absolute term, or the term independent of x and 
 y in the second equation. 
 
 Returning to our equations, we will multiply (1) by m 
 and add the result to (2) ; we thus obtain 
 
 (Zim + Zo) X + ( Yijn + F,) y = .^^m + ^s- (3 ) 
 Assume Yim + F) = 0, which gives 
 
 m = — -. (4) 
 
 This causes (3) to become 
 
 ( Xim -\-Xo)x = ^im-{- ^2, (5 ) 
 
 which gives immediately 
 
 ^ ~ Xim + X2~'XjY2 — A'o Yi ^ ^ 
 
 Assume Aim -f- Xo =: 0, which gives 
 
108 SIMPLE EQUATIONS. 
 
 » = -| (7) 
 
 This value of m causes (3) to become 
 
 _ Aim + >^2 _ Ji-iXj — AyX-2 
 
 Hence, the values of x tind y are 
 
 (S) 
 
 A2X1 — AiX>2 
 
 (9) 
 
 These values of x and y may be considered as comprising 
 the solution of all simple equations combining only two un- 
 known quantities. If we wish to adapt this general solution 
 to the equations 
 
 2x + 3y = 13, 
 5x + 4y = 22 ; 
 
 we must call 
 
 ^1=13; ^0 = 22. 
 
 Fi=:3; 72 = 4. 
 
 These values substituted in (9), give 
 x = 2', y = 3. 
 
 (88.) As a still farther illustration of the method of elimi- 
 nation by means of indeterminate multipliers, we will pro- 
 ceed to the solution of three simultaneous simple equations, 
 involving three unknown quantities x, y and z; and we will 
 continue to make use of the notation by the assistance of 
 subscript numbers. 
 
 Let the equations be as follows : 
 
SIMPLE EQUATIONS, 109 
 
 A>-}- \\y^Z^z = A^, (1) 
 
 Xcx + Y.,y + Zo2 = ^o, (2) 
 
 ^3x4- F3y + Z,z = ^3. (3) 
 
 In these equations, as in those of the last example, the 
 capital letters X, F, Z, are the coefficients of their corres- 
 ponding small letters. The small numerals placed at the 
 base of these coefficients correspond to the particular equa- 
 tion to which they belong. Thus X-i is the coefficient of x 
 in the second equation ; F3 is the coefficient of y in the 
 third equation ; Zi is the coefficient of z in in the first equa- 
 tion, and so for the other coefficients. The letter A is used 
 to denote the right-hand members of the equations, or the 
 absolute terms ; the subscript numbers in this case also de 
 note the equation to which they belong. 
 
 This kind of notation, by use of subscript numbers, is very 
 natural and simple, and combines many advantages over the 
 ordinary methods. 
 
 Having explained this method of notation, we will now 
 proceed to the solution of our equations. 
 
 If we multiply (1) by ?n, and (2) by 7j, and then add the 
 results, we shall obtain 
 
 (Xiwi -f X-ra) X -f ( Ym + Fon) y 
 -j- {Ziin -\- Z'in)z = Jlym -f- A^n. 
 
 (4) 
 
 From (4) subtracting (3), we find 
 
 ( Xim -\- X<in — Xz)x+{ Yitn -\- Y^n — F3) y 
 -\-[Zim-\-Zon — Z3)s = A\m-{-Aon — A^. 
 
 In order to cause y and z to vanish from this equation, we 
 will assume 
 
 (5) 
 
 Y,m + Ym=Y,, (6) 
 
 Z^w -i-Z<<n — Zz. (7) 
 
110 SIMPLE EQUATIONS. 
 
 This assumption causes (5) to become 
 
 {Xym + X^:>a — X2)x = Aini + Ji2n — .U. (8) 
 
 Therefore, 
 
 . Ai'in -f- A-iU — Ai / Qv 
 
 Xxm -j- X<in — X^ 
 
 We must now find the values of m and w, by aid of con- 
 ditions (6) and (7) ; for this purpose we will compare them 
 with (1) and (2), (Art. 87). Now, !n order to make (6) 
 and (7) agree with (1) and (2) respectively, we must change 
 X to m, y to n; X, to F, , Xg to Z,, F, to F,, Y^ to Zj, 
 A , to F3, A^ to Z3. Making these same changes in equa- 
 tions (9) of Art. 87, we obtain 
 
 F3Z2 — Z3 Y-2 
 
 (10) 
 (11) 
 
 (12) 
 
 a3) 
 
 Substituting these values of m and 71, in (9), we readily 
 find 
 
 F1Z2 — Z1F2 
 
 ^Z aFi— F3Z1 
 "" FiZo— Z1F2' 
 
 Arranging the terms alphabetically, we have 
 F3Z2— F2Z3 
 
 F1Z2- 
 
 -F2Z1 
 
 F1Z3- 
 f;z2-^ 
 
 -F3Z1 
 
 -Y,Z, 
 
SIMPLE EQUATIONS. 
 
 Ill 
 
 :i;i^ 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ^ 
 
 tn 
 
 ^ 
 
 1 
 
 g 
 
 >^" i 
 
 I I 
 
 N o 
 
 ^" I 
 
 + : 
 
 « o 
 
 >^' I 
 
 ^ i 
 
 4- ^ 
 
 CO "^ 
 
 tS3 - 
 
 
 
 + + 
 
 
 N 
 
 I I 
 
 I 
 
 I I 
 
 + + + + 
 
 + + 
 
 
119 SIMPLE EQUATIONS. 
 
 (89). We will now proceed to point out some remark- 
 able relations in the combinations of the letters marked 
 with subscript numbers, as given by equations (15), (16), 
 and (17). 
 
 I. The denominator, which is common to the three 
 expressions, is composed of six distinct products, each 
 consisting of three independent factors. Three of these 
 products are positive, and three are npgative. 
 
 II. The letters forming the different products of this com- 
 mon denominator being always arranged in alphabetical 
 order, X, F, Z, we remark that the subscript numbers of 
 the first product are 1, 2, 3. Now, if we add a unit to 
 each of these numbers, observing that when the sum 
 becomes 4 to substitute 1, we shall obtain 2, 3, 1, which 
 are the subscript numbers of the second product. Again, 
 increasing each of these by 1, observing as before, to write 
 1 when the sum becomes 4, we find 3, 1,2, which are the 
 subscript numbers of the third product. If we increase 
 each of these last numbers by 1, observing the same law, 
 we shall obtain 1, 2, 3, which are the subscript numbers 
 belonging to the first product. A similar method of chang- 
 ing has already been noticed under Art. 85. 
 
 What we have said in regard to the subscript numbers of 
 the positive products, applies equally well in respect to the 
 negative products. 
 
 III. The numerator of the expression for x, may be 
 derived from the common denominator by simply substi- 
 tuting J? for Xj observing to retain the same subscript 
 numbers. 
 
 The numerator of the expression for y may be derived 
 from the common denominator by substituting A for F, 
 observing to retain the same subscript numbers. 
 14 
 
SIMPLE EQUATIONS. 113 
 
 In the same way may the numerator of the expression 
 for z be found by changing Z of the denominator into A, 
 retaining the same subscript numbers. 
 
 (90.) We will now proceed to show how these expres- 
 sions, for X, y, and c, can be obtained by a very simple and 
 novel process, which is easily retained in the memory, and 
 which is applicable to all simple equations involving only 
 three unknown quantities. 
 
 Writing the coefficients and the absolute terms in the 
 same order as they are now placed in equations (1), (2), 
 (3), we have 
 
 ^l 
 
 Fi 
 
 Zi =: A, 
 
 x> 
 
 Fa 
 
 Zo = A. 
 
 X; 
 
 Fa 
 
 Za = A; 
 
 i 
 
 w 
 
 Now, all the products of the common denominator can 
 be found by multiplying together by threes, the coefficients 
 which are found by passing obliquely from the left to the 
 right, observing that if the products obtained by passing 
 obliquely downwards, are taken positively, then those form- 
 ed by passing obliquely upwards must be taken negatively, 
 and conversely. This is in accordance with the property 
 of the negative sign. In the present case the products 
 formed by passing obliquely downwards, are taken posi- 
 tive. 
 
 In this sort of checker-board movement, we must ob- 
 serve that when \ve run out at the bottom of any column, 
 we must pass to the top of the same column ; and when 
 we run out at the top, we must pass to the bottom of the 
 same column. 
 
 This method is most readily performed upon the black- 
 board, by drawing oblique lines connecting the successive 
 factors of the different products. 
 
114 SIMPLE EQUATIONS. 
 
 We will trace out this sort of oblique movement. 
 
 Commencing with .Xi, we pass obliquely downwards to 
 Foj and thence to Z3, and thus obtain the positive product 
 of Xr Yi Z3. 
 
 Commencing with X^, we pass obliquely downwards to 
 F3, and since w'e have now run out with the column of Z's 
 at the bottom, w^e pass to Z], at the top of the column, and 
 thus obtain the positive product X2F3Z1. 
 
 Again, commencing with X3, we pass to Fi, and thence 
 obliquely downwards to Zo, and find the positive product 
 XzYiZ'Z. 
 
 Now, for the negative products we make similar move- 
 ments obliquely upwards. 
 
 Thus, commencing with Xi, we pass to Y3, and thence 
 obliquely upwards to Zo, and find the negative product 
 X1F3Z,. 
 
 Commencing with A'o, we pass obliquely upwards to Fi, 
 and thence to Z3, and find the negative product X^ Fi Z3. 
 
 Again, commencing with A3, we pass obliquely upwards 
 to Fi, and thence to Zi, and thus obtain the negative pro- 
 duct A3F2Z1. 
 
 Having thus obtained the denominator which is common 
 to the values of x, y, z; we may find the numerator of 
 the value of a-, by supposing the c/Ts to take the place of 
 the A's, and then to repeat our checker-board movement. 
 By changing the F's into the ^^'s, we shall find the numera- 
 tor of the value of y ; and by changing the Z's into jJ's 
 we shall find the numerator of the value of z. 
 
 (91.) We will now illustrate this method of solving 
 simple equations containing only three unknown quantities, 
 by a few examples. 
 
SIMPLE EQUATIONS. \]i) 
 
 2x -j- 3y -{. 4z = 16, -\ 
 Given ^3x -{- 5y -\~ Iz = 26, > to find x, y, and z. 
 4x -f 23/ + 3z = 19, ) 
 
 We will first find the common denominator. 
 
 Positive Products. Negative Products. 
 
 2X5X3=30 2X2X7 = — 28 
 
 3x2x4 = 24 3x3x3 = — 27 
 
 4X3X7 = 84 4X5X4= — 80 
 
 138 — 135 
 
 — 135 
 
 3 = common denominator. 
 
 We have for the numerator of x the following operation 
 
 Positive Products, Negative Products. 
 
 16X5X3 = 240 16X2X7 = — 224 
 
 26X2X4 = 208 26x3x3 = — 234 
 
 19x3x7 = 399 19X5X4=— 380 
 
 847 — 838 
 
 838 
 
 9 = numerator, for x. 
 
 To find the numerator for y, we have 
 
 Positive Products. Negative Products. 
 
 2X26X3 = 156 2x19x7 = — 266 
 
 3X19X4=228 3x16x3 = — 144 
 
 4X16x7=448 4x26X4= — 416 
 
 832 — 826 
 
 — 826 
 
 6 = numerator, for y. 
 
116 SIMPLE EQUATIONS. 
 
 To find the numerator for c, we have 
 
 Positive Products. Negative Products. 
 
 2x5x19= 190 
 
 
 2X2X26=— 104 
 
 3X2X16=: 96 
 
 
 3X3X19 = — 171 
 
 4X3X26 = 312 
 
 
 4X5X16 = — 320 
 
 598 
 
 
 — 595 
 
 — 595 
 
 
 
 3 = 
 
 = numerator, for z. 
 
 Hence, x 
 
 — ti 
 
 3 
 
 = 3. 
 
 y 
 
 = 1 
 
 = 2. 
 
 z 
 
 3 
 
 = 1. 
 
 When some of the coefficients are negative, we must ob- 
 serve the rule for the multiplication of signs. 
 
 r 2a; + 42/ — 3r = 22, ^ 
 2. Given ^ 4x — 2y + 5r= IS, > to find x, y, and z, 
 ( 6x-{-ly— z = 62, ) 
 
 To find the common denominator, we have 
 
 Positive Products. Negative Products. 
 
 2X — 2X — 1= 4 2X 7x 5= — 70 
 4X 7X — 3=-84 4X 4X— 1= 16 
 6X 4X 5= 120 6X — 2X — 3=— 36 
 
 40 -90 
 
 -90 
 
 — 50 = common denominator. 
 
SIMPLE EQUATIONS. 117 
 
 Positive Products. Negative Products. 
 
 22X- 
 
 -2X — 1= 44 
 
 22 X 
 
 7X 5= — 770 
 
 18 X 
 
 7X — 3 = — 378 
 
 18 X 
 
 4X — 1== 72 
 
 63 X 
 
 4X 5= 1260 
 
 63X- 
 
 -2X — 3^— 378 
 
 
 926 
 
 
 -1076 
 
 
 — 1076 
 
 
 
 — 150 = numerator, for x. 
 
 — 50 
 
 Proceeding in a similar way, we find the values of y and z. 
 
 to find X, y, and z. 
 'J ) 
 
 We will arrange the coefficients, omitting the unknown 
 quantities, observing also to write for such terms as are 
 wanting. 
 
 This arrangement being made, we have 
 1 ^ = a 
 1 1 = a 
 i 1 = a 
 
 Positive Products. Negative Products. 
 
 1X1X1= 1 lXOXi = 
 
 0X0X0=0 OxiXl = 
 
 iXiXi = ^V 1X1X0 = 
 
 2 A = common denominator. 
 
 For the numerator of x, we have 
 
118 simple equations. 
 
 Positive Products. Negative Products. 
 
 0X1X1= a axOXi= 
 
 aXOxO= aX\Xl = — \a 
 
 aX}sX\ = la aXlXO= 
 
 jfl = numerator, for x. 
 
 Hence, x=%a-^%\=^ \la. 
 
 By a similar process is the value of y and z fouml. 
 
 C x-\-a{y-\-z)=m^ ^ 
 4. Given }^ y -\-b{x -\- z)=n, > to find rr, y, and 2. 
 
 These coefficients, being properly arranged give. 
 
 1 a 
 
 a 
 
 1= m 
 
 b 1 
 
 b 
 
 = n 
 
 c c 
 
 1 
 
 = P 
 
 siTivE Products. 
 
 
 Negative Products 
 
 1X1X1= 1 
 
 
 \XcXb=—hc 
 
 b XcXa=abc 
 
 
 ftXaXl=— aft 
 
 c XaXb=abc 
 
 
 cXlXa=— ac 
 
 \-\-2abc — ab — ac — be 
 
 ■ab — ac — be 
 
 1 -\-2abc — ab — ac — bc = common denominator. 
 For the numerator of .t, we have 
 
simple equations. 119 
 
 Positive Products. Negative Products. 
 
 mxlxl= m mxcxh:= — bcia 
 
 n xc xa = (ten n xax\=- — an 
 
 p xaxb =: ahp ^ x 1 X o =: — nj; 
 
 m -j- acn -\- ahp — hem — an — ap 
 
 hem — an — ap 
 
 m-\-acn-{-ahp — hem — an — ap = numeralor of x. 
 
 Tj m-\-acn-\-abp — hem — an — ap 
 
 1 -|- 2ahc — ah — ae — he 
 
 If to this expression for x we apply the principle of per- 
 mutation, as already explained, by advancing the letters one 
 place lower in the alphabetical scale, we shall find 
 
 n -f- hap -j- hem — can — hp — hm 
 
 l'-\-2hca — he — 6c — ea 
 Again, permuting this expression, we have 
 
 p-\- ehm -j- ean — ahp — cm — C7i 
 
 l-\-2eab — ca — ch — ah 
 
 This solution is far shorter than the one given on page 
 94, and the expressions for x, y, and z, are far more sim- 
 ple. 
 
 We may remark, that the denominators of the above ex- 
 pressions are common, as they must of necessity be, in vir- 
 tue of the general results given by Equations (15), (16), (17), 
 on page 111. 
 
 5. A, B, and C, owe together (a) $2190, and none of 
 them can alone pay this sum ; but when they unite, it can 
 be done in the following ways : first, by ]3's putting 5 of 
 his property to all of A's ; secondly, by Cs putting j of 
 his property to all of B's ; or by A's putting I of his 
 property to all of Cs. How much was each worth 1 
 
120 SIMPLE EQUATIONS. 
 
 Let X, y, and c, represent -what A, B, and C, were re- 
 spectively worth. 
 
 Then we shall have these conditions, 
 X + ^y = a, 
 y + f c = a, 
 z + |x = a. 
 
 Clearing these of fractions, and arranging the coefficients, 
 we have 
 
 7 3 = 7a 
 
 9 5 = 9a 
 
 2 3 = 3a 
 
 Positive Products. Negative Products. 
 
 7 X 9X 3=1=189 7 X X 5=0 
 
 X X 0= X 3 X 3=0 
 
 2x3x5= 30 2X9X0 = 
 
 219 = common denominator. 
 
 Positive Products. Negative Products. 
 
 7a X 9 X 3 = 189a 7a X X 5 = 
 
 9aX0x0= 9ax3x3 =— 8la 
 
 3a X 3 X 5 = 45a 3a X 9 X 9= 
 
 234a —81a 
 
 —81a 
 
 153a ::= numerator of x- 
 
 153a _ 153X2190 
 "219 ~ 219 
 
 For the numerator of y, we find 
 
 Hence, x = -^ = ^^^ = 1530. 
 
simple equations. 121 
 
 Positive Products. Negative Products 
 
 7 X 9a X 3 = 189a ' 7 X 3a X 5 = — 105a 
 0x3aX0== 0x7ax3= 
 
 2 X 7a X 5 = 70a 2 X 9a X = 
 
 259a — 105a 
 
 105a 
 
 154a = numerator of y. 
 
 154a 
 Hence, y = ^^ = 1540. 
 
 For the numerator of z, we have 
 
 Positive Products. Negative Products. 
 
 7x9x3ai=189a 7x0x9a = 
 
 
 
 0x0x7a= 0x3x3a = 
 
 
 
 2x3X9a= 54a 2x9x7a = 
 
 — 126a 
 
 243a 
 
 — 126a 
 
 — 126a 
 
 
 117a = numerator of z. 
 
 
 Hence, z = ^19- = 11^0- 
 
 
 Collecting the results, we find that 
 
 
 A was worth $1530, 
 
 
 B " " $1540, 
 
 
 C " " $1170. . 
 
 
 The student will find, after a little practice in this method 
 that it is much more simple than would at first sight seem 
 
 Whenever some of the coefficients are zeros, as in th« 
 3d and 5th examples, the work is much abridged, as in this 
 caee some of the products must become zero 
 15 
 
INVOLUTION. 
 
 CHAPTER IV. 
 
 INVOLUTION, EVOLUTION, IRRATIONAL AND 
 IMAGINARY QUANTITIES. 
 
 INVOLUTION. 
 
 (92.) The process of raising a quantity to any proposed 
 power is called Involution. 
 
 When the quantity to be involved is a single letter, it is 
 involved by placing the number denoting the power above 
 it a little to the right. (Art. 11.) 
 
 After the same manner we may represent the power of 
 any quantity, by enclosing it within a parenthesis, and then 
 treating it as a single letter. 
 Thus, 
 
 the second power of mx =^ {mxy, 
 the third power of a + 6 = (a -f- by, 
 the fourth power of 3m -{- y= (3m + y)*, 
 &c., &c. 
 
 CASE I. 
 
 (93.) To involve a monomial, we obviously have this 
 
INVOLUTION. 123 
 
 RULE. 
 
 I. Raise the coefficient to the required power^ by actual 
 multiplication. 
 
 II. Raise the different letters to the required power by 
 multiplying the exponents^ which they already have, by the 
 number denoting the power, observing that if no exponent 
 is written , then one is always understood. To this power 
 prefix the power of the coefficient. 
 
 Note. — If the quantity to be involved is negative, the 
 signs of the even powers must be positive, and the signs of 
 the odd powers negative. (Art. 29.) 
 
 1. What is the square of Zax^l 
 Here the square of 3 equals 
 
 3== 3x3 = 9. 
 Considering the exponent of a, in the expression ax^, i 
 one, ve find a'x^ for the square of ax^. 
 Therefore we have 
 
 {'iax^Y=2aHK 
 
 2. What is the fifth power of — 2ab^ ? 
 
 Ans. {—2ab^)^= — ?,2a^b'' 
 
 3. What is the fourth power of — - xy~^? 
 
 o 
 
 Ans. (_lx2/-')^==^^:r*3/-«, 
 which by Art. 49, is the same as 
 
124 INVOLUTION. , 
 
 4. What is the seventh power of — ar^xl 
 
 
 Ans. -a-V = — ^. 
 
 5. 
 
 What is the third power of x^jr' ? 
 
 
 Ans. x^jr' = -^ 
 
 6. 
 
 What is the nth power of — 2x-3 y^ 1 
 
 Ans. ±2"a:-3Y'' = i:-^ 
 
 7. 
 
 What is the square of — 7a; - * y - ^ ? 
 
 Ans. 49x-^y-=i?, 
 
 
 8. What is the third power of — - x'^y~^ ? 
 
 1 x^ 
 
 Ans. r x^y~^^ = 
 
 125 " 125y'^ 
 
 9. What is the seventh power of — m^xz-^ ? 
 
 7 
 
 Ans. — m^x''z~'^. 
 
 10. What is the fourth power of — - n-^y^ ? 
 
 Ans. Q^n^Y^- 
 
 CASE II. 
 
 (94) When the quantity is compound, we can write the 
 different powers by the aid of rules which we will hereafter 
 point out. (See Binomial Theorem.) 
 
 At present we will content ourselves, by involving com- 
 pound expressions by actual multiplication, according to 
 Rule under Art. 33. 
 
INVOLUTION. 125 
 
 EXAMPLES. 
 
 1. Find the seconil power of x -{• V — 
 
 x-{-y — z 
 
 X' -|- XT/ XZ 
 
 — XZ —yz-{- 
 
 Ans. = x'^-\- 2xy — 2xc+ y^ — 2yz -f- z'-. 
 
 2. Find the fifth power of a -f- ^j as well as all the lower 
 powers of the same. 
 {a-\-by = a-^b. 
 a-\~b 
 
 a- + ab 
 -^ab-\-b'' 
 
 (a-\-b)' = a^-}'2ob -\- b^. 
 a +6 
 
 a3 -f 2a'b -f- ab" 
 
 a"b -i- 2ab^ 4- 63 
 
 {a^by = a^ + Sa-b -\- 3ab- + b\ 
 a -{-b 
 
 a^ + 3a'b + 3a'h^ + ab' 
 
 a^b + 3a=6= + 3a6» + ^* 
 
 (a 4-6)^ = a^ 4- 4a86 + 60=6^ + 4a6^ + ft-*, 
 a 4-6 
 
 a^J _|_ 4a^6- + Ga^ft* + 4fl6* -f • 6* 
 
126 INVOLUTION. 
 
 3. Find the fifth and lower powers of a — b, 
 {a — b)'=a — b. 
 a — b 
 
 
 -br= 
 
 -b)*= 
 
 a^- 
 
 -ab 
 -ab + b- 
 
 -b' 
 
 
 {a- 
 
 a — 
 
 ■'2ab-j-bK 
 -b 
 
 
 
 a«- 
 
 - a'b-\-2ab^- 
 
 
 {a- 
 
 a - 
 
 - 2a% -f- 3a6*— 
 -b 
 
 b\ 
 
 
 
 a*- 
 
 -3a'b-\-3a^'^- 
 - a'b + 3a-b^- 
 
 -ab 
 Sal 
 
 3 
 
 {a- 
 
 a — 
 
 -4a'b-{-6a^b''- 
 -b 
 
 -4a 
 
 b' + b*. 
 
 
 «'- 
 
 ~ia*b-{- 6a^^ 
 
 — 
 
 4:aH'+ ab* 
 ea^^-\- 4ab*— 6» 
 
 (a_ J) «= a'-~5a*b + lOa^b^— 10a^^-\- 5ab*— b' 
 
 4. What is the cube of c — x ? 
 
 Ans. o^ — 3a^a: -j- 3aa:^ — x^. 
 
 5. What is the square of m-{-n — x'f 
 
 Ans. m^-\- 2mn — 2mx + n^ — 2nx -\- x^. 
 
 6. What is the fourth power of 3x — 2y? 
 
 Ans. 81x^—216x^1/ + 216x^y^— 96xy« + 16y*. 
 
 7. What is the square of a-\-b? 
 
 Ans. a"+2ab-i-.b^. 
 
 8. What is the square of a + 6 + c ? 
 
 Ans. ai-\- 2ab + 2ac -{-b^ + 2bc + c^. 
 
EVOLUTION. 127 
 
 EVOLUTION. 
 
 (95.) Evolution is the extracting of roots, or the 
 reverse process of involution. 
 
 When the quantity whose root is to be found is a single 
 letter, the operation is denoted by giving it a fractional ex- 
 ponent, the denominator of which denotes the degree of 
 the root. (Art. 14.) 
 
 And in the same way we may denote the extraction of a 
 root of any quantity or expression, by enclosing it within 
 a parenthesis, and then treating it as a single letter. 
 
 Thus, the second root of my= {my)'^, 
 
 the third root of x -\- y ^ {x -\- yY , 
 
 i_ 
 the fourth rpot of 2x — 3?/ = (2x — 3y) * , 
 
 J. 
 the nth root of a — ft = {a — b)" , 
 &c., &., 
 
 CASE I. 
 
 (96.) To extract a root of a monomial, we obviously 
 have the following 
 
 R U L E . 
 
 I. Extract the required root of the coefficient^ by the i/sual 
 arithmetical rule. When the root can not be accurately oh- 
 
128 EVOLUTION. 
 
 tained, it may be denoted by means of a fractional exponent, 
 the same as in the case of a letter. 
 
 II. Extract the required root of the different letters, by 
 multiplying the exponents which they already have by the 
 fractional exponent denoting the required root. To this 
 root prefix the root of the coefficient. 
 
 Note. — Since the even powers of all quantities, whether 
 positive or negative, are positive'; it follows that an even 
 root of a negative quantity is impossible, and an even root 
 of a positive quantity is either positive or negative. 
 
 We also infer that an odd root of any quantity has the 
 same sign as the quantity itself. 
 
 EXAMPLES. 
 
 1. What is the square root of 64a^6'*x^ ? 
 
 In this example, the square root of the coefficient, 64,13 
 ±8, where we have used both signs. 
 And, 
 
 (0^6^)^= a6V, 
 
 .-. (64a«6V)^==h8a6V. 
 
 2. What is the cube root of 64:a^x^ ? 
 
 Ans. Aax'. 
 
 3. What is the fifth root of — 32x/ ? 
 
 Ans. — 2x^y^. 
 
 4. What is the seventh root of — ax~^ 1 
 
 Ans. — a'^x~-^=z — — 
 
 a 
 ■ X^ 
 
 6. What is the square root of — 4a^5*? 
 
 Ans. Impossible. 
 
V EVOLUTION. 129 
 
 6. What is the cube root of 21a^b^'' 1 
 
 Ans. 3«6'. 
 
 7. What is the fourth root of 16a -^x-^ ? 
 
 3- L L 26' 
 
 Ans. =b2a-*6*x-'=i-^— 7. 
 
 (97.) By comparing the operations of this rule, with those 
 of rule under Art. 93, we see that involution and evolution 
 of monomials may both be performed by one general rule, 
 of multiplying the exponents of the respective letters by 
 the exponent denoting the power or root. We will there- 
 fore give the following promiscuous examples, which will 
 require the aid of one or both of these rules. 
 
 EXAMPLES. 
 
 1. What is the cube root of the second power of 8a'6^ 1 
 If we first raise 8a ^6^ to the second power, it will become 
 (8o369)2= 64a«&'^ 
 extracting the third root, we find 
 
 (64a«6»«^^=4a=i«, 
 for the result required. 
 
 Again, first extracting the cube root of Sa^i^, it becomes 
 
 raising this to the second power, it becomes 
 
 the same as before. 
 
 (98.) Hence, the cube root of the square of a quantity, 
 is the same as the square of the cube root of the same quan- 
 tity. 
 
 And in general, the nth root of the mth power of a quan- 
 tity^ is the same as the mth power of the nth root of the 
 same quantity. 
 
 17 
 
130 
 
 EVOLUTION. 
 
 Therefore, a^ may be read, the fourth power of the fifth 
 root o{ a, or the fifth root of the fourth power of a. 
 
 And in the same way, {a -\-b)'^ is read, the third power 
 of the square root of the sum of a and 6, or the square root 
 of third power of the sum of a and b. 
 
 2. What is the value of (— 3a6 V) 
 
 3 ? 
 
 Ans. 3^ah*^x\ 
 
 3. What is the value of {^a-'^b'xy ? 
 
 Ans. ±32a-'b'^x^. 
 (99.) Surd quantities may be made to assume several 
 equivalent forms which require to be read differently. As 
 
 3 
 
 an example, the surd a~* may be written six different 
 ways, as follows : 
 
 — I — ) s 
 
 1. (M^) ; 2. ((«,^)') ; 3. ((a-)^) ; 
 
 4. ((ah-^; 5. ^(a-'))'; 6. ((a^)-)'. 
 
 These six expressions are read as follows : 
 
 1. The reciprocal of the fifth root of the third powei 
 of a. 
 
 2. The reciprocal of the third power of the fifth root 
 of a. 
 
 3. The third power of the fifth root of the reciprocal 
 of a. 
 
 4. The third power of the reciprocal of the fifth root 
 of a. 
 
 5. The fifth root of the third power of the reciprocal 
 of a. 
 
EVOLUTION. 131 
 
 6. The fifth root of the reciprocal of the third power 
 of a. 
 
 CASE I. 
 
 (100.) To extract any root of a polynomial, we liave the 
 following general 
 
 RULE. 
 
 I. Havhig arranged the polynomial according to the 
 powers of some one of the letters^ so that the highest fower 
 shall stand first, extract the required root of the first term, 
 which will be the first term of the root sought. 
 
 II. Subtract the power of this first term of the root from 
 the polynomial .J and divide the first term of the remainder.) 
 by the first term of the root involved to the next inftrior 
 power, multiplied by the number denoting the root; the quo- 
 tient will be the second term of the root. 
 
 III. Subtract the power of the terms already found from 
 the polynomial, and using the same divisor proceed as before. 
 
 This rule obviously verifies itself, since, whenever a new 
 term is added to the root, the whole is raised to the given 
 power, and the result is subtracted from the given polyno- 
 mial : and when we thus find a power equal to the given 
 polynomial, it is evident that the true root has been found.. 
 
 1. What is the fifth root of 
 
 a5 -j- 50^6 -f lOa^i^ _^ \Qa^b^-\- ba¥ -f b' ? 
 
 OPERATION. 
 
 ROOT. 
 
 a» + 5a^6 + lOa^ft- + lOa^ft > 4- dab* -f- b' {a -\-b 
 a* 
 
 5a*) 5a*6 
 
 ^a -\- by = a** -f 5a*b -{- lOa^b^ + lOa^^ + 5ab* + b' 
 
132 EVOLUTION. 
 
 EXPLANATION. 
 
 We first found the fifth root of the first term a', to be ff, 
 which we plated to the right of the polynomial for the first 
 term of the root. Raising a to the fifth power and subtract- 
 ing it from the polynomial, we have Dab for the first term 
 of the remainder. 
 
 Since the number denoting the root is 5, we raise the first 
 term of the root, a, to the fourth power, which thus becomes 
 a', this multiplied by the number derK^ting the root, gives 
 5a-* for our divisor. 
 
 Now, dividing 5a'6 by 5a', we get b, which we write for 
 the second terra of the root. 
 
 Involving this root to the fifth power by actual multipli- 
 cation, as was done in Ex. 2, Art. 94, we have 
 
 (a + by= a^+ 5a'b -f lOa^^^-j- 10a''b^-\- 5ab*+ 6* ; 
 which subtracted from the given polynomial, leaves no re- 
 mainder, so that we know that a -\- b \s the true root. 
 
 2. What is the square root of 
 
 4x^ — 16x3 + 24a- — 16a: + 4 ? 
 
 OPERATION. 
 
 ROOT. 
 
 4x' — 16a:3+24j2_l6x+4(2x=^ — 4x-H2 
 
 4:X~) — ]6x^ 
 
 (2x*— 4xf = 4x* — 16x3+i6x=^ 
 
 4x2)8x2 
 (Sr*— 4x-f2)=^=4x^ — 16x3+24x2— 16x-f 4 
 
EVOLUTION. 133 
 
 3. What is the square root of 
 
 16a;* + 24x=' + S9x- -f- 60x + 100 ? 
 
 Ans. 4x^ + 3x + 10. 
 
 4. What is the cube root of 
 
 a« + 3a^ — 3a* — lla^ + 6a~ + 12fi — 8 ? 
 
 Ans. a- -(- a -s— 2- 
 
 5. What is the sixth root of 
 
 Ans. a — 6. 
 
 6. What is the fourth root of 
 
 a*— 4c36 + 6a26- — 4o53 + b* ? 
 
 Ans. a — b. 
 
 (101.) If we carefully observe the law by which a poly 
 nomial is raised to the second power, we shall, by reversing 
 the process, be enabled to deduce a rule for the extraction of 
 the square root of a polynomial, which will be more simple 
 than the above general rule, and of more interest, since the 
 arithmetical rule is deduced from it. 
 By actual multiplication, we find 
 (a+6)2 = a2+2a6+62, 
 (a+6+c)- = a-+2«&+Z^--f2(«+6)c-4-c% 
 {a+b4-c+dy 
 
 = a2+2a6+62+2(a+6)c + c-+2(a+&-fcV-f-aS 
 (a-l-6+c+d+e)2 
 
 ^ ( a2-h2a6+6'^+2(a+6)c+c2 ) 
 
 I -\-2{a-\-b + c)d+d'-{-2{a+b+c-\-d)e+e'. i 
 &c. &c. 
 
 From the above, w^e discover, that 
 
 (102.) The square of any polynomial is equal to the square 
 of the first term, plus twice the first term into the second^ 
 plus the square of the second ; plus tvnce the sum of the first 
 two into the third, plus the square of the third; plus twice 
 
134 EVOLUTION. 
 
 tke Stan ofthejirst three into the fourth^ plus the square of 
 the fourth; and so on. 
 
 (103.) Hence, the square root of a polynomial can be 
 found by the following 
 
 RULE. 
 
 I. Jlfter arranging the polynomial according to the powers 
 of some one of the letters^ take the root of the first term for 
 the first term, of the required root ^ and subtract its square 
 from the polynomial. 
 
 II. Bring down the next two terms for a dividend. Di- 
 vide its first term hy twice the root just found, and add the 
 quotient, both to the root, and to the divisor. Multiply the 
 divisor, thus increased, into the term last placed in the root, 
 and subtract the product from the dividend. 
 
 III. Bring down two or three additional terms, and pro- 
 ceed as before. 
 
 EXAMPLES. 
 
 1. What is the square root of 
 
 OPERATION. 
 
 ROOT. 
 
 a-'+2a6H-6-+2(a+6)c+cH-2(flH-i+c)d+d-^[a+6-hc4-d. 
 
 2fl& + 6--' 
 2a6+62 
 
 2(a+6)-|-c 2(a+6)c4-c2 
 2(a+6)c+c2 
 
 2(«+6+c)4-d. 2{a-\-b-\-c)dJ^d^ 
 
EVOLUTION. 135 
 
 2. What is the square root of 
 
 OPERATION. 
 
 RO(T 
 
 4:X^-i-12x^-{-5x* — 2a:3-|-7a;- — 2x-\-l (2x3+30;^ — x+ 1 . 
 4x« 
 
 4x'+3x- 12x^+5x-' 
 12x^+9x^ 
 
 4x^H-6x- — X — 4X'' — 2x3+7x'^ 
 — 4x^ — 6x3-(- x-^ 
 
 4x3+6x2— 2x+l 4x3-1-6x2— 2x+l 
 
 4x3+6x2— 2x+l 
 
 
 
 3. What is the square root of 
 
 X* — 2xY — 2a;- + 2/" + 2y- + 1 ? 
 
 Ans. X- — y- — 1. 
 
 4. What is the square root of 
 
 9xy — 30xhj3 + 25xy ? 
 
 Ans. 3x-y2 — p,xy. 
 
 5. What is the square root of 
 
 a^ + 2ab — 2ac + i- — 25c + c' 1 
 
 Ans. a + 6 — c. 
 
 6. What is the square root of 
 
 4m-' — 36m7i + 81n-'? 
 
 Ans. 2m — 9/j. 
 
 In these examples, and in all others where an ece?i root 
 is extracted, the terms of the root may have all their signs 
 changed, and still satisfy the questions. 
 
136 EVOLUTION. 
 
 (104.) We will now endeavor to find a particular rule 
 for the extraction of the cube root of a polynomial. 
 B-y actual multiplication, we find 
 {a-\-by = a^-\-3a-b+3ab^-\-b^, 
 {a-\-b-\-cY 
 
 = as-\-3(rb-\-3ab'-\-b-'-\-'3{a-\-bYc-\-3{a-\-b)c'-\-c', 
 {a+b+c+dy 
 
 &c., &c. 
 
 (105.) From which we discover that 
 
 The cube of any polynomial is equal to the cube of the first 
 teryn, plus three times the square of the first into the second^ 
 plus three times the first into the square of the second, plus 
 the cube of the second; plus three times ^ the square of the 
 sum of the first two into the third, plus three times the sum 
 of the first two into the square of the third^ plus the cube of 
 the third; plus three times the square of the sum of the first 
 three into the fourth, plus three times the sum of the first 
 three into the square of the fourth, plus the cube of the 
 fourth ; and so on. 
 
 (106.) Now we may reverse the above process, that is, 
 we may extract the cube root of a polynomial by the fol- 
 lowing 
 
 RULE. 
 
 I. Having arranged the terms of the polynomial accord- 
 ing to the powers of some one of the letters, seek the cube 
 root of the first term, which place at the right of the poly- 
 nomial for the first term of the root, also place it at the left 
 by itself, for the first term of a column, headed, first 
 
EVOLUTION. 137 
 
 COLUMN. Then multiply it into itself , and place the pro- 
 duct for the first term of a column^ headed, secoi^d column. 
 Again, multiply this last result, by the same first term of 
 the root and subtract the product from the first term of the 
 polynomial, and then bring down the next three terms of the 
 polynomial, for the first dividend. Md the first term of 
 the root just found to the first term of the first column, the 
 sum will constitute its second term, which must be multi- 
 plied by the first term of the root, and the result added to 
 the first term of the second column, for its second term, 
 which we will call the first trial divisor. The same first 
 term of the root must be added to the second term of the 
 first column, forming its third term. 
 
 II. Divide the first term of the first dividend by the first 
 term of the trial divisor, the quotient must be added to the 
 root already found-, for its second term, it must also be 
 added to the last tenn of the first column, the result will be 
 its fourth term, which must be multiplied by the second term 
 of the root, and the product added to the last term of the 
 second column, which sum will give its third term, which in 
 turn must be multiplied by the second term of the rooty and 
 the product subtracted from the first dividend. 
 
 III. To the remainder bring down three or four of the 
 next terms of the polynomial for a second dividend. Pro- 
 ceed with this second term of the root, precisely as was done 
 with the first term, and so continue until the entire polyno- 
 mial has been exhausted. 
 
 IS 
 
138 
 
 EVOLUTION. 
 
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EVOLUTION. 
 
 139 
 
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140 
 
 EVOLUTION. 
 
 5. What is the cube root of the polynomial 
 
 a;6 — 6x^ 4- iDx^ — 20x'' + 15x^ —6a; + 1 1 
 
 Ans. x2 — 2a;+l. 
 
 6. What is the cube root of the polynomial 
 
 a» + 12a'x'' — Sa'x' — 6a''x ? 
 
 Ans. a^ — 2ax. 
 
 7. What is the cube root of 
 
 a' — 3a' -j- ea' — la' -\- 6a^- — 3a + 1 1 
 
 Ans. a^ — a-\-l. 
 
 8. What is the cube root of 
 
 x^ + 6x' + 21x' + 44x3 + 63ar» + 54a: + 27 1 
 
 Ans. x^-{-2x-\-3. 
 
 (107.) From the above rule, for extracting the cube root 
 of a polynomial, we can easily deduce the rule which we 
 have given in the Higher Arithmetic for the extraction of 
 the cube root of a number. 
 
 This rule is also particularly interesting because of its 
 close analogy to the method of finding the numerical roots 
 of a cubic equation, as explained in a subsequent part of 
 this work. 
 
SURD QUANTITIES, ^ 
 
 IRRATIONAL OR SURD QUANTI- 
 TIES. 
 
 (108.) An Irrational Quantity, or Surd, is a quan- 
 tity affected with a fractional exponent or radical, without 
 which, it can not be accurately expressed. 
 
 Thus, 
 
 v'B is a surd, since the s-unu e root of 3 can not be accu- 
 rately found 5 also 8^ 4% V4, V5, &c., are surd quanti- 
 ties. 
 
 REDUCTION OF SURDS. 
 
 CASE I. 
 
 (109.) To reduce a rational quantity to the form of a 
 surd, we have this 
 
 RULE. 
 
 Raise the quantity to a power denoted by the root of the 
 required surd; then the corresponding root of this poicer, 
 expressed by means of a radical sign or fractional exponent, 
 mil express the quantity under the proposed form. 
 
 EXAMPLES. 
 
 1. Reduce 5a to the lorm of the cube root. 
 Raising 5a to the tl.ird power, we have 
 (5a)'=r25a^; 
 extracting the cube root, it becomes 
 
 5a= Vl25^=(125a^)^ 
 
142 
 
 SURD QUANTITIES. 
 
 X 
 
 2. Reduce — to the form of the fifth root. 
 
 3. Reduce — to the form of the fourth root. 
 
 Ans. — =1 
 
 4. Reduce — to the form of the nih root 
 6* 
 
 oot. 
 
 1 
 a- __ ja-"^ 
 
 CASE II. 
 
 (110.) To reduce surds expressing different roots to equi- 
 valent ones expressing the same root. 
 
 Reduce the different indices to common denominators ; then 
 raise each quantity to a power denoted by the numerator of 
 its respective exponent; afterwards take the root denoted hi/ 
 the common denominator. 
 
 EXAMPLES. 
 
 1. Reduce ^/3, V4, and V5 to surds expressing the same 
 root. 
 
 Changing the radicals into fractional <jxponents, they be- 
 come i, I, 5, which reduced to a common denominator, are 
 
SURD QUANTITIES. 143 
 
 tV? tV) fV* Now, raising the quantities 3, 4, and 5, to 
 powers denoted respectively by 6, 4, and 3, we find 3*, 4", 
 53, or, which is the same, 729, 256, 125. Taking the 12th 
 root of these results, they become 
 
 (729)^'^, (256)^'^, (125)^'^. 
 
 1 ? 
 
 2. Reduce a^ and x^ to surds expressing the same root. 
 
 Ans. ) ^ '^ J 
 
 ( x^ =(ar*)«. 
 
 3. Reduce x^, y^, m^ to surds exprt? sing the same root 
 
 x'^ = {x')', 
 
 Ans.;^!^^^.)^^ 
 
 1 i 
 
 4. Reduce V2, V3, V4 to surds expressing the same 
 root. 
 
 ' V2 = (220)«'°, 
 V4=:(4'=^)«''. 
 
 (•A:M- Iir 
 
 (111.) To reduce surds to their simplest form. When 
 ever a surd can be separated into two factors, one of which 
 is a perfect power, it can be simplified by this 
 
144 
 
 SURD QUANTITIES, 
 
 RULE. 
 
 Having separated the surd into two factors, one of which 
 is a perfect power, take the root of the factor which is a 
 perfect power, and multiply it by the surd of the other fac- 
 tor. 
 
 EXAMPLES. 
 
 1. Reduce v^SSS to its simplest form. 
 
 We can separate 288 into the factors 144X2, of which 
 144 is a perfect square who=e root is 12 ; therefore 
 
 v/288= v/l44x2= n'mIx v/2 = 12v/2. 
 
 2. Reduce Vx^y — a-x- to its simplest form. 
 
 Ans. Vx'-y — a-x' = x '^y — a?. 
 
 3. Reduce V— 32a*6 to its simplest form. 
 
 Ans. V— 32a^6 = — 2a V6. 
 
 4. Reduce (a-x^i/~')' to its simplest form. 
 
 Ans. (c-x^y-') =iax^y-'--. 
 5. Reduce (?7i^nx^y^) to its simplest form. 
 
 f 
 
 Ans. mxy{nxy) 
 
 (112.) When a surd is in the form of a fraction, it may 
 be simplified by the following 
 
 RULE. 
 
 Multiply both numerator and denominator by such a quan-' 
 tity as will render the denominator a perfect power. 
 
SURD QUANTITIES. 145 
 
 Reduce -i/ — to its simplest form. 
 
 Multiplying both numerator and denominator by 11, we 
 have 
 
 2. Reduce \/ — 7 to its simplest form. 
 
 3 /-T— 3 
 
 
 1 
 
 3. Reduce (— ) to its simplest form, 
 
 \xy I xy 
 
 (_1L— 2v 3 
 J to its simplest form 
 
 ADDITION AND SUBTRACTION OF SURDS. 
 
 RULE. 
 
 (113.) Reduce the surds to their simplest form ; theii^if 
 the surd part is the same in both, add or subtract the ration 
 al parts, and annex the common surd part to the result; 
 hut when the surd parts are different, they can only be ad- 
 ded or subtracted by the aid of the signs -|- or — . 
 19 
 
146 
 
 SIRU QIjANTITIES. 
 EXAMPLES. 
 
 1. What is the. sum of v/54 and \/24 ? Also, what is 
 the difference of the same surds ? 
 
 By reduction we have 
 
 v/ 54 = ^9X6 = 3 x/ 6 
 ,/ 54 = v/^x6 = 2 v/ 6 
 
 Therefore, v/54 -fv/ 24 =5v/6. 
 
 And, v/54 — v/ 24 = ^ 6. 
 
 2. What is the sum and difference of \/a^b and X/ab* 1 
 
 The sum ={a 4- h\i/ah. 
 
 The diff =(a — ?))Vfl6. 
 
 Ans. 
 
 3. What is the sum of (36x=^)' and (252/)' ^■ 
 
 Ans. (6x -|- 5)s/y. 
 
 4. What is the . i of (8a:)^ {xy') ^ and i21x*f 7 
 
 Ans. (2 4-3x4-3/'-')Va:. 
 
 5. What is the sum of (a6V)' and {m.y)^ 1 
 
 Ans. a:(a6"a:)^-j- 3/-(m^)*. 
 
 MULTIPLICATION AND DIVISION OF SURDS, 
 
 RULE. 
 
 (114.) Reduce the surds to equivalent ones expressing the 
 same root, (Case II. Art. 110,) then multiply or divide as 
 required. 
 
 EXAMPLES. 
 
 1. What is the product of v/8 by V16 ? 
 
 By Case 11, we find 
 
 Therefore, 
 
 ^8 = (8-^)*=(512)*. 
 V16 = (16=^)^=(256)*. 
 v/8x V16 = (512X256)*=4V32. 
 
SURD QUANTITIES. 147 
 
 2. What is the product of 4Vfl6 by 3^/by 
 
 I 7 
 
 are binomial surds. 
 
 Ans. 12Va-6y. 
 
 3. Divide 4 V32 by V16. 
 
 Ans. v/8. 
 
 4. Divide "^a'-b-^ by ^aM. 
 
 5. Divide v^4tt"6-^ by ^2^2^ 
 
 Ans. a{ieab^y. 
 
 EXTRACTION OF THE SQUARE ROOT OF A BINOMIAL SURD. 
 
 (115.) When one or both of the terms of a binomial are 
 surds, it is called a binomial surd. 
 Thus, 
 
 a±Vb 
 Vc±Vd 
 
 (116.) Before we proceed to the extraction of the square 
 root of a binomial surd, we will establish the following 
 le7nmas. 
 
 LEMMA I. 
 
 The square root of a rational quantity can not consist 
 of the Sinn of five parts^ one of which is rational and the 
 other irrational. 
 
 For if possible, suppose we have the relation 
 
 s/a = x-\- v/y, (1) 
 
 where z is rational. 
 
 Squaring both members of (1), we find 
 
 a = 3ca-h2x^/y-f y (2) 
 
 From (2) we obtain 
 
 a — x* — y /o\ 
 
 ^/y = ^^. (3) 
 
US. 
 
 SrUD EQUATIONS. 
 
 Equation (3) gives an irrational quantity equal to a ra- 
 tional one, Avhieh is impossible, therefore the condition (1) 
 is impossible, hence the above lemma must be correct. 
 
 LEMMA II. 
 
 In any equation, cousistincr of rational quantities and ir- 
 rational quantities, the rational quantities on each side are 
 equal, <!S aho are the h rational quantities. 
 
 Suppose we have the equation 
 
 a -H v/6 = X -I- v/y. (1) 
 
 Then if a is not equal to x, let us have o = x ± wi, this 
 value of a substituted in (1), gives 
 
 x±:7n-\- Vh = x+ Vy-, (2) 
 
 or im-j- -y/i=: v/j/. (3) 
 
 Equation (3) shows that the square root of y is partly 
 rational and partly irrational, which is impossible (Lemma 
 I). Therefore it is absurb to suppose that .r differs in value 
 from 0, hence x ^ a. Consequently y/b = ^/y. So that 
 the above lemma is correct. 
 
 LEMMA III 
 
 // 
 
 a -\- y/b = X -{- y/y, then icill ^ a — ^/ft =^x — y/y. 
 
 If we square both the members of the equation 
 
 find 
 
 Va -|- v/ft = x-f- v/y 
 
 (1) 
 
 a + v/6 = a;-+ 2x y/y + y. (2) 
 
 Equating the rational, as well as the irrational parts of 
 (2), (Lemma II), we have 
 
 a = x»+ y, (3) 
 
 y/b = 2xVy- (4) 
 
SURD QUANTITIES. 149 
 
 Subtracting (4) from (3), we have 
 
 a — ^b = x- — 2xVy-\-y- (5) 
 
 Extracting the square root of both members of (5), we get 
 
 Va — Vb = x-Vy. (6) 
 
 So that if (1) is true, then also will (6) be true, which es- 
 tablishes the above lemma. 
 
 (117.) We are now prepared to proceed to the extraction 
 of a binomial surd. 
 
 Assume 
 
 Va-{-Vb = x-{-Vy. (1) 
 
 Then, (Lemma III) 
 
 ^a — Vb=x—Vy- (2) 
 
 Equations (1) and (2), when squared, become 
 
 a-\-^b = x^-\-2xVy-\-y. (3) 
 
 a — ^b=x'^ — 2x^y-\-y. f4) 
 
 Taking the sum of (3) and (4), and dividing the result by 2, 
 we obtain 
 
 a=x2-f-y. (5) 
 
 If we multiply together equations (1) and (2), we get 
 
 Va'' — h = x^ — y. (6) 
 
 Adding (5) and (6), and dividing by 2, we have 
 
 a±Va^-b^ 
 
 2 ^ ' 
 
 Subtracting (6) from (5), and dividing by 2, we find 
 
 a — v/a' — b /Q\ 
 
 2 =y. (^) 
 
 Extracting the square root of both members of (7) and (S), 
 we get 
 
150 
 
 y/y- 
 
 SURD QUANTITIES. 
 
 
 
 (9) 
 
 _) a—Va'^ — b I 
 
 (10) 
 
 Taking the sum of (9) and (10) , we have 
 
 \ a-{-^a^ — b ( j^) a—Va^ — b ( ^ ^^^ 
 
 x-j-vy- 
 
 Subtracting (10) from (9)^ we get 
 
 Va-_6 
 
 (12) 
 
 For the left-hand members of (11) and (12) , substitute theii 
 vahies given by (1) and (2), and we then have 
 
 v^a+vfe 
 
 = ) a-{-^a^ — b I -^\ a—^^a' — b 
 
 (A) 
 
 2 
 
 — Va'^ — b 
 
 (B) 
 
 By using the double sign dcj we may combine in one for- 
 mula, both (A) and (B). 
 
 h 
 
 v/.":r;7r= W + ^°^-M ±) a-^a'-b ( .(C) 
 '( 2)1 2 \ 
 
 (118.) We will now show the use of formulas (A) and 
 (B) by the following 
 
SURD QUANTITIES. 151 
 
 EXAMPLES. 
 
 1. What is the square root of 7 — 2v/10 / 
 Reducing the factor 2, to the form of the square root 
 (Art. 109), and then introducing it under the radical sign, 
 we have 7 — 2 v/10 = 7 — x/40, which, when compared 
 with the general form a—Vb, gives a = 7 ; & = 40, these 
 values of a and &, substituted in (B), give 
 
 ,__V^=l ) ^ ( 7-v^49- 40 ) /LzL^jL^ 2. 
 
 Therefore, we have 
 
 v/7Tr27rO =v/5*— v/2. 
 
 2. What is the square root of 6 +v/20 ? 
 [n this example we have a = 6 ; 6 = 20, which substi- 
 tuted in formula (A) , gives 
 
 
 v/6 -|-v/20=v/5 + 1. 
 
 3. What is the square root of 2{x -f- 1) + 4 v/J- ? 
 
 Ans. v/2x+s/2. 
 
 2 
 
 r 
 
 (6+V/36- 
 
 -20 
 
 a —Va''— b 
 
 U— v/36 
 
 — 20 
 
 2 
 Therefore, 
 
 1 ^ 
 
 
152 
 
 SURD QUANTITIES. 
 
 4. What is the square root of 6 — 2>yo ? 
 
 Ans. v/5 — 1. 
 
 5, What is the square root of 7 -|- 4s/3 ? 
 
 Ans. 2 +v/3. 
 
 TO FIND MULTIPLIERS WHICH WILL CAUSE SURDS TO 
 BECOME RATIONAL. 
 
 CASE I. 
 
 (119.) When the surd consists of but one term, we can 
 proceed as follow^s : 
 
 1 rn—l 
 
 Suppose the given surd is x"\ if we multiply this by a; m * 
 
 1 m—l 
 
 by rule under Art. 114, we shall have x "'Xx m =x, a ra- 
 tional quantity. 
 
 Hence, to cause a monomial surd to become rational by 
 multiplication, we have this 
 
 RULE. 
 
 Multiply the surd by the same quantity^ having such an 
 exponent, as when added to the exponent of the given surdj 
 shall make a unit. 
 
 EXAMPLES. 
 
 1. How can the surd x^ be made rational by multiplica- 
 tion. 
 
 In this example, § added to the exponent i, gives 1, 
 
 3 
 
 therefore we must multiply by x% performing the operation, 
 we have 
 
 1 2 
 
 x^Xx ^=x. 
 
 3 
 
 2. Multiply x^ so that it shall become rational. 
 
 3 a 
 
 Ans. x*Xa:*=a^ 
 
SUKD QUANTITIES. 153 
 
 3. Multiply x~^ so that it shall become rational. 
 
 4 11 
 
 Ans. x~'' Xx "^ =x. 
 CASE II. 
 
 (120.) Wh^n the surd consists of two terms, or is a bino- 
 mial surd. 
 
 Suppose it is required to multiply -/« + \/& so as to pro- 
 duce a rational product ; we know from Art. 35, Theorem 
 III, that 
 
 ( v/a + \/6) X ( s/rt — v/6) = a — h. 
 Hence, to cause a binomial surd to become rational by 
 multiplication, we have this 
 
 RULE. 
 
 Change the sign which connects the two terms of the bi- 
 nomial surd, from -j- to — , or from — to -j-, and this re- 
 sult, multiplied by the binomial surd, mil give a rational 
 product. 
 
 EXAMPLES, 
 
 1. Multiply v/3 — s/2 so as to obtain a rational product. 
 
 Ans. (v/3— v/2)X( v/3-(-v/2)^ 3 — 2 = 1. 
 
 2. Multiply 4-f-\/5 so that the result shall be rational. 
 
 Ans. (4+^/5)X(4— v/5) = n. 
 
 3. How canv^a-{-6 — ^a — b be made rational by mul- 
 tiplication ? 
 
 Ans. ( v/a+6 — Va — b) X ( V^b^ ^a — b)= 2*- 
 
 4. How can y/l — 1 become rational by multiplication ^ 
 
 Ans. (s/7 — l)x(v/7 + l)=:6. 
 
154 
 
 SURD QUANTITIES. 
 
 (121.) If the surd consist of three or more terms of the 
 square root, connected by the signs plus and minus, it can 
 be made rational, by first multiplying it by itself after chang- 
 ing one or more of the connecting signs. 
 
 EXAMPLES. 
 
 1. If it is required to make v/5 — v'3+i^2 rational by 
 multiplication, we should first multiply by -v/5 + v/3-}--v/2j 
 by which means we obtain 
 
 ^/5— v/3+ ^/2 
 
 v/5-f ,/3+ v/2 
 
 5— V/15+ ^10 — 3+N/6 
 -j-v/15+ v/10 — v/6+2 
 
 5 4-2\/10 — 3 +2=2v/10-|-4 
 
 Again, multiplying 2 x/lO + 4 by 2\/ 10 — 4, we get 
 (2v/10 + 4) X (2v/10 — 4) = 24. 
 
 2. Multiply 2 4-\/3 — v/2 so that it shall become ra 
 tional. 
 
 FIRST OPERATION. 
 
 2 + ^3— v/2 
 2 + v/3 4- ^-2 
 
 4+2v/3 — 2^2 + 3— v/6 
 + 2^3-1-272 +^6 — 2 
 
 4-f4^/3 -f3 — 2=4y3-f5. 
 
 SECOND OPERATION. 
 4v/3 +5 
 
 4^/3—5 
 
 48 
 
 + 20v^3 
 
 — 20v/3 — 25 
 
 48 
 
 — 25=23. 
 
SURD QUANTITIES. 155 
 
 3. Multiply v/5 + 72 — y3 + 1 so that its product shall 
 be rational. 
 
 FIRST OPERATION. 
 
 v/5 + v'S— v/3+ 1 
 V5 — v/2 + v/3 + 1 
 
 ~5~+^10— V/15+ ^/5+ ^/6— ^/2+n/3 
 _2— yl0+yl5+ n/5 + v/6+v/2— v/3 
 
 3 
 
 1 
 
 1 +2^/5+2^/6. 
 
 SECOND OPERATION. 
 
 l+2v/5+2v/6 
 1—2/5 +2v/6 
 
 14_2y5 +2^/6— 4v/30 
 _20 — 2v/5+2v/6 -h4v/30 
 24 
 
 5 +4V6. 
 
 THIRD OPERATION. 
 5-[- 4V6 
 
 _5^ 4V6 
 
 — 25 — 20v6 
 96 4- 20v6 
 
 ^ 71. 
 (122.) To reduce fractions, having polynomial surds for 
 a numerator or denominator or both, so that either the nu- 
 merator or denominator may be free from radicals. 
 Suppose we wish to transform the fraction 
 
 1_ 
 
 V3 + V2 + 1 ' 
 into an equivalent fraction, having a rational denominator. 
 
156 
 
 SURD QUANTITIES. 
 
 It is evident that this transformation can be eflfected, pro- 
 vided we multiply both numerator and denominator by such 
 a quantity as will cause the denominator to become free of 
 radicals, so that the operation is reduced to the finding a mul- 
 tiplier which will make v/3 + ^/2 -[- 1 rational. 
 
 We will first multiply by — ^ 3 + v/2 + 1. 
 
 OPERATION. 
 
 V3+V/2 + 1 
 _^3 4-v/2 + l 
 
 — 3— v/6— y3-f- v/2 
 
 _|_ 2-fv/6+^3+ ,/2 I 
 
 2v/2. 
 Hence, if we multiply both numerator and denominator ol 
 
 by — v/3+ y/2-\-l it will become 
 
 1+^2— v/3 
 
 y3H-v/2+l J ^ ' ^ ' 2s/2 
 
 Again, multiplying both numerator and denominator of 
 
 — -!— ^^ ^— by y2, we finally have — . The 
 
 2^/2 -^ ' -^ 4 
 
 denominator is now rational. 
 
 (123.) Hence, to transform a fraction, ha^-ing surds in its 
 numerator or denominator or both, into an equivalent frac- 
 tion, in which the numerator or denominator may be free of 
 
 RULE. 
 
 Multiply the numerator and denominator by such a quan- 
 tity as will cause the numerator or denominator, as the re- 
 quired case may be^ to become rational. 
 
SURD QUANTITIES. 15" 
 
 EXAMPLES. 
 
 1. Reduce — ^^I^ — to a fraction havino- a rational nu- 
 
 4 ° 
 
 merator. 
 
 Multiplying both numerator and denominator by 5 — V3, 
 we have 
 
 5 +v/3_ (5 -f-s/3)(5 — ^/3)_ 22 _ 11 
 
 4 4(5 —v/3) 20 — 4v/3 10-2 x/3 
 
 /g j O 
 
 2. Reduce — r — ; — -, to a fraction having a ration- 
 
 al denominator. 
 
 Multiplying both numerator and denominator by 
 v/.5 -j-v/3 — v/2, we get 
 
 5 +v/15 — y/lO + 2^/5 + 2v/3— 2v/ 2 
 6 -t- 2v/15 • 
 
 Again, multij)lying both numerator and t'enonihiator of 
 this last fraction, by C — ~\/15, it btroines 
 
 4v/30 — 4^/15— 6^/10+10v/6— 8v/3— 12^/2 
 — 24 ' 
 
 or changing the signs of both numerator and denominator, 
 it becomes, after striking out the factor 2 from each, 
 
 6^/2 + 4v/3 — 5v/6 -f-3N/10 -f 2v/15 — 2^/30 
 12 
 
 3. Reduce ^ — to an equivalent fraction having 
 
 1 -r- \/Z 
 
 a rational denominator. 
 
 . ^/14 — /l) — s^T -l-v/5 
 Ans. z ' . 
 
4. Reduce 
 
 SURD QUANTITIES 
 1 
 
 y3 — n/2 -f 1 
 having a rational denominator. 
 
 to an equivalent fraction 
 
 Ans. 
 
 2 — v/2 -hx/6 
 
 5. Reduce — p-— ^ , first to a fraction having a ration- 
 
 al denominator, and then to a fraction having a rational 
 numerator. 
 
 Ans. 
 
 ■v/6 -^-Vx b — X 
 
 y/b — y/x y/ab -^\/ax -\-y/bx — X 
 
IMAGINARY QUANTITIES. 259 
 
 IMAGINARY QUANTITIES. 
 
 (124.) We have already shown, that (see Note to the 
 Rule under Art. 96,) an even root of a negative quantity is 
 impossible. Such expressions are called imaginary. 
 
 ^~^' J 
 
 are all imaginary quantitie*:. 
 
 Surd quantities, though their values can not be accurately 
 found, can, nevertheless be approximately obtained j but 
 imaginary quantities can not have their values expressed by 
 any means, either accurately or approximately. They must, 
 therefore, be regarded merely as symbolical expressions. 
 
 (125.) We will confine ourselves to the imaginary ex- 
 pressions arising from taking the square root of a negative 
 quantity. 
 
 The general form of imaginaries of this kind, is 
 
 ^ — a =v«x — 1 =yaX -^ — 1, 
 substituting h for Va, we have 
 
 so that all imaginary quantities arising from extracting the 
 square root of a minus quantity are of the form 
 
160 
 
 IMAGINARY QUANTITIES. 
 
 (126.) If we put v^ — 1 =z c, we shall always have 
 c-'= — 1, 
 
 c^ 1, 
 
 V-l. 
 
 And in general. 
 
 = 1, 
 
 c4m + 2^ _ 1, 
 
 m being any positive integer whatever. 
 
 (]27.) From which we easily de;duce the following prin- 
 ciples. 
 
 1. ( + v/ir^)x(+v/^^)=— ^a^= — a. 
 3. (+^/I^7)x(— >/^^^) =+ v/ «'= + «• 
 
 ^- (— ^/— "7)X(— ^/^^) = —Vab. 
 6. (-f v/ir7)x(_v/^i) = -\-^ab. 
 
 The above is in accordance with the usual rules for the 
 multiplication of algebraic quantities, and must be consid- 
 ered as a definition of this symbol, and of the method of 
 using it, and not as a demonstration of its properties. 
 
 (128.) The student must not infer from what has been 
 said, that imaginary quantities are useless. So far from 
 being useless, they have lent their aid in the solution of 
 questions, which required the most refined and delicate 
 analysis. 
 
 (129.) We will now, in order to become more familiar with 
 the operations of imaginaries, perform some examples in 
 
IMAGIXAKY QIANTITIES. jQ] 
 
 MULTIPLICATION CF IMAGINARIES. 
 
 1. Multiply 4 v/ 131 j_ ^ IZ2 by 2v iry_x/Zr3. 
 
 OPERATION. 
 
 2--'— 1 — ^ -^ 
 — 8— 2v/2+4v/3+v/G. 
 
 2. Multiply 4 + v/ZTs by 2 — v/^2. 
 
 OPERATION. 
 
 4+ v^^^ 
 
 2— ^/^2 
 
 8-f2v/— 3— 4v/— 2+v6. 
 3. Multiply 3 — v/^^1 by 4 + ^/^^ 
 
 OPERATION. 
 
 3— ^/::^^ 
 
 4+ v/=^ 
 
 4. Multiply 
 
 12— 4n/— 1 
 
 
 -|-3v/— 1 -f 1 
 
 
 12— %/- l-^l = l3_ 
 
 V_l. 
 
 .^ v^ — 3 into itself. 
 
 
 OPERATION. 
 
 
 i — i^^^3 
 
 
 i-i^/~3 
 
 
 ^-^^^=^3 
 
 
 -i^-3-l 
 
 
 
 jV-3 
 
 i_^x/_.3-5=_i_ 
 
 21 
 
 
162 IMAGINARY QUANTITIES. 
 
 (130.) Wc will now ptMibrm some examples in 
 
 DIVISION OF IMAGINARY QUANTITIES. 
 
 1 . Divide 4 + ^'^2 by 2 — ^^^. 
 
 OPERATION. 
 
 4-f n/— 2 
 
 Multiplying numerator and denominator by 2 -|-'^ — 2, i* 
 becomes 
 
 = 1_^n/_2. 
 
 In the same way we find 
 
 1 + ^^- 
 
 3. 
 
 4. 
 
 1-N/-1 
 
 6n^^=^3 
 2v/^^ 
 3-v'~~l 
 
 = V. 
 
 = 1^/3. 
 
 
 4-f-2v/— 1 
 
 (131.) We will also add a couple of examples of the ex- 
 traction of the square root of imaginary binomial surds. 
 
 1. Extract the square root of 3 -J-2n/ — 1. 
 Comparing this expression with the general formula (A) 
 Art. 117, we havo a = 3 : b = — 4 : hence, 
 
IMAGINARY QUANTITIES. \Ci?> 
 
 * . { v/13 — 3 ) ■ 
 
 ■V— 1 
 
 2. Extract the square root of 3 — 2-^ — 1. 
 
 All the difference between this example and the last, is 
 in the sign which connects the two terms, so that we need 
 only change the sign which connects the two terms of the 
 answer to the last, in order 1o obtain the answer of this. 
 (Compare formuU.s (A) and (B), Art. 117.) 
 
 Hence, 
 
 V' " ' j 2 ^ ^ 2 
 
 If we add the cinswers of these tvvo questions, we shall 
 
 ha 
 
 ve 
 
 + 2v/^i-y 3 - 2v/^ =2 j ^^ii±^ 
 
 = v^2(v/13 -4-3). 
 Jn the same way we find 
 
 (132.) Before closino; this chapter , we unit shoiv the in- 
 Ur fr elation of the jcllowing syrahols -, — , -. 
 
 We know frum the nature of multiplication, that niuh 
 liplied by a finite quantity, that is, repeated a finite num- 
 ber of times, must still remain equal 0, hence we have this 
 
 condition 
 
 0X^ = 0. (1) 
 
 Dividing both members of (1) by ^, we find 
 
 0=1 (^) 
 
]G4 
 
 SYMBOLICAL EXPRESSIONS. 
 
 Therefore the symbol -- will always be equal to 0, as 
 
 long as .7 is a finite quantity. 
 
 (133- ) Since the quotient arising from dividing one 
 number by another becomes greater in proportion as the 
 divisor is diminished, it follows that when the divisor be- 
 comes less than any assignable quantity, then the quotient 
 will exceed any assignable quantity. Hence, it is usual for 
 
 a 
 
 mathematicians to say, that — is the representation of an 
 infinite quantity. The symbol employed to represent infi- 
 nity is QQ, so that we have 
 
 i^CC- (3) 
 
 (134.) Dividing both members of (1) by 0, we find 
 
 
 A. 
 
 (4) 
 
 This being true for all values of A shows that - is the 
 symbol of an indeterminate quantity. 
 
 To illustrate this last symbol, we will take several exam- 
 ples. 
 
 1. What is the value of the fraction -, when x:=a 1 
 
 ox — ab 
 
 Substituting a for a:, our fraction will become 
 
 I = ::=- =1071 indeterminate qxiantity. 
 
 hx — ah ah — ah 
 
 If, before substituting a for r, we divide both numerator 
 and denominator of the given fraction by x — a, (Art. 55,) 
 we find 
 
 X- — a- X -\- a 
 hx — ah h 
 
SYMBOLICAL EXPRESSIONS. 16.: 
 
 Now, substituting a for x, in this reilucctl form, we find 
 x-\- a a -j- a 2a 
 
 Therefore, — is the true value of --, when x.^a. 
 
 b ox — ab 
 
 y;2 QJ^ 
 
 2. What is the value of , when x = a? 
 
 X- — 2(ix -\- a- 
 
 Writing a for x, we fiml 
 
 x^ — ax a- —a^ 
 
 XT — 2ax + a- a- — 2a- + a- 
 
 If we reduce this fraction by dividing both numerator and 
 denominator by x — o, we find 
 
 x- — 2ax -\- a- x — a 
 Now, writing a for x, in the reduced form, we find 
 
 -!—=-^=:'=(X. (Art. 133.) 
 X — a a — a U 
 
 3. What is the the value of ; — ^^ ; , when x=a? 
 
 bx — ab 
 
 When a is substituted for x, we have 
 
 3^—3aj^-\-3a-x — a'_a' — 3 a^-\-3a^ — a^ _ 
 bx — ab ab — ab 
 
 Reducing by dividing numerator and denominator by x — a, 
 we find 
 
 x^ — Sox^+Sa^x — a^ _ x- — 2ox-l-rt- 
 bx — ab h 
 
 Writing a for x, wc have 
 
 ±=i"f±f.'='i!z^f!±i' = ° = 0. (Art. 132.) 
 6 4 6^^ 
 
I6G 
 
 SYMBOLICAL EXPRESSIONS. 
 
 (135.) From the above, we conclude that whenever an 
 algebraic fraction is reduced to the form -, there exists a 
 
 factor becomes zero for the particular value of the unknown 
 quantity made use of. In the foregoing examples there was 
 very little difficulty in discovering this factor. 
 
 It is obvious that examples of this kind may be chosen 
 where it would be more difficult to find this factor. 
 
 In the fraction, 
 
 a. we shall have for its value 
 
 In this case we do 
 
 not readily discover the factor required j but if we multiply 
 the numerator and denominator each by v^|(a- + x^) -j-x, 
 
 it will become 
 
 ^{(I' — X^) 
 
 We now discover that the factor sought is a — x. Divi- 
 ding numerator and denominator each by a — x, it becomes 
 
 ^c-l-rr) 
 
 Now, when 
 
 v'^(a-(- x')-{-x' 
 a, this last expression will become 
 
 Hence, we conclude that indeterminate expressions of the 
 above kind, when properly reduced, will take one of the 
 following form?. 
 
 = a finite quantity. 
 
 B 
 
 no value. 
 
 — =00= aw infinite quantity. 
 
QUADRATIC EQUATIONS. 16' 
 
 CHAPTER V 
 
 QUADRATIC EQUATIONS. 
 
 (136.) We have already (Art. 66), defined a quadratic 
 equation^ to be an equation in which the unknown quantity 
 does not exceed the second degree. 
 
 The most general form of a quadratic equation of one 
 unicnown quantity, is 
 
 fix'-+ hx-= c. (1) 
 
 Dividing all the terms of (1) by a, (Axiome IV,) we find 
 
 x-^+^r = -^ (2) 
 
 a a 
 
 where, if we assume ^= -, and S = -, we shall have 
 a a 
 
 x'^-\-^x=B (3) 
 
 Equation (3) is as general a form for quadratics as equa- 
 tion (1). 
 
 In (3), ^ and B can have any values either positive or 
 negative. 
 
 (137.) When ^ — 0, equation (3) will become 
 
 x^ = B, (-1) 
 
 which is called an incomplete quadratic equation, since om 
 of the terms in the general forms (1) and (3) is wanting. 
 
168 
 
 QUADRATIC EQUATIONS. 
 
 (138.) When B = 0, equation (3) will become 
 X' -\- Jix = 0, 
 
 which divided by x is reduced to 
 
 x-^Ji=0, 
 which is no longer a quadratic equation, but a simple equa- 
 tion. 
 
 (139.) If ^^ =: and j5 = at the same time, equation 
 (3) will become 
 
 x^=0, 
 which can only be satisfied by taking a;= 0, 
 
 INCOMPLETE QUADRATIC EQUATIONS. 
 
 (140.) We have just seen that the general form of an 
 incomplete quadratic equation is 
 
 = B. 
 
 (1) 
 
 If we extract the square root of both members of this 
 equation, we shall (Art. 96,) have 
 
 x = dzVB. (a) 
 
 Equation (a) may be regarded as a general solution of 
 incomplete quadratic equations. 
 
 (141.) To find the value of the unknown, when the 
 equation which involves it, leads to an incomplete quadratic 
 equation, we have this 
 
 RULE. 
 
 I. Clear the equation of fractions by the same rule as for 
 simple equations. (Art. 70.) 
 
 II. Tlicn transpose and unite the like terms., if necessary., 
 observing the rule under Art. 73, and loe shall thus obtain^ 
 after dividing by the coefficient of x-, an equation of the 
 
QUADRATIC EQUATIONS. 169 
 
 form of a;-= B. Extracting the square root of both mem- 
 bers, we shall Jind x= z^^/B. 
 
 1. Given ^±^+7 = 9, to find a:. 
 
 This, \vhen cleared of fractions, by multiplying by 19, 
 becomes 
 
 a;' 4- 2 4- 133=171, 
 transposing and uniting terms, we find x^=^ 36. If we 
 compare this w'ith our general form, we shall see that 
 B=^ 36. Extracting the square root, we have a:=: it 6, 
 or as it may be better expressed, x = 6 or a: = — 6. 
 
 or- 3 , 1 346 ^ „ , 
 
 2-Given — +- = ^g^,tofindx. 
 
 This cleared of fractions, becomes 
 
 147 -f- 343x'^= 346x2, 
 transposing and uniting terms 3x-'= 147, 
 
 dividing by 3 x^= 49, 
 
 extracting the square root, we find x = ±7. 
 
 3. Given x- -— = 44, to find x. 
 
 do 
 
 Ans. x=: ± 12. 
 
 4. Given S + 5x'-= ^ + 4x-+ 28, to find x. 
 
 Ans. xz= =h 5. 
 
 X- 
 
 5. Given2+~— 7=1--|-13, to findx. 
 o J 
 
 Ans. x = =t 9. 
 (142.) We must be careful to interpret the tlouble sign 
 
 :, correctly, the meaning of which is, that the quantity 
 22 
 
170 
 
 QUADRATIC EQUATIONS. 
 
 before which it is placed may be either plus, or it may be 
 minus. It does not mean that the quantity can be both 
 plus and minus at the same time. 
 
 (143.) If an equation involving one unknown quantity 
 can be reduced to the form a;"=J\'', the value of x can be 
 found by simply extracting the nth root of both members, 
 thus, 
 
 (144.) Where it must be observed (Art. 96.) that when 
 n is an even number, the value of x will be either plus or 
 minus for all positive values of J\^, but for negative values 
 of JV the value of x will be impossible. When n is an odd 
 number, the value of x will have the same sign as ^A'^has. 
 
 (145.) If the equation can be reduced to the form x"'=zJV^ 
 then X can be found by raising both members to the mXh. 
 power, thus : x=^.K"K , 
 
 (146.) Where x will be positive for all values of J\\ pro- 
 vided m is an even number, but when m is an odd number 
 then X will have the same sign as JV. 
 
 (147.) Finally, when the equation can be reduced to the 
 form IL 
 
 We must first involve both members to the mth power, 
 and then extract the ni\\ root, or else we may first extract 
 the nth root, and then involve to the ??ith power. (Art. 98.) 
 
 Thus, 
 
 m 
 X^=J^n. 
 
 EXAMPLES. 
 
 1. Given = — -, to find x. 
 
 v/x + 4 Vx + 6 
 
 This, when cleared of fractions, becomes 
 
QUADKATIC EQUATIONS. 171 
 
 .. -r 34 v/o: + 16S = j^ + 42 v/x + 152, 
 transposing and uniting terms, we have 
 
 8v/x = 16, 
 ilividing by 8, v'a: = 2, 
 
 raising to the second power, x = 4. 
 
 2a 
 
 2. Given y,' x 4- "^ a 4- x ^= , , to find x. 
 
 vfl -(-a; 
 
 This equation, when cleared of the fractions, by multiply 
 ing by '^ a -\- x, becomes 
 
 v'ax -1- x- -{- a -\- x = 2a, 
 v^aa; -}- x'^=a — x, 
 squaring both members, 
 
 ax + X- = a- — ■2ax-\- a;'*, 
 3aa:= a~ 
 a 
 
 3. Given 3 + x^= 7, to find x. 
 
 Ans. a: = -b8. 
 
 4. Given {y" — b) = a — rf, to find y. 
 
 Ans. y=:{{a-dY+by'. 
 
 5. Given ^x—^2 = 16 — v/x, to find x. 
 
 Ans. x = 81. 
 
 6. Given (x -fa) = — -, to find x. 
 
 (x-a)* 
 
 Ans. x = rh(2a2 + 2a6 + 6= 
 
 7 
 
 7. Given ^ = , to find x. 
 
 v/x — >/a: — a ^ ^ 
 
 . «(lzbcr 
 
172 
 
 QUADRATIC EQUATIONS. 
 
 8. Given ^^x+^x— ^x—Vx = -\/—^ — .tofindx. 
 
 2 V X-\-^/X 
 
 9. Given 
 
 Ans. a; = — . 
 16 
 
 = -— -, to nnd X. 
 
 1— N/l_a^i 1+v/l— x2 a;^ 
 
 Ans. a:=zb-. 
 2 
 
 COMPLETE QUADRATIC EQUATIONS. 
 
 (148.) We have already seen, that 
 
 aa;-+fex=c, (A) 
 
 is the most general form of a quadratic equation, vrhere 
 a = the coefficient of the first term ; 
 h = the coefficient of the second term ; 
 c = the term independent of r. 
 If we multiply the general quadratic equation (A), by 4fl, 
 it will become 
 
 4a2a;2 -f 4a6a; = 4rtc. (1) 
 
 Adding 6^ to both members of (1), it becomes 
 
 4aV -I- 4,abx -^b^= b' + 4ac. (2) 
 
 The left-hand member of this equation is a complete 
 square, equal to {2a:x-\-bY. The process by which we so 
 transform an equation as to cause one of its members to be- 
 come a complete square, is called Completing the Square. 
 This may be effected by the following 
 
 RULE. 
 
 Let the quadratic equation be reduced to this form^ 
 ax--\-bx = c. Then multiply each member by four times 
 the coefficient of the first term., after which add to each 
 member the square of the coefficient of the second term. 
 
QUADRATIC EQUATIONS. 173 
 
 1. Complete the square of the equation a;--|-3a?= 4. 
 Multiplying each member by 4, we have 
 
 4:X'-\~ 12a: = 16. 
 Adding the square of 3 = 9, to each member, we find 
 
 4x^4- 12x-f- 9 = 25. 
 The left hand member is now a complete square, equal 
 to {2x -f- 3)'^, so also is the right hand member, 
 
 2. Complete the square of 18x- — 3x = 1. 
 Multiplying each member by 4 X 18 := 72, we have 
 
 1296a,-2— 216x = 72. 
 Adding 3-= 9, to each member we finally have 
 1296x2—216x4-9 = 81, 
 each member of which is a complete square. 
 
 3. Complete the square of 6x2 — 7x= — 2. 
 
 Ans, 144x^— 168x + 49 = 1. 
 
 4. Complete the square of lOx^ — 99x= 10. 
 
 Ans. 400x2— 3960x -|- 9801 = 10201. 
 Raving completed the square of a quadratic equa 
 tion, if we extract the square root of each member, the result 
 will be a simple equation, but as the square root of a quantity 
 may be either positive or negative, it follows that our result 
 will be equivalent to two distinct simple equations. Thus, 
 returning to our general equation, ax'-\- bx = c, which, 
 when its square was completed, became 4:a'-^x--\- 4abx-\- b'= 
 62-|~ 4"C, we have, by extracting the square root of each 
 member, 
 
 2ax-\-b= ±:^b^-\-^ac. 
 If we make use of the + sign, we have 
 
 2ax-f-6 = v^62_|_4„(, 
 
174 
 
 QUADRATIC EQUATIONS. 
 
 If we use the — sign, we have 
 
 2ax -\- 6= — ^/6■^-|-4ac. 
 
 Hence, a quadratic equation must, in general, yield two 
 distinct values for the unknown quantity. The above results 
 give at once 
 
 2a 
 Uniting these values by the aid of the ambiguous sign i, 
 which is read plus or minun, not plus and minus, we have 
 
 — 6d=^6'+4ac 
 2^i ■ 
 
 (B) 
 
 (149.) This may be regarded as a general solution of all 
 quadratic equations, and it is obvious that we may derive 
 from it a general rule which will apply to all quadratic 
 equations, so as not to be under the necessity of actually 
 going through Avith all the preliminary steps of completing 
 the square. The following is such a 
 
 RULE. 
 
 Having reduced the equation to the general form ax^-\- 
 hx == c, we can find x, by taking the coefficient of the second 
 term with its sign changed, plus or minus the square root 
 of the square of the coefficient of the second term increased 
 by four times the coefficient of the first term into the term 
 independent of x, and the whole divided by twice the coeffi- 
 cient of thefi/rst term. 
 
 35 y. 
 
 1 . Given 4x = 46, to find the values of x. 
 
QUADRATIC EQUATIONS. Yl', 
 
 This, when cleared of fractions, becomes 
 4x=— 36+x = 46x. 
 Transposing and uniting terms, we have 
 
 4x2— 45x = 36. 
 This compared with the general form 
 ax^-{- hx=c. 
 gives a = 4; 6 = — 45; c = 36. 
 The square of the coefficient of the second term 
 = (—45)-= 2025. 
 Four times the coefficient of the first term into the term 
 independent of x, 
 
 = 4X4X36 = 576. 
 Therefore, taking the square root of the square of the 
 coefficient of the second terra increased by four times the 
 coefficient of the first term in.o the term independent of 
 T, we get 
 
 rt= v^2025 -f 576 = ± ^2601 = dr 51 . 
 This added to the coefficient of the second term with the 
 sign changed, gives 
 
 45 ±51, 
 which must be divided by twice the coefficient of the first 
 term. Hence, 
 
 45 rb 51 
 
 If we take the upper sign, we get 
 
 . = 1^+^=12. 
 
 O 
 
 If we take the lower sign, we find 
 
 45 — 51 3 
 
 X 
 
 S 4' 
 
 3 
 
 Therefore, x = 12, or 
 
 ' ' 4 
 
 Either of wliich values of x, will verify tie cquUlon. 
 
176 QUADRATIC EQUATIONS. 
 
 2. Given =9 , to find the values of x. 
 
 X — 4 2 
 
 This, when reduced to the general form, becomes 
 
 r»— 18a;=— 72. 
 
 Squaring 18, we get 
 
 (18)2=324. 
 Four times the first coefficient multiplied into — 72, gives 
 
 4X — 72 = — 288, 
 which added to 3^4, gives 36, the square root of which is 
 ±6. 
 
 Therefore, x = — - — = 12 or 6. 
 
 3. Given v/3x— 5 = '^^'+^^^ ^ to find the values of x. 
 
 X 
 
 Squaring both members, we have 
 
 7x2-}-36x 7x + 36 
 
 3x — 5 = . 
 
 x^ X 
 
 This, cleared of fractions, becomes 
 
 3x2— 5x = 7x4-36. 
 
 Transposing and uniting terms, we have 
 
 3x2 — 12x=36. 
 
 This divided by 3, gives 
 
 x2_4x=12. 
 
 ^^ - 4=b^/(4)2 + 4Xl2 4zb8 p _ o 
 Therefore, x = ^\ ~~2~~ ' ^^~^' 
 
 3 3 27 
 
 4. Given 1 =^i to find the values of x. 
 
 X* — 3x ' x2-f- 4x 8x 
 
 This, by reduction, becomes 
 
 9x=^— 7x=116. 
 
QUADRATIC EQUATIONS. 177 
 
 ™, , 7±v^7='+4x 9X116 7±65 , ., 
 
 Therefore, x= ^■— =_^-=:4, or— 3 J. 
 
 X'-\- 12 X 
 5. Given — 1— - = 4x, to find x. 
 
 Tliis reduced, becomes 
 
 x' — '7x= — l2. 
 
 T,. , 7±^^72 + 4x— 12 7±1 
 Therefore, x = ~ -= — - — = 4, or o. 
 
 (150.) An equation of the form 
 
 ax^-{-hx" = Cj (A) 
 
 can be solved by the above rule, which indeed will agree 
 with the form under consideration in the particular case of 
 71 = \. 
 
 If, in the above equation, we write y for a:", and conse- 
 quently 3/' for x'^, it will become 
 
 which is precisely of the form of (A), Art. 148. Conse- 
 quently, 
 
 "= i;r^- 
 
 Re-substituting x^ for y, we have 
 
 „_ — 6d=^fc^ + 4ac 
 ^ 2^ ' 
 
 This value of x, must hold for all values of the constants 
 n, a, 6, and c, whether positive or negative, integral or 
 fractional. 
 
 23 
 
178 
 
 QUADRATIC EQUATIONS. 
 
 EXAMPLES. 
 
 1. Given ar*-]-ax^ = 6j to find x. 
 
 This becomes y'^-\-ay = 6, when for x* we write y. 
 
 — a± ^a\-\-4:b 
 •'•3/ = H — = ^' 
 
 hence, 
 
 — a=t^a-+46 
 
 ' 2. Given Sx** — 2a:" = 8, to find x. 
 
 n 2 ±10 _ 
 
 x^ = -^ = 2, 
 
 3. Given2(l+a: — x-)— ^l + x — o:^ — -, to find 
 
 If for 1 -f- ^ — 2;2, we put y', our equation will become 
 2f-y=-l, 
 or 
 
 I8y'-9y = -l, 
 9=b3 1 1 
 
 y = 
 
 36 
 
 3'"' 6' 
 
 hence 
 
 2^= 9'°^' 3-6- 
 
 Re-substituting 1+x — rr^, for j/^, we have, when we 
 take the first value of y", 
 
 1 
 
 1 +x — x^= 
 
 9' 
 
 9x2— 9x = 8, 
 9±3v/41 1 , 1 
 
 18 
 
 :+^^41,or?-1^41. 
 
QUADRATIC EQUATIONS. 179 
 
 When vre take the other value of y'^, we have 
 
 or 36i'— 36t = 35, 
 
 36i24v'n 1 , 1,,, 1 1 „, 
 ■■■' = 72 =2 + 3^"'"2-3^"- 
 
 Collecting these four values of x, we find 
 a:=i — Av/41, 
 
 4. Given ja;2 — -J ~\-la" — ^1 = -, to find the \'ia- 
 
 lues of X. 
 
 This equation is easily put under this form 
 
 - va-x* — a*=x' vx* — a*. 
 
 X X 
 
 This squared, becomes 
 
 a*— - = X*— 2ax Vz'—a'^ a^x^—-. 
 x' ^ x-i 
 
 By transposing, we have 
 
 x* — a* — 2aa: Vx* — a*-\- a'^x^= 0. 
 
 Extracting the square root, we find 
 
 >/a:' — a* — aa: = 0, 
 or v^x^ — a*= ax. 
 
 Squaring, we find 
 
 
 a:^— a*=a2x2, 
 
 or 
 
 x*—a'^x^= a\ 
 
 Hence, 
 
 2 
 
 Consequently, 
 
 
180 
 
 QUADRATIC EQUATIONS. 
 
 = -f-^^-^/ 
 
 (151.) We have seen that the general form of a quadra- 
 tic equation, ax'-j- bx =- c, gave, for the value of the un- 
 known, the following expression : 
 
 — h ±v/62_L4ac 
 
 a:= , 
 
 2a 
 
 When a = 1, the equation ax~-\-hx = c, becomes 
 
 x'-\-bx=c. (C) 
 
 And the above expression for the unknown, will become 
 
 — &±v^6-+4( 
 
 l-v^g+ 
 
 (D) 
 
 Now, since all quadratic equations may be made to assume 
 the form of (C), by dividing all the terms by the coefficient 
 of a:-, it follows that formula (D) must, when properly 
 translated into common language, give a general rule for the 
 solution of all quadratic equations. The following is the 
 
 RULE. 
 
 Having reduced the equation to the form x^ -\-hx = c,we 
 can find x by talcing half the coefficient of the second term, 
 with its sign changed; plus or minus the square root of the 
 square of the half of the coefficient of the second tertn in- 
 creased by the term independent of x. 
 
 EXAMPLES. 
 
 1. Givena:*—10x = — 24, to finder. 
 
 In this example, half the coefficient of the second term is 
 5, which squared and added to — 24, the terra independent 
 of X, is 1. Extracting the square root of 1, we have ±1. 
 
 Therefore, x = 5 i 1 = 6, or 4. 
 
QUADRATIC EQUATIONS. 181 
 
 X 7 
 
 2. Given =- -, to find x. 
 
 a: + 60 3x — 5 
 
 This cleared of fractions, becomes 
 
 3a;^ — 5a:z=7a:+420. 
 Transposing and uniting terms, we have 
 
 3x- — I2x = 420. 
 Dividing by 3, we have 
 
 x2 — 4z=140, 
 . • . a; = 2 ± 12 = 14, or — 10. 
 
 ^ „. x + 12 , X 26 ^ . , 
 
 3. Given = — , to find x. 
 
 X ^x+12 5' 
 
 Ans. a; = 3, or — 15. 
 
 4. Given 3x^ -}- 42x3 = 3321, to find x. 
 
 Ans. 3, or (—41)*. 
 
 (152.) Equations containing two or more unknown 
 quantities, which involve in their solution 
 
 quadratic EQUATIONS. 
 
 Given < ^ , Jl^^^ r ? to find x and y. 
 I y'i — 7^ = 90000. S 
 
 From the first of these equations, we find 
 ^__300y_ 
 y— 125' 
 Substituting this value of x in the second equation, it be 
 
 y-_,J^r= 90000. 
 
 y-125 
 Which, when expanded, is 
 
182 
 
 QUADRATIC EQUATIONS. 
 90000/ 
 
 = 90000. 
 
 "^ / — 2b0y + 15625 
 This, cleared of fractions, and terms united, becomes 
 
 yi _ 250 f — 164375/ -\- 22500000^/ = 1406250000. 
 This may be written as follows 
 
 (/ — 125y)- — 180000(/— 125^/) = 1406250000. 
 Solving by rule for quadratics, considering / — 125t/ as 
 the unknown quantity, we have 
 
 Hence, 
 
 125y = 90000 d= 97500. 
 
 1251/ = 187500, or /— 125y= — 7500. 
 
 The first of these gives 
 
 125 db 875 
 
 :500, or —375. 
 
 The second gives 
 
 125 ±25x^—23 
 
 Both of which values are imaginary. 
 
 Having found y, we can substitute it in the equation 
 _ Z00y_ 
 
 ^— ^pri25' 
 
 and thus obtain the values of x. 
 
 2. 
 
 Given 
 
 
 x^= 
 -ex'' 
 
 
 (1)? 
 (2)5' 
 
 to find 
 
 X and 
 
 y 
 
 From (2) 
 
 , we get 
 
 
 
 
 
 
 
 
 
 
 x* 
 
 /- 
 
 c' 
 
 
 (3) 
 
 
 bic 
 
 h substituted in 
 
 (1), 
 
 we have 
 
 
 
 
 
 
 
 y'- 
 
 
 ay 
 
 — a? 
 
 
 (4) 
 
 
 
 /- 
 
 -2cy'4- 
 
 c» 
 
 
QUADRATIC EQUATIONS. 183 
 
 Clearing (4) of fractions, it may then be put under the 
 form 
 
 (y6 _ cy^y. — 2a- {y" — cy^)= a'-c-. (5) 
 
 Solving this by quadratics, considering y^ — cy^ as the un- 
 known quantity, we have 
 
 3/6 _ cy3 = a2 dr a ^a* -f c-. (6) 
 
 Again, solving (6) by quadratics, considering y^ as the un- 
 known, we have 
 
 Extracting the cube root of (7), it becomes 
 
 The value of y, (8), or better the value oi y^, (7), when 
 substituted in (3), will give x. 
 
 v-\-w-{-x-^y-\-z= 56 (!)■ 
 
 .vw — x — y — z= 207 (2) 
 
 3. Gi\'en<(wx — v — y — z= —9 (3) 
 
 \xy — V — w — z= — 19 (4)1 
 
 yz — V — w — x= 38 (5) 
 
 (8) 
 
 , to find V, ty, 
 ar, y, and z. 
 
 vw-\-v + w = 263, (6)=zz(l)-|-(2) 
 
 y,x-{-w-{-x= 47, (7)=(l)-f-(3) 
 
 xy-{-x + y= 37, (8)=(l)4-(4) 
 
 yz^y-\-z= 94. (9)=(l)+(5) 
 
 By adding a unit to both members of equations (6), (7), 
 
 (8), (9), they may be put under the following forms : 
 
 (v -|-l)(wj+ 1)=264, (10) 
 
 (tx^-f l)(x+l)= 48, (11) 
 
 (x+l)(y-f 1)= 38, (12) 
 
 (y + 1)(2 + 1)= 95. (13) 
 
184 
 
 QUADRATIC EQUATIONS. 
 
 If we add 5 to both members of (1) it may be written 
 as follows : 
 (r+l)4-(u.-4-l)-f-(x+l)+(y+l)+(z+l)=61. (14) 
 
 We shall now use equations (10), (11), (12), (13) and 
 (14), which are symmetrical instead of the original equa- 
 tions. 
 
 264 
 
 w-\-l 
 
 v-\-r 
 
 48 2 
 
 (15)=(10)^(v + 1) 
 
 + i=i;rqrr=fi^^+^^' (i6)=(ii)-(t/;-fi) 
 
 38 209 
 
 3/+1 
 
 + 1 v+l' 
 
 (17)=(12)^(x+l) 
 
 z^l=^=Uv-\-l). (I8)=(13)--(y + l) 
 y-f-l 11 
 
 Substituting these values oft« + l, x-\-lj y+l, - + 1) 
 in (14), we have 
 
 264 
 
 209 
 
 This reduces to this form, 
 18, 
 
 ^^;(.+ l) + iI3_61. 
 
 (20) 
 
 Clearing of fractions, we have 
 
 I8(v+1)2 — 671 (v + l)= — 5203. (21) 
 
 This quadratic solved, gives 
 
 v+l = 11, or 26/3. 
 
 These two Values of v+ 1> ^^ing substituted in (15) 
 
 (16), (17), (18), will give two sets of values for w; + 1 
 
 x-^l, y+lj s+1. These values when found are, 
 
 v-f 1 = 11, or26T\. 
 
 M,-|-l=24, 10y»3. 
 
QUADRATIC EQUATIONS. 
 
 185 
 
 a: + 1 = 2, or 41; 
 y + 1=^19, or 7ii. 
 z-f-l= 5, or lly^. 
 
 
 
 Ay=25^ 
 
 V 1^=23/ 
 
 
 V'-= ^^3 
 
 h= 1,V 
 
 or 
 
 <^^= 31. 
 
 ;3/ = isA 
 
 
 )y= 611 
 
 r= 4, ) 
 
 
 ( z = 10|^ 
 
 4. Given x'^"—2x^"-\-x"=.6, to find a:. 
 This is readily put under this form 
 
 If we make y = x^" — a;", equation (1) will become 
 
 r— 2/ = 6, (2) 
 
 .••y = i±^ (3) 
 
 Re-substituting for y, we have 
 
 X-"— a.'" = 3, (4) ) 
 
 or x-"—x''—--2. (5) \ 
 
 Now, in (4) and (5) substituting c for a:", and we have 
 
 z'-z = 3, 
 
 From (6), we have 
 
 c= > ± V13. 
 From (7), we find 
 
 Re-substituting x" for c, we find 
 
 a:"=i± ^n/13, 
 
 a:"= i db 5 v^ — 7. 
 Taking the 7ith roots of (10) and (11), we find 
 
 (G)( 
 
 (8) I 
 {9)S 
 
 (10) 
 (H) 
 
 Ans 
 
 24 
 
186 QUADRATIC EQUATIONS 
 
 5. (jrnen ^ ' -^ ^ ^ 
 
 x^+r'=6 (2) 
 
 Squaring (1), we have 
 
 x'-\-2xy-\-y''=a~. 
 Subtracting (2) from (3), we get 
 2xy = a- — b. 
 Subtracting (4) from (2), we find 
 
 ■j^ — 2xy-\-y' = 26 — a-. 
 Extracting the square root of (5) , we get 
 
 x — y—± N/2fe — a". 
 Taking half the sum of (1) and (6), we get 
 
 to find X and y. 
 
 a 1 
 
 x = -:k-^2b — a^. 
 2 2 
 
 Subtracting (7) from (1), we find 
 
 (I 1 /— , :, 
 
 •^ 2 2 
 
 (3) 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 (x + y = « (1)} 
 
 6. Given < , ^ , >, to find a; and y. 
 
 (a-3-j-3/^' = 6 (2)) 
 
 We will indicate our operations upon the successive equa 
 tions, by the method explained under Art. 80. 
 
 'ixy[x-^y) = iv^ — h. 
 
 n-' — h 
 3xy — . 
 
 a-y = 
 
 3a 
 
 x^+2xy-\-y'' = a- 
 
 4a> — 46 
 ^'^ = —30-- 
 
 (3)-(iy 
 (4)=(3)-(2) 
 
 (5)=(4)-(l) 
 (6)=(5)--3 
 
 (S)= (6)X4 
 
QUADRATIC EQUATIONS. 187 
 
 4/) /j3 
 
 ^-2xyJry' = —^^. (9)=(7)- (8) 
 
 (46 — flsH 
 
 3c 
 
 in)J-2Wm 
 
 3a \ • ' ' 2 
 
 a 1 46-a3 i (i)_(io) 
 
 ix+y = a (1) 
 
 7, Given { ^ 1 If find x and v. 
 
 ar^+4r^i/+6xV+4x3/3+y =:a^ (3)=(1)^ 
 
 4x1/ (a-2+/)+6a:y = a^-h. (4)=:(3)— (2) 
 a--^+2x3/+y-2=a2. (5)=(1)^ 
 
 Transposing 2xy of (5)\ve get 
 
 .x-^4-7/-^=:a-^— 2x3/. (6) 
 
 Substituting this value of x- -j- 3/* in (4)^ we get 
 
 4x7/(a2 — 2xy) +6xV' = «' — ^- C' ) 
 
 This becomes, by putting z for xy, and transposing, 
 
 2c2— 4a'r = 6 — a', (8) 
 
 ■.•. = «=±v/3^'. (9) 
 
 Hence, 
 
 X2/ = a-^±\/-+-. (10) 
 
 4x3/ = 4a^±4\/yii\ (Il)=(l0)x4 
 
188 
 
 QUADRATIC EQUATIONS. 
 
 
 X — y = =h 
 
 =1^1 -3.= T4/l±^«r. (15)=W^) 
 
 8. Given 
 
 ( x- — y= = a (1) 
 Ix'y-^xf^h (2) 
 
 a^* + y- = - • 
 
 h , a 
 2xy^2 
 
 to find a:, and y. 
 
 {'i)={2)-rxy 
 
 (4) 
 
 __(3)+(l) 
 
 2xy 2 
 
 
 4a;Y 4 
 This read.ly gives, 
 
 4x'2/' -{- cL-x'-y- = b. 
 Consequently, 
 
 ^ ' 2 
 
 (6)=(4)X(5) 
 
 xy = ± ^ — '. j 
 
 /— rt^x/fl'-f-166^ 
 
 (8) 
 
 This value of xy, introiluced into (4) and (5), we obtain 
 2 
 
 x= izl doh 
 
 ^^/a^-\-16b! 2 
 
 (9) 
 
QUADRATIC EQUATIONS. 189 
 
 -bl^ g V-^l^ (10) 
 
 Cxy+a:V=135 (1) ? . . , 
 
 x'Y +2xV + xV" =18225 (3)==(1)- 
 
 2xY = 13122 (4)=(3)_(2) 
 
 a:Y = 6561 (5)=(4)-^2 
 
 xy=±3 (6)=-V(5) 
 
 xY = ±:21 (7)=(6)3 
 
 a:^-f-y^ = ±5 (8)=r(l)-^(7) 
 
 2x3/= ±6 (9)=:(6)X2 
 
 a:^-2x2/4-y^= =F 1 (10)=(8)-(9) 
 
 X— y=:±%/— lorrhl (11)=>/(10) 
 
 x-' + 2x3/ + y^=ill (12)=(8)+(9) 
 
 x + 7/ = ±v/ll orrbv^— 11 (13)=:v/(12) 
 
 x=l{±Vll±V—^)or i{±^^-n±i) (i4)=^-^-ii^^ 
 
 , , (13)— (11) 
 
 i/=i(±:v/ll=Fv/— Dor J(±v/-11=F1) (lo)=^ ~ 
 
 10 Given ^^'^"'^'"^'^""^ ^\^ ? .tofind xandy, 
 
 '''•'''''" ^i/x(2/x-f-l)-x^4-x=6 (2)5' 
 
 Subtracting (1) from (2), we find 
 
 y'^r^ -{- yx — yx^ = 3. (3) 
 
 Dividing (3) by (1), we get 
 
 y=l. (4) 
 
 This value of y substituted in (1), gives 
 x=3. 
 /xy— 8xV+16x^ N 
 
 \ =90x.y+60(x-2/0-720(y-l) (1)( 
 11. iiiven< / o . I .X lo r ■■ 
 
 ) (r— 4y-H)x _^__12 ^2)V 
 
 ( 6 X J 
 
 to find X and y. 
 
190 
 
 QUADRATIC EQUATIONS. 
 
 Multiplying (2) by bx{y'^ + 4y + 4), it becomes 
 
 I =15z^'^+60xy+60x— 60/— 240?/— 240 ) ^ ^ 
 
 Subtracting (1) from (3), we have 
 
 = Ibxy" — 30x2/ + 48O3/ — 960. (4) 
 
 Dividing (4) by Ibxy + 480, it becomes 
 
 0.= y-2, (6) 
 
 .•.y = 2. (6) 
 
 This value of y substituted in (2), gives 
 
 x=4. (7) 
 
 rxy+z=5 (1)^ 
 
 12. Given ? xyz + 2'^=]5 (2) > 5 to find x, y 
 
 ( xy*-j-x2y— 2x-i-2c=8 (3) ) and z. 
 Dividing (2) by (1), we find 
 
 z = 3. (4) 
 
 Substituting this value of z in (1) and (3) and they be- 
 
 xy = 2. 
 
 
 (5) 
 
 xy(x+2/) = 2 + 2x. 
 
 
 (6) 
 
 Dividing (6) by (5), we find 
 
 
 
 xH-y=l-f-x, 
 
 
 (7) 
 
 .'. y=h 
 
 
 (8) 
 
 Dividing (5) by (8), we get 
 
 
 
 x=2. 
 
 
 (9) 
 
 rx{y+z) = a (1)) 
 
 
 
 13. Given] 2/(2:+-) = & (2) > 
 
 to find X 
 
 , y, and z 
 
 (z{x-\-y) = c (3)) 
 
 
 
 Before proceeding to the solution of these equations, we 
 will remark, that they are symmetrica], and consequently 
 
QUADRATIC EQUATIONS. 191 
 
 all the derived equations will either contain all the letters 
 similarly combined, or else they will appear in systems of 
 three equations each, which can be deduced from each other 
 by simply permuting. 
 
 If we take the sum of (1), (2), and (3), after expand- 
 ing them, we shall have 
 
 2xy-{-2yz-^2zx=:a-\-b~^c. (4) 
 
 In this equation all the letters enter symmetrically ; 
 therefore it will not give rise to any new equation by per- 
 mutation. 
 
 If we subtract twice (3) from (4), we get 
 
 2xy = a-\-b-~c. (5) 
 
 By permutation, we derive from (5) these two equations : 
 2yz = b-\-c-a. (6) 
 
 2zx=c-\-a — b. (7) 
 
 Equations (5), (6), and (7) readily give 
 
 a-\~b — c 
 ^V = -^ (8) 
 
 y-.^'j^. (9) 
 
 .. = i±p.^ (10) 
 
 Taking the continued product of (8), (9) and (10), we 
 have 
 
 ^a..a.i \ a + 6— c ) ^^ ( b-\-c—a ) ^^ ( c-\-a—b 
 
 X -V- X --.— • (11) 
 
 ( ^ ) ( ^ ) ( 2 
 
 This equation containing all the letters symmetrically com- 
 bined, can give no new condition by permutation. 
 Dividing (11) by the square of (9), wc have 
 (a+6-c)(c4-a-6) 
 
 ^ 2{y^^ ^^^^ 
 
192 
 
 QUADRATIC EQUATIONS. 
 
 By permuting, we derive from (12) these two equations : 
 
 r 
 
 (64-c — a)(a+6 — c) 
 
 (13) 
 (14) 
 
 2(c+a — &) 
 
 ^.- (c+^-&)(^+c-q) 
 2(a+6 — c) 
 
 Taking the square roots of (12), (13) , and (14), we find 
 (a^_6_c)(c+o — 6) ) i 
 
 a; = rh 
 
 y = ± 
 
 2(6+c — a) 
 (6-fc — fl)(a-|-6 — c) 
 
 2 = rb 
 
 (15) 
 (16) 
 (17) 
 
 2(c+a — 6) 
 
 (c+g — 6)(6-fc — g) ) ^ 
 ^ 2(g-f6-c) ^ • 
 
 This question is a good illustration of the beautiful method 
 of deriving one quantity from another, of a similar nature, 
 by simply permutating. 
 
 14. Given, the two equations 
 (x'-f x")(l+x'a;"+a:'V'+x'a:"2+a:'-x"«)-fx'a:" = g, 
 x'x"{x' -^x"){x'-\-x"-\-x'x"){x'-\-x"-^x'x"+x'-'x" -\'x'x"") I 
 
 to find x' and x". 
 
 If, in these equations, we make successively the substitu- 
 tions x'4- x" = 3/', x'x" = y" ; y'+y"= 2', y'y"= z" ; 
 z'^z"=w', z'z"==w", we shall finally have 
 w'-\-w" = 0, 
 w'w" = b. 
 
 The quantities sought, x', x", will be determined by means 
 of these four quadratic equations : 
 
 w^ — aw-\-b =0. 
 z -^w'z-\-w"=0. 
 
QrADRATIC EQIATIOXS. 193 
 
 cc- — y'x -^y" = 0. 
 The first of tliesc equations determines w' iind lo" ; the se- 
 cond z' and z" ; the third y' and y" ; and, finally, the 
 fouvth x' and x" . We thus suciessively obtain 
 
 a ± n/^;~- 
 
 -46 
 
 2 
 
 
 _.„,'_ts/i«'^ 
 
 — 4:10" 
 
 2 
 
 
 C'±v/C'^ 
 
 -Az" 
 
 2 ' 
 
 w' 
 
 (1 =FN^a- — 46 
 
 2 
 
 ^1 
 
 _2o'=FN^w'-— 4t/;" 
 
 
 2 
 
 y' 
 
 ^_^'z:pVz'-^—4:z" 
 
 2 
 
 x' 
 
 ,_y^^y'— 4.v" 
 
 ,_iy' ±^Vj— 4//" 
 ''- 2 ' " 2 
 
 and there are, consequently, for a:', as well as for a;", six- 
 teen different values. If we had solved the first two equa- 
 tions by the common method, we should, after a laborious 
 elimination, have obtained an equation of the 16th degree. 
 If a = 371, and b = 13530, then will one set of values 
 be, x' = 2 and x" = 3. 
 
 Cx'+xy-]-y'=a', (1)^ 
 15. Given ^ y--\-yz -f- z*=6^, (2) > to find a-, y, z. 
 iz'-\-zx-\-x''=c\ (3)) 
 
 =a2+62-f-c^ 
 
 (4)=(l) + (2)+(3) 
 
 4(x^-i-2/^-|-2^f+4(a:-^+2/^-|--^)(2-y+yz-fca:) ) ,-._,, ^ 
 
 -^{xy+yzJr~:ty^{a' + lr+c'y. ^ ^^>'-^'*^ 
 
 r'+3aV + 2/-* + 2a:'y-{-2y^x=:a'. (6)=(1)^ 
 
 y^+33/V + c' + 2i/3z + 2z^3/^6^. (7)=(2)^ 
 
 c-* + -iz^x'- + r' + 2z^^a: + 2x'»z = c\ (S)=(3)- 
 
 irx'-j-y'^z'y-\^{T^-\-y^-^z%xy+yz-^zx) ) (9):=2(6) 
 
 -2(x3/+y;=+zx)^=2(aM-6' + c^). i +2(7)+2(8) 
 25 
 
194 
 
 QUADRATIC EQUATIONS. 
 
 (10)=(5)-(9) 
 
 or, which is the same thing, 
 
 {:ri/+yz+zxf=?,{a:^h'^})'c'^c'a^)-^{a^+¥-\~c^). (11) 
 {xy- {-yz-{.zx)-= ) 
 
 6(.T2/ + 7/c4-cx)-3:6/c. (13)=(12)X6 
 
 (14)=^4)X2 
 
 4(x+.r/+c)^2K+6'^+c^)+6/c. (15)=(14)+(13) 
 
 2(a:+2/+c)=± V2{a'-\-b''~\-c^6k. ( 16)=v/(15J 
 2(x^ + r + - + x:v-Hz-f c.) 1 .n)=(4)+(12) 
 
 2x{x^y-i-z) = a'—b^-\-c'+/c. (I8)=(17)-(2)x2 
 
 x= -^!=L±SL^. (19)^(18)^(16) 
 
 ± v'2(a2+62^c3)+6/c 
 
 Having found the value of x, we may find the values of y 
 and c, by simply permuting the letters in the above expres- 
 sion, (19). Since the expression for k is symmetrical, i* 
 must remain constantly the same. Consequently the deno- 
 minator of the expression for x, (19), will not, during this 
 permutation, change its value. 
 
 In this way we find 
 
 y 
 
 db ^/2(a'-{-62-^-c•*)+6A:' 
 
 (20) 
 (21) 
 
QUADRATIC EQUATIONS. 
 
 195 
 
 
 F 
 
 
 "o 
 
 
 C 
 
 ^^ ^ 
 
 1+ 
 
 ,_^ 
 
 + 
 
 I'- 
 
 + 
 
 l"^ 
 
 i-c 
 
 <o 
 
 j'-O 
 
 r« 
 
 ■<5 
 
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 + 
 
 1+ 
 
 '4- 
 
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 i^ 
 
 's 
 
 55 
 
 j3 
 
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 ^ ;+ 
 
 
 + 
 
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 + 
 
 
 « 
 
 1 1 
 
 — 
 
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 _ 
 
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 -= « 
 
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 — 1 'Z^^ 
 
 1^ 
 
 ^ 
 
 ?7~" 
 
 -In 
 
 ^ 
 
 're : ,■ 
 
 
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 c 
 
 , 
 
 1 '^ 
 
 •^ : 1 
 
 ! '^ 
 
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 ^ 
 
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 1- 
 
 
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 1« 
 
 + 
 
 Js 
 
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 \% 
 
 ■■^ 
 
 ^ 
 
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 ''■ , 
 
 
 h 
 
 4- 
 
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 4- 
 
 3. i-^ 
 
 i jc ■ ] 
 
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 Si 
 
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 ^ + 
 
 
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 ^ 
 
 
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 ■P 
 
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 <s 
 
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 + 
 
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^^^^ QUAUIfATIC KQLATIO.XS. 
 
 ^^ e have cho.en this example, partly from its being one 
 rather .hfficult of solution by the ordinary methods, and 
 partly because it affords an excellent opportunity for exem- 
 plifying the beauty of syn.metrical equations. Equations 
 U , (2), and (3), which are given, are not only symmetrica], 
 but they are also homogeneous. Consequently all our de- 
 rived equations will be homogeneous, and will either con- 
 tain all the different letters similarly involved, as in (4) (51 
 (9), (10), (11), (12), (13), (14), (15), (16), (17), and (IS), 
 or else there will be a system of three equations which can 
 be deduced from each other simply by permutating the let- 
 tcrs, as IS the case with the given equations (1), (2), and 
 (3), also equations (6), (7), and (8). Equations (19), 
 (20), and (21), a,e also of this nature. This perfect sym- 
 " etry of expressions, must in a great measure serve as a 
 check upon our work, preventing errors which otherwise 
 could not be so readily detected. 
 
 (153.) Questions which require for their solution a 
 
 KNOWLEDGE OF QUADRATIC EQUATIONS. 
 
 1. A widow possessed 13,000 dollars, which she divided 
 into two parts, and placed them at interest, in such a man- 
 ner that the incomes from them were equal. If she had 
 put out the first portion at the same rate as the second, 
 she would have drawn for this part 360 dollars interest j 
 and if she had placed the second out at the same rate as 
 the first, she would have drawn for it 490 dollars interest. 
 What were the two rates of interest 1 
 
 Let X = the rate per cent, of the first part. 
 
 Let 3/ = the rate per cent, of the second part. 
 
 Now, since the incomes from the two parts were equal 
 
QUADRATIC EQUATIONS. 197 
 
 they must have been to each other reciprocally as .r to y. 
 Hence, if my denote the first part, then will mx denote the 
 second part. 
 
 We shall then have 
 
 „j (:c _]- y)=13000. 
 
 13000 
 Consequently, m = '^JTy 
 
 iSOOOy ^, r. , , 
 Therefore, -^ = t^^^ first part. 
 
 iSOOOo: , , 
 
 = the second part. 
 
 x-\-y 
 
 The interest on these parts, at y and x per cent., respec- 
 tively, is 
 
 I30v- , 130x2 
 — -^ and — ; — . 
 x-\-y x-\-y 
 
 Hpnce, by the conditions of the question, we have 
 
 122^=360. (1) 
 
 X+J/ 
 
 1^=490. (2) 
 
 x+y 
 
 Dividing (2) by (1), we get 
 
 t-^1 . (3) 
 
 3/2-36- 
 
 Extracting the square root of (3), we have 
 
 y 6 
 
 Subtracting (1) from (2), we have 
 
 130(.2-/)^^3^ ^5^ 
 
 x-\-y 
 Dividing boUi numerator and denominator, of the left-hand 
 
198 
 
 QUADRATIC EQUATIONS. 
 
 member of (5), by a'+y, and also dividing both members 
 by 130, we get 
 
 --r-2/=l. 
 
 (6) 
 
 Dividing (6) by t/, we find 
 
 
 ^—1 = 1 
 
 y y 
 
 (7) 
 
 Subtracting (7) from (4), we have 
 
 
 -i-f 
 
 (8) 
 
 Clearing (8)of fractions, we obtain 
 
 
 6y =^1y — 6. 
 
 (9) 
 
 ••• 3/=6. 
 
 (10) 
 
 Adding (10) and (G), we get 
 
 
 x = 7. 
 
 (11) 
 
 Therefore the per cent, of the first part w^as 7, and of the 
 second part was 6. 
 
 2. A certain capital is out at 4 per cent. ; if we multiply 
 the number of dollars in the capital, by the number of dol- 
 lars in the interest for 5 months, we obtain $117041|. 
 What is the capital ? Ans. $2650. 
 
 3. There are two numbers, one of which is greater than 
 the other by 8, and whose product is 240. What numbers 
 are they 1 Ans. 12 and 20. 
 
 4. The sum of two numbers is = a, their product = b. 
 What numbers are they ? 
 
 Ans. fl + v^(" -' — 4ft) a — V{a' — U) ^ 
 
 5. It is required to find a number such, that if we multi 
 ply its third part by its fourth, and to the product add 5 
 
QUADRATIC EQLATIONS. 109 
 
 times the number required, the sum exceeds the number 200 
 by as much as the number sought is less than 280. 
 
 Ans. 48. 
 
 6. A person being asked his age, ansAvcred, " My mother 
 was 20 years old when I was born, and her age multiplied 
 by mine, exceeds our united ages by 2500." What was 
 his age ? ^ Ans. 42. 
 
 7. Determine the fortunes of three persons, A, B, C, from 
 the following data : For every $5 which A possesses, B 
 has $9, and C $10. Farther, if we multiply A's money 
 (expressed in dollars, and considered merely as a num- 
 ber) by B's, and B's money by C's, and add both product? 
 to the united fortunes of all three, we shall get 8832. 
 How much had each ? 
 
 Ans. A $40, B $72, C $80. 
 
 8. A person buys some pieces of cloth, at equal prices, 
 for $60. Had he got three more pieces for the same sum, 
 each piece would have cost him $1 less. How many pieces 
 did he buy 1 Ans. 12. 
 
 9. Two travellers, A and B, set out at the same time, 
 from two different places, C and D ; A, from C to D ; and 
 B, from D to C. On the way they met, and it then appears 
 that A had already gone 30 miles more than B, and, accord- 
 ing to the rate at which they travel, A calculates that he 
 q^n reach the place D in 4 days, and that B can arrive at 
 the place C in 9 days. What is the distance between C 
 and D ? Ans. 150 miles. 
 
 fifth powers 17050. What are the numbers ? 
 
 Ans. 3 and 7. 
 
200 
 
 QUADRATIC EQUATIONS. 
 
 11. The sum of two numbers is 47, and their product 
 546. Required the sum of their squares. 
 
 Ans. 1117. 
 
 12. The sum of two numbers is 20, and their product 
 99. Required the sum of their cubes. 
 
 Ans. 2060. 
 
 13. Divide the number a into two such parts, that the 
 sum of their reciprocals may equal b. What are the parts 1 
 
 Ans 
 
 
 9 
 14. Divide - into two such parts, that the sum of their 
 2 ^ ' 
 
 reciprocals may equal 1. What are the parts 1 
 
 Ans. 3 and -. 
 
 15. Given the sum of the squares of two numbers =a. 
 and the sum of their reciprocals = 6; to determine the num- 
 bers. 
 
 Sum of numbers = TaZ'- + 2 ± 2(a62_|_ i)n ', 
 
 Ans 
 
 Difference 
 
 lb 
 
 ■ 2^2{ah- + l) 
 
 J 
 
 16. Find the values of x from the equation 
 3x4-25 7 
 
 21 + 2X 
 
 7 + a:. 
 
 Ans. X = — 7, or — 6 i . 
 
QUADRATIC EQUATIONS. 201 
 
 17. Find the values of x from the equation 
 
 13 ^ 4 
 
 Ans. x=:2 rb^ — 1. 
 
 18. A and B can together perform a piece of work in 
 two days, and it wouhl take A, alone, three days longer to 
 perform it than it would B alone. In what time can A and 
 B respectively perforin it 1 
 
 . ^ A would require 6 days. 
 '■<B " " 3 " 
 
 19. A, B, an^ C agree to contribute $730 towards build- 
 ing a school-house, which is to be at the distance of 2 miles 
 from A, and f of a mile further from C than from B. They 
 agree that their shares shall be reciprocally proportional to 
 their distances from the school-house. When it was found 
 that A paid $98 more than B paid. What was B's distance 
 from the school-house ? Ans. 2} miles. 
 
 20. I have a certain number in my thoughts ; this I mul- 
 tiply by 2 J, add 7 to the product, multiply this sum by 8 
 times the number ; I then divide by 14, and from the quo- 
 tient subtract four times the number, and thus obtain 2352. 
 What number is it 1 Ans. 42. 
 
 21. Find two numbers such, that their sum and product 
 
 together may be = 34, and the sum of their squares exceed 
 
 the sum of the numbers themselves by 42. What are the 
 
 numbers 1 
 
 4 and 6 ; or, 
 Ans. ' 
 
 i(— 11 -l-v/— 59), i(_ll_v/_59). 
 
 22. It is required to find a number, consisting of three 
 digits, such, that the sum of the squares of the digits, with- 
 out considering their position, may be = 104 j but the 
 26 
 
202 
 
 QUADRATIC EQUATIONS. 
 
 square of the middle digit exceeds twice the product of the 
 other two by 4 ; further, if 594 be subtracted from the num- 
 ber sought, the three digits become inverted. What num- 
 Der is it ? Ans. 862. 
 
 23. Find two numbers such, that their sum, their pro- 
 duct, and the difference of their squares may be equal. 
 
 Ans. ^±W5',^,±W5. 
 
 24. What two numbers are they, whose sum is 3, and 
 the sum of whose fourth powers is 17 1 
 
 ( 2 and 1 ; or, 
 
 Ans. \ , , 
 
 ; J(3 4-n/— 55), and J(3— v^— 55). 
 
 25. What two numbers are they, whose product is 3, and 
 the sum of whose fourth powers is 82 ? 
 
 ( zir 1, and dr 3 ; or, 
 
 Ans. \ , , 
 
 ( =b^— 1, and =F^— 9. 
 
 26. A and B can together perform a piece of work in 3 
 days, and it would take B alone to do it 8 days longer than 
 it would take A. How many days would A alone require 
 to perform it ? Ans. 4 days. 
 
 Properties of the roots of quadratic equations. 
 
 (154.) We have seen that all quadratic equations tun be 
 reduced to this general form. 
 
 x' -f flx = 5. (1) 
 
 This, when solved, gives 
 
 Therefore the two values of x are 
 
 (2) 
 
QUADRATIC EQUATIONS. 203 
 
 |+\/j^6. (3) 
 
 (155.) Now, since -={ - | is always positive for all 
 
 real values of a, it follows that the sign of the expression 
 
 \- b, depends upon the value of b. 
 
 (156.) When b is positive, or when b is negative and less than 
 
 — , then will [- 6 be positive, and consequently y 7+^ 
 
 will be real. 
 
 (157.) When b is negative, and numerically greater than 
 
 — , then \-b will be negative, and consequently V T H~ ^ 
 
 will be imaginary. 
 
 CASE I. 
 
 When Y \-b in real. 
 
 1. If rt is positive, and - is numerically greater than 
 
 Y ^ — |- 6, then will both values of x be, r.cfil and negative 
 2. When a is either positive or negative, and - is nume 
 
 rically less than y^- -f &, then will both values of x hi 
 real, the one positive and th« other negative. 
 
204 QUADRATIC EQUATIONS. 
 
 3. When a is negative and - is numerically greater than 
 y — -j- i, then both values of a; will be real Q.m\ positive. 
 
 CASE II. 
 
 n/";+- 
 
 When \/ — \- a IS imaginary. 
 
 In this case both values of x are imaginary for all values 
 of a. 
 
 (158.) When b is negative, and numerically equal to 
 
 — , then both values of x become = — -. 
 
 (159.) If we add together the two values of .t, we 
 have 
 
 (-iWf+^)^-(-i-^/^^)=-<.. 
 
 If we multiply them, we find 
 
 From which we see, 
 
 That the sum of the roots of the quadratic equation a:-+ax 
 = h is equal to — a. 
 
 And the product of the roots is equal to — h. 
 
 Hence the roots of the equation. 
 
 x^ — (n + ^2)a: = — Tiro, 
 are ri and r^. 
 
 (160.) We can also deduce these properties as follows • 
 
 If, in the equation x'^-\-ax=z 6, we suppose the two roots 
 of a; to be r^ and rs, we shall have 
 
QUADRATIC EQUATIONS. 205 
 
 ri+ar, = b. (1) 
 
 rl-^ar^=b. (2) 
 
 Subtracting (1) from (2), we find 
 
 ,-_,-4_a(ro-;-0 = 0. (3) 
 
 Dividing (3) by /••,. — ?-i, it bi^comes 
 
 r. + n + a = 0; (4) 
 
 . • • ri-\-ri = — a. (5) 
 
 Multiplying (4) by ri, we get 
 
 r;ri + ?••■; -f ar, = (\ (6) 
 
 Subtracting (1) from (6), we get 
 
 r,r, = — /j. (7) 
 
 Equations (5) and (7) correspond with the properties just 
 found, Art. 159. 
 
 (161.) We have seen that every quadratic equation, when 
 solved, gives two values for the unknown quantity. These 
 values will both satisfy the algebraic conditions, and some- 
 times they will both satisfy the particular conditions of the 
 problem, but in most cases but one value of the unknown is 
 applicable to the problem ; and the value to be used must 
 be determined from the nature ol the question. 
 
 We will illustrate this principle by the solution of some 
 particular questions, 
 
 1 . Find a number such that its square being subtracted 
 from five times the number, shall give 6 for remainder. 
 Let x= the number sought. 
 Then, by the conditions of the questions, we have 
 
 5x— .r*=6. (1) 
 
 Changing all the signs of (1), it becomes 
 
 j-3_5:c3= — 6. (2) 
 
 which, wdien solved by the rule for quadratii-s, gives 
 
206 
 
 QUADRATIC EQUATIONS. 
 
 5±1 
 
 3, or 2. 
 
 Taking the first value of x = 3, we find its square to be 9. 
 
 Five times this value of a;, is 5 X 3 = 15. 
 
 And 15 — 9 = 6; therefore the number 3 satisfies th' 
 question. 
 
 The number 2 will satisfy it equally well, since its squan 
 = 4, which, subtracted from five times 2 = 10, gives fo 
 remainder 6. 
 
 2. Find a number such that when added to 6, and th. 
 sum multiplied by the number, the product will equal th« 
 number diminished by 6. 
 
 Let a; = the number sought ; then, by the conditions of 
 the question, we have 
 
 {x-\-6)x=x — 6. ' (1) 
 
 Expanding and collecting terms, we find 
 
 x-^-\-5x= — 6. (2) 
 
 This solved gives 
 
 -5±1 
 
 Here, as in the last question, we find that both values of 
 X will satisfy our question. 
 
 If we take the first value, x = — 3, we find that the num 
 ber — 3 added to 6 gives 3, which multiplied by — 3 give? 
 — 9 ; and this is the same as — 3 diminished by 6. 
 
 If we take the second value, x = — 2, we find that the 
 number — 2 added to 6 gives 4, which multiplied by — 2 
 gives — 8 ; and this is the same as — 2 diminished by 6. 
 
 3. Find a number which subtracted from its square, shall 
 give 6 for remainder. 
 
 Let x= the number, then we have 
 
QUADRATIC EQUATIONS. 
 
 207 
 
 X' — X:=e. 
 
 (1) 
 
 x-^^^-3,or_2. 
 
 
 This gives 
 
 If we take 3 for the number, its square is 9, from which 
 subtracting 3, we have 6. 
 
 Again, taking — 2 for the number, its square is 4, from 
 which subtracting — 2, we have 6. 
 
 So that both values of x satisfy the conditions of the ques- 
 tion. 
 
 4. A and B travel from the same place, and in the same 
 direction. The first day A travels but 1 mile, the second 
 day he goes 3 miles, the third day 5 miles, and so on in 
 arithmetical progression. After A has been gone 8 days, 
 B follows, travelling uniformly at the rate of 30 miles each 
 day. How many days after B starts will they be to- 
 gether ? 
 
 Let X = the number of days sought. 
 Then will z-f-8= the number of days which A travelled, 
 (a:4-8) = distance travelled by A. 
 36a: = •• " B. 
 
 Hence, 
 
 {x -I- Sy- = 360-. 
 
 This, solved by the usual method of quadratics, gives 
 j-=:4, or X =: 16. 
 
 From which we learn that they were twice together. First 
 B overtakes A at the end of 4 days, and then in 12 days 
 more A overtakes B. 
 
 In this case both answers arc aj^plicable. 
 
 . By selling a watch for $24, I lose as much per cent 
 as tiie watch cost me. What was the cost of the watch '? 
 
208 
 
 QUADRATIC EQUATIONS. 
 
 Let X = the number of dollars the watch cost. 
 Then will x — 24 = the loss incurred by the sale. 
 The loss per cent, will be 
 
 (x — 24)100 
 
 Therefore, by the question, we have this equation : 
 100(a; — 24 )_ 
 
 X 
 
 This gives x = 40, or a: = 60. 
 
 In this case, also, both answers are applicable. 
 
 6. There is a number consisting of two digits, of which 
 the right-hand digit is 3 greater than the left-hand digit, 
 and the number itself is equal to the square of the right- 
 hand digit. What is the number ? 
 
 Ans. 25, or 36. 
 
 7. A number consists of two digits, of which the right- 
 hand digit is double the left-hand digit. The number ex- 
 ceeds the square of the right-hand digit by 8. What is the 
 number 1 Ans. 12, or 24. 
 
 8. A and B speaking of their ages, A said he was 15 
 years older than B, and that the square of his age was equal 
 to 64 times B's age. What were their ages ? 
 
 A =-- 40 and B =: 25 ; or. 
 A = 24 and B =-. 9. 
 
 Ans. 
 
 9. By the law of universal attraction we knoic, that the 
 attraction of different bodies, at different distances, varies di- 
 rectly as their masses and inversely as the squares of theii 
 distances from the attracted point. 
 
 The above law being admitted, it is required to find a 
 point in the right line which joins the centres of the two 
 spherical bodies, whose masses arewiiand 
 
QUADRATIC KQIATIONS. 209 
 
 (ills point will be attracted wltli equal force by each of the 
 bodies. 
 
 p~ i -j-~— — ] ——————— 
 
 Pi mi })' p 1712 pi 
 
 Let mi and 7n-2 be the position of the bodies. 
 
 Let the distance between the centres of the two bodies mi 
 and m^y = d. 
 
 Also, let p denote the point sought". 
 
 Put a;---mip = the distance from the body wii to the 
 point sought, measured from mi towards the right. 
 
 Then d — x = m-ip = the distance from the body mo to. 
 the point, measured from mo tow^ards the left. 
 
 Now, having reference to the above law, we know that 
 the attractions of the two bodies upon the point p will be to 
 each other as the expressions 
 
 mi ma 
 
 But, by the questions, these forces of attraction are equal ; 
 therefore we have' this condition : 
 rni_ mo 
 
 x^ — {d-xY' ^ 
 
 Extracting the square root of both members of (1), we 
 have 
 
 y/mi _ d b y/ma . . 
 
 X d—x ' ^ ' 
 
 This reduced, by rules for simple equations, gives 
 Hence, 
 
 mip = x = ^1 — ■ Xd. (3) 
 
 v/wii ± s/ma 
 
 m^p = d — x= xd. (4) 
 
 y/mi zfc v/ma 
 
 27 
 
210 QUADRATIC EQUATIONS. 
 
 If we use the upper signs, we get 
 
 niip 
 
 y/lTii -\- y/mo 
 
 (A) 
 
 m^ip = ^--- X d. 
 
 y/nii -\- y/m-z 
 
 By taking the lower signs, we have 
 
 mxp 
 
 y/nii — \/m.i2 
 
 '■} 
 
 ///o» = X d. 
 
 (B) 
 
 We will now interpret these expressions for different nu- 
 merical values of c/, wii, mo, and, in order that the following 
 reasoning may be rigidly correct, it is necessary to suppose 
 all the matter of the bodies W] and mo to be concentrated 
 at the centres of the bodies. 
 
 CASE I. 
 
 When d = a finite quantity. 
 And mi > ma. 
 
 In this case, we evidently have 
 
 v/mi + v/mo 
 N/m2 
 
 >i and <1 
 
 Consequently, the first set of values, denoted by (A), give 
 m\p = a positive quantity which is <d, but >-. 
 
 imp = a positive quantity which is <-. 
 
QUADRATIC EQUATIONS. 21] 
 
 These values give for the point sought, a position between 
 wii and 7712, but nearer wio than mi. 
 Again, 
 
 
 a negative quantity. 
 
 v////l y/m-2 
 
 Therefore, the second set of values, denoted by (B), give 
 mijj = a positive quantity which is ></. 
 jn'2p = a negative quantity. 
 Now, since the distances from wo, measured towards the 
 left, ar£ consit'iered as positive, the distances in an opposite 
 '.lirection must be regarded as negative. 
 
 Plence, these second values give for the point a position 
 on the right of ??io. 
 
 CASE 11. 
 
 W'ie?i d = a finite quantity, 
 .ind THi'C.rni. 
 
 In this case 
 
 v/Wli , 
 
 ~, — ; — : — ^2' 
 
 >i and <1. 
 
 Consequently the first set of values, denoted by (A), give 
 
 . d 
 
 mfp = a positive quantity >-. 
 
 . ^d 
 map =z a positive quantity >-. 
 
 And the point lies between mi and m-j, nearer 7ni than Wj. 
 
212 
 
 QUADRATIC EQUATIONS. 
 
 Vmi 
 
 v/mi — s/ni-2 
 
 a negative quantity, 
 a positii-e quantity > 1. 
 
 Tlierefore these second values give for the point a position 
 on the left of wii. 
 
 This case is obviously the same as Case I., when we in- 
 terchange the bodies mi and wia. 
 
 CASE III, 
 
 When d = a finite quantity. 
 And mi = m^. 
 
 In this case, 
 
 v/mi 
 
 ■v/Wii-j-x/'mg 
 
 Consequently, the first set of values, denoted by (A), give 
 
 d 
 mip = -. 
 
 And the point is equi-distant from nii and 7nQ. 
 Again, 
 
 ■v/mi 
 
 -v/mi 
 
 -=dr !z=±an infinite quantity, (Art. 133. 
 
 =^ - ' =^an infinite quantity. (Art. 133, 
 
 Vwii — -/wa 
 
QUADRATIC EQUATIONS. 213 
 
 Therefore, the second set of values, denoted by (B), give 
 for the point a position at an infinite distance either to the 
 right or left. 
 
 CASE IV. 
 
 Wien d =0. ^nd mi\. 771-2. 
 
 In this case, we have 
 
 imp = 0, 
 m-2p = 0, 
 
 for both sets of values ; consequently there is but one point 
 which is equally attracted by both bodies, and that point is 
 the comiron centre of the two bodies. 
 
 CASE V. 
 
 W/ien d^O. Jind mi = mo. 
 
 The first set of values evidently become 
 
 imp = 0, 
 
 m-ip = 0. 
 
 Which shows that the point is in the common centre of the 
 two bodies. 
 
 The second set of values give 
 7ni;p = -=- an indeterminate quantity. (Art. 134.) 
 
 TTi^p = - = an indeterminate quantity. (Art. 134.) 
 
 So that the point may be any where on the line which join.s 
 the centres of the bodies. Since the two centres are united, 
 every line which passes through this common point may be 
 regarded as joining those centres j consequently every point 
 
214 QIADUATIC EQUATIONS. 
 
 in space is, in this particular case, equally attracted by each 
 body. 
 
 From the above discussion, we see that the analytical ex- 
 pressions are faithful to give all the particular cas^s which 
 are possible to arise from giving particular values to the 
 constant quantities which enter into the conditions of the 
 question. 
 
 (162.) We will now add a couple examples for the pur- 
 pose of illustrating the case in which the roots are imagi- 
 nary. 
 
 1. Find two numbers whose sum is 8, and whose pro- 
 duct is 17. 
 
 Let x = one of the numbers, then will 8 — x= the 
 other number. 
 
 The product is (8 — x)x --=8x — x"^, which, by the con- 
 ditions of the question, is 17. 
 
 Therefore, we have this equation of condition, 
 
 xi — Sx=—ll. (1) 
 
 This, solved by the usual rules for quadratics, gives 
 x = 4 db "^ — 1 5 for one of the numbers, 
 and 8 — (4 ± V—l) = (4 =F v/_l,) for the other num- 
 ber. 
 
 ( 4 _j_\/] I 
 
 Therefore, the numbers are < I 
 
 (4=F^/— 1, 
 
 both of which are imaginary ; we are therefore authorized 
 to conclude that it is impossible to find two numbers whose 
 sum is 8, and proihict 17. 
 
 We may also satisfy ourselves of this as follows : Since 
 the sum of the two numbers is 8, they must average just 4 j 
 h'.'uce the greater must exceed 4 just as much as the less 
 
QUADRATIC EQUATIONS. 215 
 
 falls short of 4. Therefore any t\YO numbers whose sum !•? 
 8 may be represented by 
 
 4 — a-. 
 Taking their product, we have 
 
 {4-\-x){4: — x)=ie — x. 
 Now, since x- is positive for all real values of x, it fol- 
 lows that the product 16 — x- is always less than 16 ; that 
 is, no two real nu77ibers whose sum is 8, can be found such 
 that their product can equal 17. 
 
 If we put the expression for the product, which we have 
 just found equal to 17, we shall have 
 16 — x- = 17, 
 
 consequently, x = zb ^ — 1. 
 
 And,4 + a: = 4zb^/^,) ^ , r , u i, 
 
 . > the same values as lound by the 
 
 4_a; = 4zpv/_i,^ 
 
 first method. 
 
 These values, although they are imaginary, will satisfy 
 the algebraic conditions of the question ; that is, their sum is 
 
 (4±N/iri)+(4:^v/Zn:) = 8, 
 and their product is 
 
 (4 ± ^/=^l) X (4 T v/— 1) =z 17. 
 
 2. Find two numbers whose sum is 2, and sum of their 
 reciprocals 1. 
 
 Denoting the numbers by x and y, we have the following 
 relations : 
 
 These, solved by the ordinary rules, give 
 
216 
 
 QUADRATIC EQUATIONS. 
 
 (2) 
 
 x = 1±n/— 1, 
 
 Both lirj:\ values are imaginar} ; consequently the condi- 
 tions of the question are absurd. 
 
 We may also show the impossibility of this question as 
 follows : The sum being 2 the numbers may be denoted by 
 
 l + o:,) 
 
 1 — a:. > 
 Taking the sum of their reciprocals, we have 
 
 1+x^l— x' 
 which, when reduced to a common denominator, becomes 
 2 
 
 1 — x2 
 
 The denominator of this expression cannot be greater 
 than 1 ; for all real values of x, the expression must ex- 
 ceed 2. Therefore, it is imfossihle to find two numbers 
 whose sum shall equal 2, and sum of their reciprocals equal 1. 
 
 (163.) From what has been said, we conclude that when, 
 in the course of the solution of an algebraic problem, we 
 fall upon imaginary quantities, there must be conditions in 
 the problem which are incompatible. 
 
 Under Art. 128, we remarked that imaginary quantities 
 had been advantageously employed as aids in the solution 
 of many refined and delicate problems of the higher parts 
 of analysis ; here we notice their utility in pointing out the 
 impossibility of questions, which otherwise, with only a su 
 perficial investigation, might be supposed possible. 
 
ARITHMETICAL PROGRESSION. 21' 
 
 CHAPTER VI. 
 
 flATIO AND PROGRESSION. 
 
 (164.) By Ratio of two quantities we mean their relation. 
 When we compare quantities, by seeing how much greater 
 one is than another, we obtain arithmetical ratio. Thus : 
 the arithmetical ratio of 6 to 4 is 2, since 6 exceeds 4 by 2 ; 
 in the same way, the arithmetical ratio of 11 to 7 is 4. 
 
 In the relation a — c=ir., (1) 
 
 r is the arithmetical ratio of a to c. 
 
 The first of the two terms which are compared is called 
 the antecedent; the second is called the consequent. Thus, 
 referring to (1), we have 
 
 a = antecedent. 
 c = consequent, 
 r = ratio. 
 From (1), we get by transposition, 
 
 « = c + r, (2) 
 
 c = a — r. (3) 
 
 Equation (2) shows, that in an arithmetical ratio the an 
 tecedent is equal to the consequent increased by the ratio. 
 
 Equation (3' in like manner shows, that the consequent 
 is equal to the antecedent diminished by the ratio. 
 28 
 
218 ARITHMETICAL PROGRESSION. 
 
 (165.) When the arithmetical ratio of r,ny two terms is 
 the same as the ratio of any other two terms, the four terms 
 together form an aiithmetical proportion. 
 
 Thus, if a — c = r ; and a' — c'= r, then will 
 
 a — c = a' — c', (4) 
 
 which relation is an arithmetical proportion, and is read 
 thus : 
 
 a is as much greater than c, as a' is greater than c'. 
 Of the (our quantities constituting an arithmetical pro- 
 portion, the first and fourth are called the extremes, the 
 second and third are called the means. 
 
 The first and second, together, constitute the first coup- 
 let ; the thiril and fourth constitute the second couplet. 
 From equation (4), we get by transposing, 
 
 a + c'= a'-j- c, (5) 
 
 which shows, that the sum of the extremes, of an arithmeti 
 cal proportion, is equal to the su)n of the means. 
 If c = a', then (4) becomes 
 
 a — a'=a' — c', (6) 
 
 which changes (5) into 
 
 a-\-c'=2a'. (7) 
 
 So that, if three terms constitute an arithmetical pro 
 portion, the sum of the extremes will equal twice the mean. 
 
 (166.) A series of ([r.antities which increase or decrease 
 by a constant difference form an aiithmetical progression. 
 When the series is increasing, it is called an ascending jyro- 
 gression; when decreasing it is called a descending progres- 
 sion. 
 
 Thus, of the two series 
 
 1, 3, 5, 7, 9, 11, &c. (8) 
 
 27, 23, 19, 15, 11, 7, &c. (9) 
 
ARITHMETICAL PROGRESSION. 219 
 
 The first is an ascending progression, whose ratio or com- 
 mon difference is 2 ; the second is a descending progression, 
 whose common difference is 4. 
 
 (167.) If a = the first term of an ascending arithmeti- 
 cal progression, whose common difference = J, the succes- 
 sive terms will be 
 
 fl = first term, 
 a-\- d= second term, 
 a-\-2d = third term, 
 a + 3d= fourth term, V (\Q\ 
 
 a -\- {n — l)d = Tith term. 
 If we denote the last or nth term by /, we shall have 
 
 l = a-{.{n—\)d. (11) 
 
 From (11) we readily deduce 
 
 a = l — in—\)d, (12) 
 
 " = '-=^+1. (14) 
 
 When the progression is descending, we must write — d 
 for d in the above formulas. 
 
 Suppose, in an arithmetical progression, a; to be a term 
 which is preceded by q terms ; and y to be a term which is 
 followed by q terms ; then by using (11) we have 
 
 x^a-\-qd^ (15) 
 
 y = l—qd. (16) 
 
 Taking the sum of (15) and (16), we get 
 
 x + y = a-\-L (17) 
 
220 ARITHMETICAL PROGRESSION. 
 
 That isj the sum of any two terms equi-distant from the 
 extremes is equal to the sum of the extremes, so that the 
 terms will average half the sum of the extremes ; conse- 
 quently, the sum of all the, terms equals half the sum of the 
 extremes multiplied by the number of term^. 
 
 Representing the sum of n terms by s, we have 
 
 s = -l-xn. (18) 
 
 From (18) we easily obtain 
 
 ; = -—/. (19) 
 
 n 
 
 l=-~a. (20) 
 
 (21) 
 
 2s 
 
 Any three of the quantities 
 
 a = the first term, 
 
 d = common difference, 
 
 n = number of terms, 
 
 / = last term, 
 
 s = sum of all the terms, 
 
 being given, the remaining two can be found, which must 
 give rise to 20 different formulas, as given in the following 
 table for Arithmetical Progression. 
 
 (168.) We have not deemed it necessary to exhibit the 
 particular process of finding each distinct formula of the fol- 
 lowing table, since they are all derived from the two fun- 
 damental ones, (1) and (7) , by the usual operations upon 
 equations not exceeding the second degree. It will furnish 
 a good exercise for the student to deduce all these formulr.s 
 by the aid, only, of formulas 1 and 7. 
 
ARITHMETICAL PROGUESSIOX. 
 
 2-2\ 
 
 ' No. 
 
 Given. 
 
 Requi- 
 red. 
 
 Formulas. 
 
 Corr. , 
 
 17 
 19 
 
 20 
 
 18 
 
 1 
 
 2 
 3 
 
 4 
 
 a, dj n 
 a, rf, s 
 
 a, n, 5 
 
 / 
 
 l = a-\-{n — l)d 
 
 / = — Jd±v/2d*+(a — i(i)- 
 
 I— - — a 
 n 
 
 s {n-l)d 
 
 n~^ 2 
 
 5 
 6 
 
 ! 7 
 8 
 
 a, d, / 
 a, ?j, / 
 
 S 
 
 s=zln[2a-\'i^ji—l)d] 
 
 ' 2 ' 2d 
 
 s = hi{a+l) 
 
 s = i7i\2l — {n^l)d\ 
 
 8 
 5 
 
 1 9 
 
 !io 
 11 
 
 'l2 
 
 a, n, 5 
 a, /, 5 
 n, /, s 
 
 d 
 
 71 — 1 
 
 ^ 25 — 2an 
 Hn-1) 
 
 2,. _ / _ a 
 2nl—2s 
 n(7i-l) 
 
 12 
 10 
 16 
 14 
 
 113 
 14 
 
 i 15 
 16 
 
 a, d, I 
 0, d, s 
 a, /, V 
 d, /, 5 
 
 n 
 
 
 d — 2a J 2s , /2a — dV- 
 2s 
 
 2l + d , , //2/+d\2 2. 
 ^^- 2d ^V( 2d )-d 
 
 17 
 18 
 
 19 
 20 
 
 = 
 
 d, n, / 
 
 d, n, 5 
 
 r/, /, s 
 n, /, i 
 
 a 
 
 a=/ — (n— l)d 
 5 (n— l)d 
 "-n 2 
 
 1 
 4 
 
 2 
 3 
 
 a = idd=N/(/-f-^dr — 2d^ 
 
 2^ ; 
 
 n 
 
222 ARITHMETICAL PROGRESSION. 
 
 (169.) From the nature of an arithmetical progression, 
 we discover that if we subtract the common difference from 
 the last term, we shall obtain the term next to the last ; if 
 we subtract from the last term twice the common difference, 
 we obtain the second term from the last. Hence the terms 
 of an arithmetical progression will be reversed if we inter- 
 change the values of a and /, and at the same time change 
 the sign of d. Thus, the general form of an arithmetical pro* 
 gression is 
 
 0, ct+rf, a+2rf, / — 2rf, / — (/, I. 
 
 Changing a to /, / to «, and changing the sign of d^ we 
 have 
 
 l^ I — d) / — 2(/, a+St/, a-\-d^ a, 
 
 which is precisely the same progression as the first, with 
 the terms arranged in a reverse order. The above change 
 has, of course, no effect upon the number of terms, nor upon 
 the sum of all the terms. 
 
 Therefore, in any of the formulas of the preceding table 
 we are at liberty to make the above named changes. As an 
 example, we will take from the table formula 2, which is 
 / — — Jrf =b ^2ds-{-{a—ldf. 
 
 Now, changing / to a, a to /, and changing the sign of d. 
 it becomes 
 
 u .= Id ± V(/+;(i)s — 2rf^, 
 which is formula 19. 
 
 In the same way, formulas 14 and 16 may be deduced 
 from each other. Such formulas as may be derived from 
 each other by the above changes we shall call correlative 
 formulas. It is evident that some of the formulas of the ta- 
 ble have no correlative. Thus, formulas 13 and 15 are not 
 altered by the above changes. Those formulas which have 
 correlative formulas have them referred to in the table, un- 
 der column headed Corr. 
 
ARITHMETICAL PROGRESSION. 223 
 
 1. The first term of an arithmetical progression is 7, the 
 common difference is |, and the number of terms is 16. 
 What is the last term ? 
 
 To solve this, we take formula 1 from our table, which is 
 l = a-ir{n — l)d. 
 Substituting the above given values for a, d, and 7i, we find 
 ; = 7 + Kl6— l)=10f. 
 
 2. The first term of an arithmetical progression is |, the 
 common difference is |, and the last term is 3|. What is 
 the number of terms 1 
 
 In4.his example w-e take formula 13. 
 I — a 
 
 which in this present case becomes 
 
 n = ^i^-|-l =26. 
 
 3. One hundred stones being placed on the ground in a 
 straight line, at the distance of two yards from each other, 
 how far will a person travel who shall bring them one by one 
 to a basket, placed at two yards from the first stone '? 
 
 In this example a— -4; d=4: ; 7i = ]00; which values 
 being substituted in formula 5, give 
 
 6> =z 50 j 8 -f 99 X 4 I = 20200 yards, 
 
 which, divided by 1760, the number of yards in one mile, 
 we get 
 
 s = 11 miles, 840 yards. 
 
 4. What is the sura of n terms of the progression 
 
 1,3,5,7,9, ? 
 
 Ans. .s- = «'■'. 
 
524 GEOMETRICAL PROGRESSION. 
 
 5. What is the sum of ti terms of the progression 
 1>2,3,4,5, ■? 
 
 A„s. .^±±n. 
 
 GEOMETRICAL RATIO. 
 
 (170.) When we compare quantities by seeing how many 
 times greater one is than another, we obtain geometrical 
 ratio. Thus the geometrical ratio of 8 to 4 is 2, since 8 is 
 2 times as great as 4. Again, the geometrical ratio of 15 
 to 3 is 5. 
 
 In the relation, - =r, (1) 
 
 c 
 
 r is the geometrical ratio of a to c. 
 
 As in arithmetical ratio, 
 
 a = antecedent^ 
 
 c = consequent^ 
 
 r = ratio. 
 
 From (1), we get by reduction, 
 
 a — cr^ (2) 
 
 .=2. (3) 
 
 Equation (2) shows, that in a geometrical ratio the ante- 
 cedent is eqiial to the consequent multiplied by the ratio. 
 
 Equation (3) shows, that the consequent is equal to the 
 antecedent divided by the ratio. 
 
 (171.) When the geometrical ratio of any two terms is 
 the same as the ratio of any other two terms, the four terms 
 together form a geometrical proportion. 
 
 Thus, if -= r; and -. = r, then will 
 c c 
 
 1=?' w 
 
GEOMETRICAL TROGRESSIOX. 225 
 
 which relation is a geometrical proportion, and is generally 
 Written thus : 
 
 a : c : : o' : c', (5) 
 
 which is read as follows : a is to c, as a' is to c'. 
 
 Of the four quantities which constitute a geometrical pro- 
 portion, as in arithmetical proportion, the first and fourth 
 are called the extremes, the second and third are called the 
 7neans. 
 
 The first and second constitute the^r^^ couplet ; the third 
 and fourth constitute the second couplet. 
 
 From equation (5), or its equivalent (4), we find 
 
 ac' = a'c, (6) 
 
 which shows, that the product of the extremes of a geome- 
 trical proportion, is equal to the product of the means. 
 
 If c = a', then (5) becomes 
 
 a : a' : \ a' : c', (7) 
 
 which changes (6) into 
 
 ac = a 
 
 (8) 
 
 so that if the two means which constitute a geometrical pro 
 portion be equal, then the product of the extremes will equal 
 the square of the mean. 
 
 (172.) Quantities are said to be in proportion by inver- 
 sion, or inversely, when the consequents are taken as ante- 
 cedents, and the antecedents as consequents. 
 
 From (5), or its equivalent (4), which is 
 
 we have, by inverting both terms, 
 c ^c_[ 
 a a'' 
 Therefore, by Art. 171, 
 
 c : a : : c' : a'. (10) 
 
 29 
 
226 GEOMETRICAL PROGRESSION. 
 
 W/iich shows ^ tJiat if four quantities are in proportion they 
 will be in proportion by inversion. 
 
 (173.) Quantities are in proportion by alternation^ or al- 
 ternately., when the antecedents form one of the couplets, 
 and the consequents form the other. 
 
 Resuming (4), 
 
 -=^- . (11) 
 
 c c 
 
 c 
 Multiplying both terms of (11) by —, it will become 
 
 o 
 
 a c 
 
 a' c' 
 
 Therefore, by Art. 171, 
 
 a : a' : : c : c'. (12) 
 
 Which shows, that if four quantities are in proportion they 
 
 will be so by alternation. 
 
 (174.) Quantities are in proportion by co7nposition,\\hen 
 
 the sum of the antecedent and consequent is compared either 
 
 with antecedent or consequent. 
 
 Resuming (4), 
 
 c c' 
 If to (13) we add the terms of the following equation -=:-, 
 
 each of whose members is equal to unity, we have 
 a-fc _ a'-\-c' 
 c ~ c' ' 
 Therefore, by Art. 171, 
 
 a-\-c : c : : a'-\-c' : c'. (14) 
 
 Which shows, that if four quantities are in proportion they 
 will he so by composition. 
 
GEOMETRICAL PROGRESSION. 227 
 
 (175.) Quantities are said to be in proportion by division, 
 when the difference of antecedent and consequent is com- 
 pared with either antecedent or consequent. 
 
 c c' 
 If we subtract the equation - = -., each member of 
 c c 
 
 which is equal to 1, from equation (4) , we find 
 
 c c 
 
 Therefore, by Art. 171, we have 
 
 a — c : c : : a' — c' : c'. (15) 
 
 Which shows, that if four quantities are in proportion, they 
 vAll be so by division. 
 Equation (4) is 
 
 a a' 
 
 c c'' 
 Raising each member to the nth power, we have 
 
 0" _ a'" 
 
 Therefore, by Art. 171, we have 
 
 a" : c" : : a'" : c'". (l6) 
 
 Which shows, that if foxir quantities are in proportion, likt 
 
 powers or roots of these quantities will also be in proportion. 
 
 If we have a : c : : a' : c', ^ 
 
 a : c : : a" : c". > 
 
 (17) 
 
 a '. c : : a : C 
 
 &c., &c. 
 
 These give by alternation, Art. 173, 
 
 &€., &c. 
 
22S 
 
 GEOMETIUCAL PBOGRESSION, 
 
 Tiierefore, by inversion, Art. 172, we have 
 
 We also have 
 
 c 
 c 
 
 c" 
 c 
 
 a c 
 
 &c., &c, 
 
 c 
 
 n c 
 
 (18) 
 
 .(19) 
 
 Taking the sum of equations (18), we have 
 
 aJ^a'-\.a"J^a"> j^S^c . _ c + c'-\- c" -\ - c'" -\- kc. 
 
 a c 
 
 Therefore, by Art. 171, we have 
 a-\-a'-\-a"-\-a"'-\-kc.:a: : c4-c' + c"+c"'+&c.: c. (20) 
 
 Which shows^ that if any number of quantities are propor- 
 tional^ the sum of all the antecedents will he to any one 
 antecedent^ as the sum of all the consequents is to its corres- 
 ponding consequent. 
 (176.) If we have 
 
 
 a : c '. : a : c , 
 
 
 
 a":c": :a"':c"', 
 
 
 we find 
 
 
 
 
 a a' 
 
 (21) 
 
 
 a" a'" 
 7'~ c'"' 
 
 (22) 
 
 Multiplying together the equations (21) and (22), we 
 have 
 
 aXa" a'Xa'" 
 
 cX c" c'Xc'" 
 
 (23) 
 
GEOMETRICAL PROGRESSION. 229 
 
 Therefore, by Art. 171, we have 
 
 aXa" : cXc" : : a' Xa'" : c' Xc'". (24) 
 
 Which shows, that if there be two sets of proportional quan- 
 tities, the products of the corresponding terms will he pro- 
 portional. 
 
 (177.) A series of quantities which increase or decrease 
 by a constant multiplier forms a geometrical progression. 
 When the series is increasing, that is, when the constant 
 multiplier exceeds a unit, it is called an ascending progres- 
 sion; when decreasing, or when the constant multiplier is 
 less than a unit, then it is called a descending progression. 
 
 Thus, of the two series, 
 
 1, 3, 9, 27, 81, 243, &c., (25) 
 
 256, 128, 64, 32, 16, 8, &c. (26) 
 
 the first is an ascending progression, whose constant multi- 
 plier or ratio is 3 ; the second is a descending progression, 
 whose ratio is ^. 
 
 (178.) If a is the first term of a geometrical progression, 
 whose ratio =^ r, the successive terms w'ill be 
 a= first term, 
 ar = second term, 
 ar-= third term, 
 aH= fourth term, V (27) 
 
 nth term. 
 
 If we denote the last or nth terra by /, we shall have 
 
 l=ar"-K (28), 
 
 If we represent the sum of n terms of a geometrical pro- 
 gression by s, we shall have 
 s = a-\-ar-^ ar^-f ar"-^ . . . .-j- ar'^' + or"-'. (29) 
 
230 
 
 GEOMETRICAL PROGRESSION. 
 
 Multiplying all the terms of (29) by the ratio r, we have 
 rs =zar -{■ ar--\- ar^-\- ar'^-^ . . . . + ar'^^-\- ar^. (30) 
 Subtracting (29) from (30), we get 
 
 (r— l)s = G(r"— 1). 
 
 Therefore, 
 
 ==a. < > 
 
 Ir-lS 
 
 (31) 
 (32) 
 
 Any three of the quantities 
 
 a = first term, 
 
 r = ratio, 
 
 n = number of terms, 
 
 /= last term, 
 
 s = sum of all the terms, 
 being given, the remaining two can be found, which as in 
 arithmetical progression, must give rise to 20 different for- 
 mulas, as given in the following table for Geometrical 
 Progression. 
 
 No. Given. 
 
 Requi- 
 red. 
 
 Formulas. 
 
 Cor. 
 
 1 
 
 2 
 3 
 4 
 
 a, r, n, 
 a, r, s, 
 a, n, s, 
 r, 7i, s, 
 
 / 
 
 ^_a-\-{r-l)s 
 
 r 
 l{s — ly-'— a{s — a) "-'= 
 I {r-l)sr^'-' 
 r"—l 
 
 9 
 11 
 
 12 
 10 
 
 5 
 6 
 
 7 
 
 8 
 
 a, r, n, 
 a, r, I, 
 
 a, n, /, 
 
 r, n, I, 
 
 S 
 
 ^__a{r"-l) 
 r—1 
 rl — a 
 
 8 
 5 
 
 r—1 
 
 „_ l{r^-l) 
 (r-l)r"-» 
 
GEOMETRICAL PROGRESSION. 
 
 231 
 
 Requl. 
 rod. 
 
 , n, /, 
 
 '■) ^h *5 
 
 r, Ij 
 
 _ {r — l)s 
 
 r" — 1 
 = rl — (r — 1) 5 
 
 a(s — ay-'— I {s — iy-'= 
 
 a, n, /, 
 
 , «, o, 
 
 ■J ") "3 
 
 n, /, s, 
 
 i^^^. 
 
 a a 
 
 s — l 
 s 
 
 ,r"-'+ - = 
 
 s — / s — / 
 
 16 
 
 14 
 
 a, r, /, 
 
 a, /, s, 
 r, /, s, 
 
 log/ — log 
 
 + 1 
 
 logr 
 log[a+(r— 1>] — log a 
 
 log r 
 log / — log a 
 
 log(s— a) — log(5— /) 
 
 log I — log [ rl — (r — l)s] 
 
 ~~ logr 
 
 +1 
 
 20 
 
 18 
 
 (179.) All the formulas of the above table are easily 
 drawn from the conditions of (28) and (32), which conditions 
 correspond with formulas (1) and (5), except the last lour 
 which involve logarithms ; we will hereafter, under Loga- 
 rithms, show how these formulas are obtained. 
 
232 GEOMETRICAL PROGRESSION. 
 
 If in a geometrical progression we change a to /, Z to a, 
 
 and r to r~'=-, the progression will remain the same as 
 
 before, taken in a reverse order. These changes being made 
 in the formulas of the preceding table, we shall discover 
 that some of the formulas, as in arithmetical progression, 
 have correlative formulas. Those having correlative for- 
 mulas, have them referred to in the table, under column 
 headed Cor. 
 
 EXAMPLES. 
 
 1. The first term of a geometrical progression is 5, the 
 ratio 4, the number of terms is 9. What is the last term 1 
 
 Formula (1), which is / = ar"-^^ gives 
 / =5X4^= 327680. 
 
 2. The first term of a geometrical progression is 4, the 
 
 ratio is 3, the number of terms is 10. What is the sum of 
 
 all the terms 1 
 
 air""— 1) 
 Formula (5), which is 5 = — , gives 
 
 s = ^\ ^ = 118096. 
 
 3. The last term of a geometrical progression is 106f f| 
 the ratio is f, the number of terms 8. What is the firs* 
 term? 
 
 Formula (9), which is o = -j^^-j, gives 
 
 106^-^ 
 
 (180.) When the progression is descending the ratio is 
 less than one, and if we suppose the series extended to an 
 infinite number of terms, the last term may be taken 
 ^ = 0, which causes formula 6 to become 
 
GEOMETRICAL PROGRESSION. 233 
 
 Which shoiDS, that the sum of an infinite number of terms 
 of a descending geometrical progression is equal to its first 
 term, divided by one diminished by the ratio. 
 
 EXAMPLES. 
 
 1. What is the sum of the infinite progression 
 
 l + i+T + i + rV + &c.? 
 In this example a= 1, r= J , and (33) becomes 
 
 2. What is the value of 0.33333 &c., or which is the same 
 thing, of the infinite series /o-f- rf o "h t/o u ~f~ ^^- *? 
 
 yV) and (33) gives 
 
 1-A " ' 
 3. What is fhe value of 0.12121212 &c., or which is the 
 same, of j\% + r^\%^ + to o H o o &c. ? 
 
 In this example a= j\%, r = j^-^y and (33) gives 
 
 4. What is the sura of the infinite series 
 
 i + i + i + .V + 8VH-&c.? 
 
 5. What is the sum of the infinite series. 
 
 30 
 
 Ans. 
 
 Ans. j. 
 
234 HAEMONICAL PROPORTION. 
 
 6. What is the sum of the series 1-1 |-~5~l~-^ ~h 
 
 X XT t' 
 
 &c., to infinity 1 
 
 Ans. 
 
 x—1 
 
 7. What is the sum of the series 1 -| j-— -\ 
 
 -f- &c., to infinity 1 
 
 x+l ' (x+1)^ 
 
 Ans. ^±i. 
 
 X 
 
 8. Suppose the elastic power of a ball, which falls from 
 a height of 100 feet, to be such as to cause it to rise 0.9375 
 of the height from which it fell ; and to continue in this 
 way diminishing the height to which it will rise in geomet- 
 rical progression, till it comes to rest. How far will it 
 have moved 1 
 
 Ans. 3100 feet. 
 
 HARMONICAL PROPORTION. 
 
 (181.) Three quantities are in harmonical proportion, 
 when the first has the same ratio to the third, as the differ- 
 ence between the first and second has to the difference 
 between the second and third. 
 
 Four quantities are in harmonical proportion, when the 
 first has the same ratio to the fourth, as the difference 
 between the first and second has to the difference between 
 the third and fourth. 
 
 Thus, if 
 
 a : c : : a — b : b — c, (1) 
 
 then will the three quantities a, &, c, be in harmonical pro 
 portion. 
 
 If a : d : : a—b : c — dj (3) 
 
HARMONICAL PROPORTION. 235 
 
 then also will the four quantities a, 6, c, and d be in harmo- 
 nical proportion. 
 
 Multiplying means and extremes of (1), we have 
 
 ab— ac = ac — be, (3) 
 
 which by transposition becomes 
 
 ab -{-bc = 2ac. (4) 
 
 In a similar way equation (2) gives 
 
 ac-\-bd:= 2ad. (5) 
 
 Suppose a, 6, c, d, e, &c., to be in harmonical progression; 
 then from (4) we have 
 
 be -{- ab = 2ac, ") 
 cd + be = 2bd, i (6) 
 
 de -{- cd= 2ce, } 
 &c. &c. 
 
 Dividing the first of (6) by abc, the second by bed, and the 
 third by ede, &c., we find 
 
 (7) 
 
 11111 
 
 From which we sec that -,7,-515-) &c., are in arith- 
 a b c a e 
 
 metical progression. (Art. 165.) 
 
 Hence, the reeiproeals of any number of terms in harmo- 
 nical 2iTOgression are in arithmetical frogression ; and con- 
 versely the reciprocals of the terms of any arithmeticar 
 progression must be in harmonical progression. 
 
236 
 
 HARMONICAL PROPORTION. 
 
 when reduced to a 
 common denominator, are 60, 30, 20, 15, 12, 10, which by 
 the above property must be in harmonical progression. 
 
 If six musical strings of equal tension and thickness, have 
 their lengths in proportion to the above numbers, they will, 
 when sounded together, produce more perfect harmony than 
 could be produced by strings of different lengths ; and hence 
 we see the propriety of calling this kind of relation, har- 
 monical or musical proportion. 
 
 (182.) If we take the arithmetical mean, the geometrical 
 mean, and the harmonical mean, of any two numbers, these 
 three means will be in geometrical proportion. 
 Let a and 6 be any two numbers, then will 
 \{a -\- h)= their arithmetical mean, 
 y/ah^ " geometrical " 
 2a6 
 
 a-\-h 
 And we evidently have 
 
 \{a + h) : ^/a6 
 
 harmonical " 
 
 v/a6 
 
 _2a&_ 
 a-\-h' 
 That is, 
 
 The geometrical mean, between the arithmetical mean 
 and the harmonical mean of two quantities, is the same as 
 the geometrical mean of the quantities themselves. 
 
237 
 
 CHAPTER Vn. 
 
 SERIES. 
 
 METHOD OF INDETERMINATE COEFFICIENTS, 
 
 (183.) Suppose we have the following conditi 
 
 ion : 
 
 .% + A3^ -\-J,x'-\-JlsX^+ &C. ^ ^j^ 
 
 =Bo+ Bix +5ox2-f B3X3+ &c. 
 If the above condition is true for all values of x, we must 
 have 
 
 
 ^^ = B.,^ (2) 
 
 X =B„. J 
 For, since the condition (1) is true for all values of x, it 
 becomes, when x = 0, ^0 = J^o. 
 
 Now, rejecting A^ from the left-hand member of (1) and 
 its equal ^o from its right-hand member, it wnll become 
 Axx■^Ji<^'^-J^^x'-\- &c.=i?i.r+7?.jX^+53a-3-|- &c. (3) 
 Dividing through by x, we find 
 
 ^i+j?ax-l-^3a:'-f &c.=^i-f B.x-^7?3r-^+&c. (4) 
 When X = Oj equation (4) becomes A\=^B\. 
 
238 
 
 SERIES. 
 
 (5) 
 
 By a similar process we can show, that ^2=1 B^; Jia^^B^; 
 and, in general, ^n = Bn^ 
 
 If we transpose all the terms of the right-hand member 
 of (1), it will become 
 ^o — Bo+{A — Bi)x-\-{Jiii~~B2)3r ) 
 + {Jl3— Ba)3^' + &c. = 0. S 
 
 (184.) Hencey when we have an equation of the form of 
 (5), true for all values of x^ it follows ^ that the coefficients 
 of the different powers of x, are respectively equal to 0. 
 
 We will now apply the above principle in the develop- 
 ment of some particular 
 
 EXAMPLES. 
 
 1 _L. 2x . 
 1. Required to expand — ^^ — Ij^^^*^ ^^ infinite series. 
 
 Assume, , 
 
 1-1- 2a: 
 
 A + A^x 4- A-^ -|-^3x' 4- &c. 
 
 Clearing this of fractions and then transposing, it becomes 
 
 I—aAx — JIi Vx- — ^2>x'-1-&c.=0. 
 — 2) —A) —a) 
 Now, since the right-hand member is equal to 0, it fol 
 lows, by the above principle of indeterminate coefficients 
 that the coefficients of the left-hand member must each 
 equal ; hence we have the following conditions : 
 ^0-1=0, (1)- 
 ^1—^0 — 2 = 0, (2) 
 ^2 — •^i — -^0 = 0, (3)1 
 .^3 — ^2-^1 = 0, (4)> (A) 
 
 A 
 
 ^„_,— ^„_2=0. 
 
 (71 + 1). 
 
From 
 
 the above we 
 
 readily 
 
 find 
 
 (1) 
 
 
 •^1 = 3, 
 
 « 
 
 
 (2) 
 
 
 ^2 = 4, 
 
 
 
 (3) 
 
 
 ^3 = 7, 
 
 
 
 (4) 
 
 
 
 
 
 (B) 
 
 ^„ = A_i+^,^o, (n + 1)/ 
 
 The value of the general coefficient ./?„, as given in group 
 (B), shows that, any coefficient is equal to the sum of the 
 two preceding ones. 
 
 Substituting these values, as given by (B),in the assumed 
 1 + 2^: 
 1 — X — x-^ 
 
 value 01 ' ■ T we iind 
 
 1 4-21: 
 ^ —1 -[-3a: -[-4a:--(- 7x3+ 11x^+18x^-1- &c. 
 
 1 — X — X- 
 
 2. Required the development of ; — - by this method . 
 
 1+x+x^ 
 
 Assume 
 
 .,z=.^o + ^lX + ^2X^ + ^3x3+^4X*+ &C., 
 
 1+x+x- 
 proceeding as in last example, we find 
 
 ^0 + ^1 ) +^. ) +^'^3) +^4) 
 
 — 1 ) +jio) +a; +^2) 
 
 Equating the coefficients to zero, we have 
 
240 
 
 SERIES. 
 
 ^0 = 0, 
 
 (1) 
 
 ^^^^,J—l = 0, 
 
 (2) 
 
 ^2+A+^0 = 0, 
 
 <3) 
 
 ^34-^2 + ^1 = 0, 
 
 (4) 
 
 ^4+^3+^2 = 0, 
 
 (5)| 
 
 
 
 (B) 
 
 ^n+'^n— l+^n-2 = 0. 
 
 (^ + 1). 
 
 Commencing with the first condition, we find ^o = 0, 
 which substituted in (2) gives ^i=: 1, these values of Jlo 
 and ^1, substituted in (3), give ^2 = — 1, now substituting 
 jii and Ji-i in (4), we find ^3 =0, continuing in this way, 
 we find Jii= 1 ; ^5= — 1, and so on; from the general 
 condition (n + 1) we find jin = — A-i — c/?„_2, that is, 
 any coefficient is equal to the sum of the two preceding co- 
 efficients taken with a contrary sign. 
 
 Hence, 
 
 l-\-x-[-x^ 
 
 r-\-x^ — x^-{- x'^ — &c. 
 
 3. Required the development of *^1 — x by this method. 
 
 Assume, v^l — x^=JiQ-{-AiX-\-A-ic--{-A^:j?-\-kc. 
 Squaring both members, we find 
 1 . /? 2 . o /7 /7 ^ +2A^o ) +2^o^3 " 
 
 S +^-; S +2A^2 
 
 Equating like coefficients, we have 
 
 ^0-^ = 1, (1) 
 
 2AA = — 1, (2) 
 
 2AA+^? = 0, (3)' 
 
 2A^3+2A-^2 = 0, (4)( 
 
 2j?oA+2j3iA+^^ = 0. (5) 
 
 X3+&C. 
 
 (C) 
 
SERIES. 241 
 
 The first condition gives 
 
 .^o = v/l = l. 
 This value of. '7o substituted (2), \ve find 
 
 2 
 In this way we find, in succession, the following values : 
 
 These values substituted in our assumed value, give 
 
 ^ . X X- 3x^ 3.5y^ 
 
 ~''~ ~2~~2l~2X6~2.4.6.8~ '^" 
 The general term of this series is 
 _3.5...(2n — 3)x" 
 2.4.6.8.... 2n ' 
 
 4. Required the development of ■ by this me- 
 
 ^ l—2x-]-x'- ^ 
 
 thod. 
 
 Ans. l+3x+5x-^-f7x^-(-9x^-{-lla;5+ &c. 
 
 l-fx 
 
 5. Required the development of 
 
 1 — X XT 
 
 Ans. l+2x-f 3z=+5r'+8ar'-|-13x''+ &c. 
 
 
 (18.5) Before closing this subject we will develop — 
 
 which will be of use hereafter. 
 Assume, 
 
 "Ln^ = ^o-f-./3iy+.%^+ +-^m3/"+ &c. ( 1 ) 
 
 X — y 
 
 Multiplying through by x — y, we obtain 
 31 
 
242 
 
 SERIES. 
 
 (A) 
 
 (n + 1) 
 (5) 
 
 — A ) -il ) -'In-l 
 
 Equating like coefficients of y, we get 
 
 yioa: = x'^ (1) 
 
 ^V — A = 0, (2)1 
 
 ^2X — .'ii^O, (3)1 
 
 - Jj^x — Jo = 0, (4). 
 
 and in general, 
 
 .'i„X — ./?„_! =0. 
 
 Equating the coefficients of y", we have 
 
 Jl,„X — J,n-i= — l. 
 
 From (1), we find 
 
 which substituted in (2), we find 
 
 .^1 = x"'-^. ■ 
 This in turn, substituted in (3), gives 
 
 and in general we have 
 
 JJ^ = x'"^-\ 
 
 In this general value of Jla write m — 1 for iij and we get 
 
 This value substituted in (5) , gives 
 
 ^,„x — 1 = — 1, or j3,n = 0, 
 and consequently all the succeeding values of Jin will be 
 reduced to zero. 
 
 These values of j?o ; -^i; ^2; ^3; &c., substituted in 
 (1), give 
 
 x — y 
 
 = x"^' -\-x"'--y-Jrx'^Y . . . +xy'"-'^+y'^\ (B) 
 
SERIES. 243 
 
 BINOMIAL THEOREM. 
 
 (186.) We have already found by actual multiplicatioi; 
 (Art. 94), that 
 (a+x)' = a-|-x, 
 
 (a+x)- = a--f2oa;+a:-^, . ,^. 
 
 {a-{-xY = a^-\-3a-x-{-3ax'+x\ ^ ^ ' 
 
 {a-\-x) ' = fi = 4-4a-'a;+6aV'+4ax^-}-a;' 
 
 Now, the Binomial Theorem teaches us the law by which 
 we may write the development of {a -\- a:)"' for any values 
 of o, x, and //;. 
 
 To determine this law, assume 
 
 (a-f x)« = .'io + .iia: + . //ox- + . ^3X^-^1- *c. (1) 
 
 We have taken the exponent of this binomial fractional, 
 in order to make the development more gen«»^i. 
 
 The assumed form ibr the development of (a -f- a:)" be- 
 ing general, must be true for all values of x. When a: = 0, 
 
 m 
 
 it becomes o" =.■?», introducing this value oi .'% in (1), we 
 have 
 
 (« -j- xf = «" -I- Ax + Ax' -f At" -f- &c. (2) 
 
 In (2), writing xj for x, and it becomes 
 
 (a -f- xi)~' = (? +./?ix, -f- ^ox^ -j- ./?3a: f -j- &c. (3) 
 Subtracting (3) from (2), we find 
 
 {a-]-xY—{a-^XiY= ( (4) 
 
 If we suppose 
 
 H = {a+xY; ui={a-\-XiY, (5) 
 
244 SERIES, 
 
 we readily find 
 
 m m 
 
 u'" — u{"^ = (a 4- x)n — {a + x.y 
 
 and 
 
 7/1" = X JCi. 
 
 (6) 
 
 (7) 
 
 Dividing the left-hand member of (4) by w" — Wi", and 
 the right-hand member by its equal a: — X], observing to 
 
 tn tn 
 
 substitute u"^ — Ui" for (a-j-a:)« — (a-j-xi)«"j as given by 
 (6), and it will become 
 
 Wl' 
 
 Ui' 
 
 (8) 
 
 \X — Xil \x X J \x — xj 
 
 Dividing both numerator and denominator of the left- 
 hand member of (8) Ity w — i^i, and performing the divisions 
 indicated in the right-hand member, and we obtain by the 
 aid of equation (B), Art. 185, the following: 
 
 M'"~'-f- Miw'"~--|- Wi'"~* i 
 
 t/,n-'-[_ UyU"-' -|- «i"-' ~ > (9) 
 
 ^i-h-^2(x-f-a:i)+^3(x^-fxx,+ x^)-f &c. > 
 
 Now, in (9), suppose a; = a;i, and consequently w = «i, 
 and it becomes 
 
 ^,-f 2A.iX -f- 3^3r^-f- 4.i,.r3 -f- &c. (10) 
 
 Re-substituting {a-\-x)n for u in (10), and it will become 
 
 (a -}- x)n 
 
 ^,-f- 2A'a- + 3^3x'-f 4^4x'-f &c. 
 
 n a-f-x 
 
 Multiplying through by a -|- x, and we obtain 
 
 (11) 
 
245 
 
 m 
 
 (a + x)" = 
 
 > (12) 
 
 +'^i ) +2^2 ) +3^3 
 
 m 
 
 Multiplying both members of (2) by— and it becomes 
 
 n 
 
 m. m 
 
 (a4-x)"=-a-4--.9ix+-Ji.2xH-8cc. (13) 
 n n n n 
 
 Equating the right-hand members of (13), and (12), we 
 have 
 
 — a « -| ^\x -\ A'lX^ -| ^3X-*-f- ^-c. 
 
 4- -^1 ) -\-2Ao ) +3^3 ) 
 
 Now, by the principle of (Art. 182), we must equate the 
 coefficients of like powers of x, by which means we have 
 
 'in 
 
 Ji\a = —a" , 
 n 
 
 n ' 
 
 3j?3a + 2^o = -^2, 
 11 
 
 pA,a^{p-i)A^, = -Aj^u 
 n 
 
 or, a,=.^Jl=:I±Aa^,. 
 
 pa 
 From this general value, we readily deduce the following 
 
246 
 
 /» fn — I 
 
 71 
 
 ^..= 
 
 mim 
 n \n 
 
 1 
 
 A, 
 
 tl-'][l-') 
 
 2.3 
 
 The general value being 
 
 A 
 
 (?-!).;>■ 
 
 2.3.4 
 
 (15) 
 
 These values of Ji\', Ji-i\ A; &.C., substituted in (2), 
 we have 
 
 r .an -x--|- &C. 
 
 (a-f-x) n=a»i-f--.0" x-f 
 
 If 71 =1, this value of (A) becomes 
 (a + x)- = n- + 7na— 'x _|_ ^(^ — ^) . a— ^x^+ &c. (B) 
 
 If 771= 1, then (A) becomes 
 
 I i 1 i._, 
 (a+ar)n = a''H «" x + 
 
 n\n 
 
 Gn~a:'' + &c. (C) 
 
 The coefficient of the (j7 + l)th term as given by (15), 
 becomes when w = 1 , 
 
 m(;7t— l)(7?i— 2)(w— 3) . . . .(771— p+2)(77t— p+1) 
 
 2.3.4 (;> — 1) -V 
 
 ! (16) 
 
SERIES. 247 
 
 (187.) The numerator of this coefficient being formed 
 of factors decreasing regularly by one, it follows that when 
 p = 771+1 it will vanish, and then the series must termi- 
 nate ; so that the number of terms of the expansion (B) 
 
 will be 771 -|- 1. But when - is fractional, or a negative 
 n 
 
 integer, the number of terms of the expansion must be 
 
 infinite. 
 
 When a or X becomes negative, then those terms of the 
 expansion will change signs, which contain odd poioers of 
 this negative quantity. 
 
 (188.) If in (B), we write a for a; and x for a, we shall have 
 (a._}_a) m^ a-'"+??ix'"- 'a -f !!i(!!^zi) a;"'-^-^- &c. (17) 
 
 Now, since the left-hand members of (B) and (17) are 
 evidently equal, their right-hand members must be ; and 
 since, when m is a positive integer, the number of terms 
 of (B) as well as (17) is equal to m -{- 1, it follows that the 
 terms of the expansion (B) must be homogeneous and sym- 
 
 (a + a;)"'= 
 a'"-f 7na'"- x^ ^^^"'~~'^ \"'-H''->r. . . . + max"'-' + x^- ^^^ 
 
 If in (A) we suppose a^x= 1, we shall find 
 
 m 
 
 ™(^') ■t-)(^^) 
 
 (E) 
 (l + l)"=2"=l-f- 4—^-^-1--^ ^-^ +&C 
 
 Tlierefore, in any expansion of a binomial, whose terms 
 are both positive, the sum of the coefficients is equal to the 
 same power, or root of 2. 
 
248 SERIES. 
 
 (189.) If in (A), we suppose a = 1 ; x= — 1, we shall 
 have 
 
 ^ ^ n 2 2.3 ^ 
 
 That isy in any expansion of a binomial ^ one of whose 
 terms is negative,, the sum of the coefficients w = ; and 
 therefore the swn of the positive coefficients must he equal 
 to the sum of the negative ones. 
 
 (190.) By inspecting formula (A), we discover that 
 the coefficients may be found in succession by the follow- 
 ing 
 
 RULE. 
 
 Multiply any coefficient of any term hy the exponent of 
 the hading quantity in that ternij and divide the product by 
 the exponent of the foil owing quantity diminished hy one^ and 
 the result will he the coefficient of the succeeding term. 
 
 APPLICATION OF THE BINOMIAL THEOREM. 
 
 (191.) We will now make an application of this theo- 
 rem ; and, first, suppose in the expression (B), of page 246, 
 we make successively m= 1, 2, 3, and 4, and the results 
 will be precisely the same as those first given on page 243. 
 If we make in succession m^ 5, 6, and 7, we shall obtain 
 the following results : 
 
 (a-fx)^=a^-|-5a4x-f-l0aV-j-10aV-|-5ax*+a:«, 
 (a+x)« = 
 
 a'''+6a'*x+15a^xa+20a3a^-fl5aV-f-6ax'^4-x«, 
 (a-fx)^ = 
 
 J a'4-7a«x-|-21aV-|-35a^x8+35aV 1 
 
 I -f21a=^x«-f-7ax«-i-x^. S 
 
SERIES. 249 
 
 EXAMPLES. 
 
 1. Required the expansion of (a -j- rr) "* . 
 
 In formula (C), make n = 2, and it becomes 
 
 {a4-x) =a -4- -a x .a 'x'^-] -.a 'x^ — &c. 
 
 ^ ~ ' ~2 2.4 ~2.4.6 
 
 = a ) 1-1 — a—'^x . a~-a:--i .a—^x^ — &c. > 
 
 \ ^2 2.4 ^2.4.6 5 
 
 Writing the different powers of a, which have negative 
 exponents, in the denominator, by which means their expo- 
 nents change signs and become positive (Art. 49), and we 
 find 
 
 i h ( . X x- . 3x' 3.5x' ) 
 
 2. Required to expand (a-f-a;) . 
 Changing ?i into 3, in (C) , and we have 
 
 Removing the factor a , and causing the different powers 
 of a to pass into the denominators, as in the last example, 
 we obtain 
 
 .A M. , a: 2a:^ , 2.5x' 2.5.8x' , , ) 
 
 ( 3a 3.6a- 3.6.9o^ 3.6,9. 12a^ ) 
 
 3. Expand (a-(-x)^. 
 
 Making n = 4 in (C), and it becomes 
 
 ^ ^ ^ 4 4.8 '^4.8.12 
 
 32 
 
250 
 Or, 
 
 SERIES. 
 
 ■ 4. Required the expansion of ( 1 4- a:) . 
 
 This example will agree with example 1, if we write 1 
 for a. Making this change in example 1, we get 
 
 5. Required the expansion of (1 -f- 1)" or y/2. 
 In the last example make a: = 1, and it becomes 
 
 (1 + 1) 
 
 3.5 
 
 11 3 
 
 "^ ~ "^2 2.4 "^2^4.6 2.4.6.8 
 
 &c. 
 
 6. Required the expansion of "^1 — x or (1 — xY • 
 In example 4, change x into — x^ and we get 
 
 (l-^f=l- 
 
 3a:^ 
 
 3.5a;'' 
 
 2.4 2.4.6 2.4.6. 
 
 &c. 
 
 This expansion agrees with the one found by indetermi 
 nate coefficients. (See Ex. 3, Page 240.) 
 
 7. Expand (a -\- x)-^. 
 
 In (B), make m = — 4, and it becomes 
 (a+x)-' 
 = 0-'*— 4a-'*a: + lOa-V— 20a-V+ 35a-V— &c. 
 
 _Ml — — ]^_^ 35x* 
 
 &c.| 
 
 8. Required to expand 
 
 a -f-2; 
 
 or (a-f-x)~'. 
 
SERIES. 251 
 
 Making m = — 1 in (B) , we find 
 — a-'— a-'x -f a-'->x-— a-*x'^-{- a" V— a-^x^-\- &c. 
 
 9, Required the expansion of — 
 
 X 
 
 In the last example, write 1 for a, and — x for x, and it 
 becomes 
 
 1 
 
 1 — a: 
 
 1 _|_ a; 4- x2+ x^+ x^-{- x^-\- x^-{- &c. 
 
 H, 6 36^ 3.763 3.7.116^ . > 
 
 10. What is the expa:!sion of (a — hyi 
 
 b 36- 3.76^ 
 
 "4a 4.8a2 ^.8.12d' 4. 
 
 11. What is the expansion of (a-j-z) *? 
 
 aH 5^^5.10a^ 5.l0.15a3"^5.l0.15.20a^ ^' \ 
 
 12. W^hat is the expansion of (a^ — x)^! 
 
 A 1^, a: x^ 3x3 3.5x> ) 
 
 '^ r 2a3 2.4a« 2.4.6a^ 2.4.6.8a»^ ' S 
 
 13. What is the expansion of (^ + 7^ — 1)^ 1 
 
 If in example 2, we change a into p, and x into gv/ZZj 
 we shall find, by reccollecting that by Art. 126, we have 
 
252 
 
 1 
 
 3 
 
 i'^+^A^^-i+lep-Y 3.e.9 
 
 p~^q^\/ — 1 — &C 
 
 14. What is the expansion of {p — q^ — 1)^ 1 
 
 Changing, in example 2, a into p^ and x into — q "^ — 1, 
 we easily find 
 
 J 
 
 {p — qV — \) 
 
 (192. ) This theorem may be applied to quantities of more 
 than two terms. 
 
 Suppose we wish the expansion of (a-j-6-|-c)'. 
 Assume 
 
 and 
 
 (a-f-6-|-cy = (a4-d)3. 
 
 Now, in (B), make x = dj and m=3, and it will be- 
 come 
 
 {a-\-dy=^-a^-\-3a^d-\-3ad'-{-d\ (1) 
 
 Now, by assumption d = h-{-c; therefore we have 
 
 and 
 
 d' = b'-\-Zb'c-\-3bc'-^c\ 
 
 These values of d, d^ and d^, being substituted in (1) , we 
 
 get 
 
 {a-\-h-\-cy = 
 
 a3+3a^(6+c)+3a(6^+26c+c^)+&^+36'c+36c2+c3 
 _ { a^4-3a=6-f3a^c+3a6'2-|-Ga6c4-3ac2 ) 
 ~ I +63+362c+36c2+c^ S 
 
SERIES, 253 
 
 We might proceed in this May to obtain the expansion of 
 algebraic polynomials of any number of terms, but abetter 
 method will be to deduce a multinomial theorem, which may 
 be done as follows : 
 
 MULTINOMIAL THEOREM. 
 
 (193.) This theorem, as we have just hinted, gives the 
 law of the expansion of 
 
 (aoH-aiX-f-cr22C*-}-a3:c3-{-&c,) " , 
 
 or of any other polynomial, having for an exponent any 
 value whatever. To determine this law, assume 
 
 (ao-}-aiX-fa2X--j-&c.)'^'=A+^iX-|-^52X-+&c. (1) 
 
 When a: = 0, we have ao'* = -'^o ; therefore we have 
 
 (ao+aix-(-a2a;-H-&c.)^=ao"'+./?ix-f-^o.r^-|-&c. (2) 
 
 Writing x^ for x, we have 
 
 (au+aia:i+a2X=4-&c,)^=flo"+^iXi-f-^2X^ + &c. (3) 
 
 Subtracting (3) from (2), we find 
 
 m 
 
 {ao-\-a^x-\-a2X'-\-kc.)" — (cfo+ai2"i+«2a-^ -f &c.)" ^ (4) 
 
 =^i(x — a:,)4-^2(x3-^x^)4-&c. 
 
 If we suppose 
 
 1 
 U=z (flo-f cix-f aox--}-&c.)% 
 
 t^i = (ao-f-aiXi-j-aaX= -I-&C.)", 
 we readily find 
 
254 SERIES. 
 
 f/m_ JJm 
 
 m, n 
 
 z={ao'^aiX-\~a2X--\-&,c.Y — {ao-\-aiXi-^aiX]-\- Scc.y 
 U»— C/i^z^rciCx — a;i)4-a2(x2 — x2)-f-&c. 
 
 Hence (4) becomes 
 
 IJm __XJ^m-- ) 
 
 Jii{x — x,)-{-Az{x^ — x\)-^M^{x^ — x])+8LC. \ 
 
 (5) 
 
 Dividing the left-hand member of (5) by 17" — t/^, and 
 its right-hand member by its equal 
 
 ai(x — a;i)-|-aa(x^ — ^\)-\- &c., 
 we get 
 
 ^,{x — x^)-+-ji,{^''~-^')-\-M^'—x'i)-^ &c. 
 
 ai(a: — xi)-|-aa(x'—x2)-|- 03(3:^—0:?)+ &«• 
 
 (6) 
 
 If we divide both numerator and denominator of the 
 left-hand member of (6) by U — Ut, it will become [see 
 formula (B), Art. 185J, 
 
 (7) 
 
 If we divide both numerator and denominator of the 
 right-hand member of (6) by x — x , it will become 
 
 ^i-f A>(a:-ha:i)H--^3(a^+xx,-|-xi')+ & c. .g. 
 
 ai-|-a.(x-|-xi)4-a3(a:--ha:xi-f-x^)-|- &c. 
 
 The expressions (7) and (8) are equal. Now, when 
 x = xi, the expression (7) becomes 
 
 ml/"'-' 
 
 nU" 
 
 n' I/" ' 
 
 which, by re-substituting the value of [7, becomes 
 
SERIES. 
 
 m 
 
 m {ao-j-aix-{-a'23p'j- &c.)'' 
 n ' ao-\-aix-^aQpc--\- &c. 
 
 When x = Xij the expression (8) becomes 
 Jli-\-2M2X-\-3Ji3X--{- &c. 
 ai-|-2aaj: -|-3a3X*-f- ^^c.' 
 
 266 
 
 (9) 
 
 (10) 
 
 Equating the expressions (9) and (10), and clearing of 
 fractions, we have 
 
 - {ao-{-aiX-i-aox-+kcY • {ai-\-2aiX-{-3a:iX''-\-kc)=^ ( 
 
 n > (11) 
 
 (oo+aiZ+caxH&c) . (^i-)-2.^2^+3.^,x3-}-&c.) ) 
 
 Multiplying (1) by—, we have 
 
 -(ao+a,a:+ayX--f&c.)'^=-(.^o+-'?!J^+^-2a;''H-&c.) (12) 
 n n 
 
 Hence (11) becomes 
 -(^oH-^ix+.7.^^+&c).(a,-h2a,a-+3a3.T2 + &c)= ) 
 («o+a,x+aox-4-&c).(^,-f-2..i.a-+3.^,a:-+&c.) ) 
 By actual multiplication (13) becomes 
 
 n hi 
 
 2^oao\ 
 
 2Anu2 
 
 3^00.3 
 
 -X- 4-^-/3^1 
 n 
 
 \ni ., , „ 
 — .T^-j- &c. 
 
 2.^202 
 3^lf/3 
 
 =Aia(i-{-2AMf}x-\-^JlzOQX'-\-i:A.\a{^x'^-Y- &c. 
 
 (1^) 
 
 -'^iffa 
 
256 
 
 Equating coefficients of like powers of x in (14), we 
 have 
 
 n 
 
 2^200-4-^101 =—Jiai4-2 — iou^. 
 n n 
 
 3«43ao-T-2./520i-j-^ia2= ^ 
 
 4^400+3^301+2^202+^103 = 
 
 -^30i + 2-^200 + 3 ^103 + 4-^004. 
 
 n ' n n n 
 
 If for Moj we use its equal ao'*? we shall find from the above 
 system of equations 
 
 m 
 >Aq = Co" } 
 
 Ji\ = — Oo" Olj 
 
 wi/m \ 
 
 j3o =— ^— oo" o = + - Co" Co, 
 
 2 n 
 
 ^,J— ^^^ "" "■' 
 
 f +- 1 Qo" aiao + -Go" 03. 
 
 V w \n / w 
 
257 
 
 These values tf ^o, .'?,, ^2, ^^3, &c., substituted in (1), 
 we have 
 
 »n m m 
 
 S,7(,7-') 
 
 ffo" « 1 + - Oo" a.' 
 
 QMM - 
 
 2.3 
 
 (+"(?-) 
 
 cro" a? 
 
 a 771 --1 
 
 n 
 
 x^-\- 
 
 'X^ 4- &c. 
 
 (A) 
 
 EXAMPLES. 
 
 1. What is the cube of \-\-x-\-x'^--\-x^-\-x^-\-Si.cA 
 
 If, in our general expression (A) of the multinomial theo- 
 rem, we make 00=^1 =«j= 03= &c.= 1; andm = 3,n=l, 
 we shall have 
 
 2. What is the square root of 14-a:+a:^+x'4-&c.? 
 
 In our general expression (A), we must have7n=l,n =2, 
 and 1 = flo = fli = «-' = f 3 = &-C. 
 
 3. What is the cube root of l-f-x-|-a:--|-a:'-f &c.? 
 
 In (A), make m = 1, 7i = 3, and 1 = oo = ai = aj:=:cr5 
 .=:&c., and we get 
 
 (1+x+xH r'-h&c.)*-l + 3^+|x»+l^x3 + &c. 
 
 33 
 
258 
 
 4. What is the cube root of 1 -\- -x -}--x'^-{—x^-\-&iC. 
 
 REVERSION OF SERIES. 
 
 (194.) Suppose we have 
 
 aiX-\-aox"-\~a2X^-\-aiX^-\-&ic.^y. (1) 
 
 The process by which we find x in terms of y, is called 
 reverting the series (1), which may be effected by the fol- 
 lowing method : 
 
 Assume 
 
 x=:^iy-}-J].2y--{~Ji3y^-{-^iy'-^Sic. (2) 
 
 Now, we find by actual multiplication, or by means of 
 the multinomial theorem, 
 
 x-^ = ^-;r + 2^,^2y' + 2A^3 } y4_^&e. 
 
 x^ = ^'y-}-3^]Jl-2y*+ &c. 
 x*=^\y'+SLc. 
 
 These values of x, x-, a:% a:"*, &c., substituted in (1), we 
 have 
 
 y*+&c. 
 
 ^-;a,|+2^,^2aa 
 
 1/3+ ^4ai 
 
 + 2^1^302 
 + 3^=1^003 
 
 = 3/- (3) 
 
 Hence, we have by the method of indeterminate coeffi- 
 cients, (Art. 182.) 
 
SERIES. 259 
 
 From the above conditions we deduce 
 
 ^,— -, 
 
 2a= — a,«3 
 
 Jl^ = ; . 
 
 These values of./?i, v^a, A^^ ^4, &c., substituted in (2), 
 \ve have 
 
 1 a„ , , 2a2 — aiQs , 
 
 -y— ^!/--l — =— i — y 
 
 l«i «? «? ^ (A) 
 
 5c,^ — 5010003+0704 , , c. 
 —- : y -\- «^c. 
 
 So^hat if (1) is true for all values of x and y, then also 
 will (A) be true for all values of x and y; and such is the 
 general relation betwpen two series when one is the rever- 
 sion of the other. 
 
 EXAMPLES. 
 
 1. Given the series a:-J-x'^-}-x'-|-x^-f-&c.= y, to find 
 its reversion, that is, to find the value of x in terras of y. 
 
 Comparing this series vdth the series (1) of this article, 
 we see that 
 
260 SERIES. 
 
 1 = fli= a_i= aj^=^ai= &c., 
 these values substituted in (A), give 
 
 a; = y — 2/2-}-i/3— y-f &c. 
 
 2. Given x — ^x'-{- {x^ '•i^+ &c.= y to find x. 
 
 In this example Ave have 
 
 fl = 1 ; a.>= — i ; 03— i ; 04= — } ; &c., 
 which values substituted in (A), give 
 
 3. Given l-\-x4-~4--—-4-—- 1- &c.=Vi to find x 
 
 ' '2 '2.3 '^2.3.4^ ^' 
 
 in terms of y. 
 
 In this example we first transpose the 1, by which means 
 
 hav( 
 
 
 2 ' 2.3 ' 2.3.4 
 This, compared with (1), we find 
 
 0^ = 1; 0, = ^; «.= ! ; ^^^ = 2^1' ^'' 
 These values cause (A) to become 
 
 " T^ 3 4 ' 
 
 or restoring the value of y', 
 
 (y -iy^ , (y-l)3 (y-ir 
 2 "^ 3 4 
 
 x = {y-l)- 
 
 + &c. 
 
 4. Given X +— -f- — -f-'T~^~ "J"^^'^^^' *° ^"^ ^' 
 
 -CO 4 
 
 Ans. x = y—'^-{-^^ — J~-{-SLC. 
 ^ 2 '^2.3 2.3.4 ' 
 
261 
 
 DIFFERENTIAL METHOD. 
 
 (195.) This method shows how to find any particular 
 term of a regular increasing series, or the sum of a certain 
 number of terms. 
 
 If we take the regulaV increasing series 
 
 oi; oj; fla; «4; 05; &c., (1) 
 
 and subtract each term from the next succeeding one, we 
 shall obtain the following series, which we shall call the^r^^ 
 order of differences : 
 
 Co — d ; C3 — Co ; 04 — as; as — 04 ; &c. (2) 
 
 Again, subtracting each term of this series from the next 
 succeeding term, and we find for the second order of differ- 
 ences 
 Gs — 2a-2-\-ai; 04 — 2as-j-a-2; 05 — 2cf4-j-f/3; &c. (3) 
 
 Subtracting again each term of series (3), of the second 
 order of differences, from its next succeeding term, and we 
 get a series of third order of differences, as follows : 
 
 04 — 3a3-|-3a.> — ai ; 05 — 3fl4-|-3cr3 — a->; &c. (4) 
 
 Subtracting once more we find,, for the fourth order of 
 difiFerenccs, 
 
 05 — 4(/4-|-6rt3 — 4ao-f-ai ; &c. (5) 
 
 If we take only the first terms of the scries (2), (3), (4), 
 (5), and represent them respectively by D, ; Do; D3 ; D\; 
 &C.J we shall have 
 
 Di =a-2 — flj. 
 Do = 03 — 2a.. -f-oi, 
 
 D3 = a4 — 3a3 + 3a2 — fli, ^ (6) 
 
 D4 = a3 — 4fl4 -\-6as — 4a.i+ai,' 
 &c 
 
•262 
 
 The coefficients of the difTerent terms which constitute 
 the right-hand members of equations (6) are the same as the 
 coefficients of the different terms of the expansion of the bi- 
 nomial (1 — 1)'', whose expanded form is 
 
 n(n— 1) 7i{n—l){n—2) n{n—l){n—2){n—3) 
 
 l-n+ 
 
 D„ 
 
 2 2.3 ' • 2.3.4 
 
 Hence, the general equation of (6) is 
 
 , n(7i— 1) n{n—l){n—2) 
 
 , fln-f-l nOn-] a?i-l ^-^ On— 2( 
 
 n{n-l){n -2){n-3 ) 
 + 2-3^4 ««-3 - &.C. 
 
 •&c. 
 
 (7) 
 
 If the terms of the right-hand members of (6) are taken 
 in a reverse order, we shall have 
 
 Dn= 
 
 When n is an even number^ 
 „(„_l)(„_2)(,.-3) 
 
 ^ 2.3.4 
 
 (A) 
 
 When n is an odd number.^ 
 -a,+na, 5^^a3+-^ :^ -'04/ 
 
 n(n— l)(n— 2)(7?— 3) 
 2.3.4 
 
 (B) 
 
 05 + &C. 
 
 EXAMPLES. 
 
 1. Required tlic first term of the fourth order of differ 
 ences of the series 1, 8, 27, 64, 125, &c. 
 
 In this example we have 
 
 ai = l- flo = 8j G3 = 27j 04 = 64 j a5=125and»=4 
 
SERIES. 263 
 
 These values substituted in the formula (A), since n is even, 
 give 
 
 2. Required the first term of the third order of differences 
 of the series 1, 2*, 3^, 4', &c. 
 
 Ans. 60. 
 
 3. Required the first term of the fourth order of differ- 
 ences of the series 1, 6, 20, 50, 105, &c. 
 
 Ans. 2. 
 
 (196.) To find the 7ith term of the series 
 
 we proceed as follows : 
 
 From the first of the equations (6) of last article, we 
 obtain 
 
 flo = Gi + Di 5 
 
 this value of co substituted in the second of equations (6), 
 gives 
 
 a3=ai + 2Dx + Do; 
 proceeding in this way we have the following : 
 
 fli = flij 
 
 0-2= fll -f--Dl5 
 
 fl3=ai + 2Di + D2, 
 
 a4=ai + 3A4-3Do4-D3, /" ^^) 
 
 as = Gi -f 4Di + 6D, + 4D3 4-^ 
 
 Where the coefficients of the terms of the value of On are 
 equal to the coefficients of the terms of the expansion of the 
 binomial (1 -j- I)""', whose expanded form is 
 
l + (n-l)+^-. 
 
 („_l)(n-2) , (n-l)(n-2)(n-3) 
 
 2.3 
 
 (n-l)(n^2)(n-3)(n -4) ^ 
 ■^ 2.3.4 "^ 
 
 (9) 
 
 ; („_i)(,-2)(„-3) ^^ 
 
 ^ 2.3 
 
 (C) 
 
 EXAMPLES. 
 
 1. Required the tenth term of the series 
 
 1, 4, 8, 13, 19, &c. 
 
 fli = l, 4, 8, 13, 19, 
 
 Di = 3, 4, 5, 6, 
 
 D2=l, 1, 1, 
 
 D3 = 0, 0. 
 
 Hence, in this example, 
 
 ai = 1 ; Di = 3; Da = 1 ; -D3 = ; and n = 10, " 
 
 which values being substituted in (C), we find 
 
 9 8 
 aio= 1+9.3 H — ~ = 64, for the tenth term required. 
 
 2. Required the nth term of the series 2, 6, 12, 20, 30, &c. 
 
 ai = 2, 6, 12, 20, 
 
 i)i = 4, 6, 8, 
 
 1)2 = 2, 2, 
 
 D3 = 0. 
 
 These values substituted in (C), give 
 
 a„ = 2 + (n-l) . 4+t:lfc:?l. 2 = n^+n=n(n+l), 
 
 which is the nth term sought. 
 
SERIES. 266 
 
 3. What is the 7ith term of the series 
 
 1, 3, 6, 10, 15, 21, &c.? 
 
 Ans. !fcti). 
 2 
 
 (197.) To find the sum of n terms of the series 
 
 O] ; 0-2', 03 ; 04 ; as; &c. 
 
 we operate as follows : 
 
 Take the new series 
 
 0; fii; fli+flj; Oi+cra-l-os; Ci+ao+a3+a4&c. (10) 
 
 Subtracting each term from its next succeeding term, we 
 
 have 
 
 oi; 0-2 ; 03 ; 04 ; 05 ; &c. 
 
 which is the same as the original series ; hence, the 7i-\-l 
 difference of the series (10), is the same as the n difference 
 of the proposed series ; therefore, if in the formula (C), we 
 change Oi into 0, n into n + l? Di into Oj, D) into Di, D3 
 into D2, &c., we shall have 
 
 -^ < .(n-l)(. -2)(n-3) ^^t 
 
 ^ 2.3.4. ^ ) 
 
 which expresses the 71 -\- 1th term of the series (10), but the 
 
 n-\- 1th term of the series (10) is the same as the sum of n 
 
 terras of the series 
 
 ai ; flj ; 03 ; 04 ; f'a ; &c. 
 
 Putting this sum equal to »S„, we shall have for the sum of 
 
 n terms of the above scries, the following expression : 
 
 , n(n-l) n(n-l)(n -2) 
 
 2 2.6 V ^D^ 
 
 2T3.4 ^^+^^'- 
 
 34 
 
^266 
 
 SERIES. 
 
 EXAMPLES. 
 
 1. Required the sum of n terms of the series 
 1, 3, 5, 7, 9, &c. 
 
 ai = 1, 3, 5j 
 i)i = 2, 2, 
 
 These values substituted in (D) , give 
 Sn=^n-{- n{7i — 1) =n , for the sum of n terms sought. 
 
 2. Required the sum of 7j terms of the series 
 1, 3, 6, 10, 15, &c. 
 
 ai = 1, 3, 6, 10, 
 A = 2, 3, 4, 
 
 1)2=1,1, 
 i>3=0. 
 
 These values substituted in (D) , give 
 
 Sn = n-\-n{7i— l)-f- 
 
 i{n—l){7i — 2)_n{7i-\-l){n-j-9. 
 
 2.3 2.3 
 
 3. What is the sum of ji terms of the series 
 
 1, 2\ 3\ 4*, &C.'? 
 
 ^"'- 5 + 2 + 3-30- 
 
 4. What is the sum of ti terms of the series 
 
 Ans. !ii!L-hi). 
 2 
 
 5, What is the sum of n terms of the series 
 1, 2', 3', 4^, &C.'? 
 
 Ans. 
 
 ;n(n-fl))2 
 
SERIES. 267 
 
 The answer of the fifth example, being the square of the 
 answer to the fourth example, it follows that 
 { 1 +2+3+4 4-5 -f . . .n } •'= P+23+3-'+4^'+53+ . . .n\ 
 
 SUMMATION OF INFINITE SERIES. 
 
 (198.) Jin Infinite Series is a progression of quantities 
 continued to an infinite number of terms, usually according 
 to some regular law% 
 
 If each term of an infinite series is greater than its prece- 
 ding term, the series is diverging. 
 
 In general, when each term is less than its preceding, 
 the series is converging^ but this is not always the case ; 
 
 for instance, the series 1-| — ~f" o H f" ^^•■) which is called 
 
 a harmonic series, is not a converging series, notwithstand- 
 ing each term is less than its preceding one ; its sum to in- 
 finity is itself infinite. 
 
 Ji neutral series is one whose terms are all equal, and 
 their signs alternately + and — , thus, 
 
 1= ^ = a — a + a — a + a — a + &c. 
 
 Mn ascending series is one in which the powers of the 
 unknown quantities ascend, as 
 
 a-{-bx-\-cx^~\-dx^-\-&.c 
 
 A descending series is one in which the powers of the 
 unknown quantity descend, as 
 
 a -|- 6r- ' +- ex-'- -(- (^ar-3 + &c. , 
 t ^ , c , d , „ 
 
268 
 
 (199.) If we take the difference between the two fractions 
 
 ' ' we shall find ^■ 
 
 pq 
 
 'r+p 
 
 r{r-\-p) p 
 
 r f-\-p 'r{r-\-p) 
 \r r-j-pT 
 
 ; hence 
 
 so that any fraction of the form — — ^ — - is equal to-th the 
 r{r-^p) p 
 
 difference between the two fractions - and — ^ : hence, it 
 
 r r-\-p 
 
 follows that if there be any series of fractions of the form 
 
 — - — -, the sum of the series will equal -th the difference 
 r{r-\-p) p 
 
 of a series of the form - and another of the form — ^ : so 
 r r+p 
 
 that whenever this difference can be found, the sum of the 
 
 proposed series can be obtained. 
 
 EXAMPLES. 
 
 1. Required the siim of n terms of the series 
 
 In this example q= I ; p = l ; and r takes successively 
 the values 1, 2, 3, 4, &c. 
 
 ,1+1+1 + '+. ...1 
 
 ^234 n 
 
 =1-^=.- 
 
 . \2^3^4^ n^n + l/y 
 
 If 71 = 00 , then the sum 
 
 n-fl 
 
 becomes = 1. 
 
SERIES. 269 
 
 2. Find the sum of ?i terms of the series 
 
 0+2-3 + 3^ + 4:7 + ^" 
 In this example (7= 1 ; /^ = 3 j and r= i, 2, 3^4, &c. 
 
 ,,i+^+,i+i+ ' 
 
 1 A 2 J 4 fi 
 
 (1 + 1+ 1+^ + ^ + ^V 
 
 \4^5^ n~?i + l~7i-f-2~n+3/ 
 
 ^1/11 1 1 1_\ 
 
 3\ 6 7i-f-l w+2 7i + 3/' 
 
 which becomes, when n is infinite — . 
 
 18 
 
 3. Find the sum of n terms of — -j- —7 -f- \- &c. 
 
 I4O 0.0 D./ 
 
 Here we find q= 1 ; p = 2. 
 1,1. 1 
 
 l\ ^3^5^"*'*2n — 1 ' =:Ul ^—\ 
 
 2;_/l4_l. _L_4. 1 \t 2\ 2n + l/- 
 
 ( \3 o"^"**2;i — l"^27i + l/) 
 
 Therefore, 
 
 '^'" ~2\^~' 2n4^1 ) ' ^"'^ "^^ ^ 2 ^^^^^^ '^'' represents 
 the sum of n terms, and S^y^ represents the sum of an infi- 
 nite number of terms. 
 
 4. Required the sum of the series 
 
 14.^-|-l_f_ i-L-&c. 
 ^3^6^ 10~ 
 
 This series divided by 2, becomes 
 
270 
 
 and the sum of this has already been found. (See example 
 1, of this Art.) Therefore, the sum of the proposed se- 
 ries is 
 
 «" = 2(l-„-fl); S- = 2- 
 
 5. Find the sum of the series of 
 ^3.4 5 
 
 3.5 
 
 4 
 
 5.7"^7:9~9ll"^^^' 
 
 /2 
 \3' 
 
 -a^+M^ 
 
 }- 
 
 ■8- 
 
 7^9 
 
 this becomes 
 
 
 
 1- 
 
 n+1 
 
 
 2n + l 
 
 . n+1 
 '2/1+1 
 
 n+1 n + 1 \; 
 
 "2/1+1 271+3/'. 
 
 (1-1 + 1 
 
 ±1-) 
 
 If we use the upper sign, the quantity within the paren- 
 thesis will = 1 J if we use the lower sign, then this quantity 
 will=0. 
 
 Hence, the above expression will become 
 
 1 2 1_/1 n + 1 \ 1_ 1 
 2 
 
 2 ■ n+1 1 
 
 3 2/1+3 2 
 
 3 2 
 
 and since p = 2, we find 
 
 12 4(2/1+3) 
 
 1/1 ^ + 1 \ 1 
 2 \2 2/1+3/ 6 
 
 6^2(2/1+3)' 
 
 Srr.= 
 
 The upper sign has place when n is even, and the lower 
 
 sign when n is odd. 
 
 4 4 4 4 
 
 6. Required the sura of 1 1 1 |-&c. 
 
 ^ 1.5~5.9~9.13^13.17^ 
 
 J_ 
 
 4n+l 
 
 Ans. Sn = l — '^-^; S^=^l. 
 
271 
 
 • 7. Required the sum of —--{- —--\- ;r-z-\- 8i.c. 
 
 l.O .6.4 O.O 
 
 9 9 2^7 
 
 .,. — 
 
 it follows that 
 
 (200.) Since ^^^_^^^ ^^+^)('^_j_2^) ^(^+^)(^,+2;,)' 
 
 r( 
 
 ? =:i/_J ? \ (A) 
 
 r+;,)(r+2p) 2p\r{r-hp) {r+p){r4-2p j' 
 
 EXAMPLES. 
 
 8. Required the sum of ^ + _|_ + _|^ + &c. 
 
 Comparing these terms with the fraction of the left-hand 
 member of (A), we discover that p = 1, and g = 4, 5, 6, 
 &c., andr= 1, 2, 3, &c. 
 
 1.2'^2.3"^3.4~^ 71(71 + 1) 
 
 _/4_ 5_ n+2 n+3 \ 
 
 i2.3'^3.4"^ ?2(// + l)"^(//. + l)(n+2)j 
 
 1.2^\2.3^3.4^4.5^ n{n-\-l)l (?i + l)(7i+2) 
 Now, by example 1, Art. 199, we know that 
 
 J_ , 1 , 1 1 - = _!i- 
 
 1.2~^2.3"^3.4 ?j(7i-fl) 7?4-l' 
 
 therefore, the series within the parenthesis 
 _ n _ I 
 "~ H^l 1.2' 
 
272 
 
 Therefore, we have 
 4 1 , n 
 
 n- f3 _3 I n^-fn — 3 
 
 1.2 1.2 ' n+1 (n4-l)(n-|-2) 2 ' (n4-l)(n+2)' 
 vvhich, diyided by 2p = 2, gives 
 
 c— ^ I n^-\-n — 3 o _5 
 
 '*~4~^2(7i+l)(n-f2)' '^«'-4' 
 
 9. rindthesumof^^ + 34^+^-A+.-A_+&c. 
 
 ^"^- '^"-4(2;^l~(2n+lX2;i+3))' "^^^ "l 
 
 10. Find the sun. of ^3 + _|-^ + 34_+^-l_^+&c. 
 
 (201.) It is obvious that this method must be applicable 
 to series, the denominators of whose terms consist of more 
 than three factors ; but our limits will not allow us to pursue 
 this subject any further. 
 
 RECURRING SERIES. 
 
 (202.) ^ recurring series is one, each of whose terms, 
 after a certain number, bears a uniform relation to the 
 same number of those which immediately precede it. 
 
 Thus, the series 
 
 l-|_2x+8a;-H-28a3 + 100x' + 35Cx^-f- &c. 
 is a recurring one, each of whose terms, after the first two, 
 can be found by multiplying the next preceding one by 
 3x, and the second preceding one by 2x'-, and taking the 
 sum of the products ; thus : 
 
273 
 
 8x==:2xX3x4-lx2x-, 
 28x3= 8x2 X 3x+2x X 2.r-, 
 100x^1= 28x^X3x4-8x^X2xS 
 356x*= 100x^x3x + 2Sx^x2x% 
 
 (^203.) The constant multipliers 3x, 2x-5 taken together, 
 constitute the scale of relation. 
 Suppose in general 
 
 to be a recurring scries depending upon the scale of rela- 
 tion ^, g, then we shall have 
 
 (A) 
 
 Ai = Aij 
 
 (1) 
 
 A-2 = Ao, 
 
 (2) 
 
 A3^pA.2-{-qAi, 
 
 (3) 
 
 Ai=pA3-\-qA,, 
 
 (4) 
 
 A5 = pAi-\-qAs, ■ 
 
 (5), 
 
 
 An = pA„^,-\-qAn-,. 
 
 {n\ 
 
 If the scale of relation consist of three parts, p, q, r, we 
 shall have 
 
 Ai = Aiy 
 
 Az = Asj 
 
 Ai = pAz-\-qAo-\-rAij 
 As = pAi -\- qAz -j- r j3o, 
 A = pAs -\- qAi -\- rAsj 
 
 (1) 
 
 (2) 
 
 (3), 
 
 (4)( 
 
 (5) 
 
 (6)1 
 
 (B) 
 
 An =pAn-\ + q-^n-l + ^*^n-3. {n ) / 
 35 
 
274 
 
 SERIES. 
 
 And in a similar way, the successive terms of a recur- 
 ring series, whose scale of relation consists of more than 
 three parts, can be found. 
 
 If we take the sum of the group of equations (A), putting 
 Sn for the sum of w terms of the series, we shall have 
 S^^,-{-Ji.:r{-p{S—J,—A.)+q {S—Jn-^—^n) (1) 
 
 This solved for Sn, gives 
 
 By adding the group (B), and reducing as above, we find 
 
 " I +q{Sn—A-^'ln-i-Jin)+r{S—A, 
 which solved for <S„, gives 
 
 (H- q —l)Jl: + {p — l)M o — Jij 
 5„=/ . .^ + :^"^"-^ \A (D) 
 
 -Jin-l—^n 
 
 + 
 
 p-rq-\-r — i 
 
 (204.) Proceeding in this way, we might find similar 
 expressions for Sn when the scale of relation consists of more 
 than three parts. 
 
 (205.) If the successive terms of a recurring series are 
 decreasing, and the series is carried to an infinite number 
 of terms, the last terms may be neglected in formulas (C) 
 and (D), as of no appreciable value, therefore, supposing 
 n = 00 in (C) and (D), they become 
 
 - p-fq-1 ' 
 
 ^ {pJrq—l)Jii-\-{p — 1)^-2 
 
 A- 
 
 (E) 
 
 • (F) 
 
275 
 
 EXAMPLES. 
 
 1. What is the sum of the infinite recurring series 
 
 1 _|_2a:-f-3x- + 4x^ + 5x' + &c. : 
 \he scale of relation being 
 
 p = 2x ; q = — x^'l 
 In this example »/3i = 1 ; ^^0 = 20;. These values sub- 
 stituted in (E), we find 
 
 e _(2x— 1)— 2a:^ 1 
 ^ 2x — x^— 1 {l—xY' 
 
 2. What is the sura of the infinite recurring series 
 
 1 + 2x -f Sx' 4- 28x-^ + lOOx^ + &c , 
 the scale of relation being 
 
 ]) = 3x ; q = 2x'1 
 We also have in this example j?i = 1 ; ^2= 2a;, which 
 values cause (E) to become 
 
 s = ^-"^ 
 
 "^ 1 — 2a: — 2x' 
 
 3. What is the sum of the infinite recurring series 
 
 1 + a; -h 5x2 -f 13^3 _j_ 4ia;i _|_ Scc ; 
 
 the scale of relation being 
 
 p=2x ; q = 3x'1 
 
 1—x 
 
 Ans. Srr, = 
 
 CO 
 
 1— 2x — 3x2 
 
 4. What is the sum of the infinite recurring series 
 1 _j_ X -f 2x-^ H- 2x' +3x' + 3x^ + 4x«-|- 4x'^ -f &c. ; 
 in which the scale of relation is 
 
 pz=x ; q = 3^ ; r = — x"? 
 
276 
 
 5. What IS the sum of the infinite recurring series 
 l_|_4a:4-Gx-4-l]x^-f 28x-*4-63a:^+ &c. ; 
 in which the scale of relation is 
 p = 2x; q = — 
 
 Ans 
 
 (l+x)»- 
 
 •^^-(l+x)^. 
 
 2x2 
 
 3x3 
 
 From these examples >ve see that the sum of an infinite 
 number of terras of a converging recurring series, is in the 
 form of a rational fraction. Conversely, all rational frac- 
 tions, when expanded by actual division, or by the method 
 of indeterminate coefficients, as accomplished under Art. 
 184, will give a recurring series. 
 
 (206.) When the scale of relation is not given, it may 
 be found by means of a few of the first terms of the series, 
 thus : 
 
 Resuming our general equation (n) of group (A) Art. 
 203, where the scale of relation consists of two parts, p 
 and 7, we have 
 
 ^,.=;.A-i + 7^'.-> (1) 
 
 Writing ;i-f-l for tz, it becomes 
 
 A+.-y^A-f-^'l-i- (2) 
 
 From these two equations we readily deduce 
 
 
 
 A- 
 
 M 
 
 (3) 
 
 (4) 
 
 If in (3) and (4) we put Ji = 3, they will become 
 AxAi — A-.'Ai 
 
 P— A^A, — A\ ' 
 AiAi — Al 
 
 g=— 
 
 A3A1 — Al 
 
 (5) 
 (6) 
 
SERIES. 277 
 
 From (5) and (6), we shall be able to find the scale of 
 relation, when it consists of but two parts, by the aid of 
 the first four terms of the series. Equations (3) and (4) 
 show that the scale may be found by using any four conse- 
 cutive terms. 
 
 By a similar process we might, by the aid of the first six 
 terms of the series, find the scale of relation when it con- 
 sists of three parts. 
 
 (207.) A geometrical series may he also a recurring 
 series. 
 
 To prove this, we will take the general form of a geo- 
 metrical series (178). 
 
 a -f ar + ar- -f ar^ -\- ar^ + ar' -\- (1) 
 
 Now, in order that this may be a recurring series, having 
 p and q for the scale of relation, we must have, (103), 
 
 ar^ = par -j- qa, 
 ar^ = par^ -j- qar, 
 ar* = par^ -\- qar-^ 
 ar^=par* -j- qar'^^ 
 
 (2) 
 
 ar" =par'^~^ -\- qar"~ 
 
 By striking out the factors common to these conditions, we 
 see that each becomes 
 
 which gives 
 
 r' = pri-q, (3) 
 
 _P 
 
 ^±i^p^-}-4q. (4) 
 
278 
 
 When these two values of r are real, and not equal, 
 there will be two geometrical series which will also be re- 
 curring, having p and q for the scale of relation. 
 
 We will denote these two values of r, by r' and r" . 
 And since it is immaterial what value we take for a, the 
 first term, we will take a' for the first term when we use r', 
 and a" when we use r" . The two series will then become 
 
 a' _f_ a V' + a 'r'^' + a' r" -fa' r" -\- a' r' ' -\- . . 
 
 ip) 
 
 (6) 
 
 a"-\- a"r"-\r a"r"^-[- a"r"^+a"r"*-\- a' V'^+ . . . . , ^ 
 each of which is a recurring series having p and q for the 
 scale of relation. If we take the sum of the correspond- 
 ing terms of the two series (5), we shall find 
 
 (u'+a') + (aV+a"r")-f(aV^ -■-«"/•"'-) ; 
 
 -\- i^.'/ '> a"r"-') -|-(aV4-f-a"r"')-j-&c., ' 
 
 which is not a geometrical series, but is, nevertheless, a 
 recurring series having p and q fpr the scale of relation. 
 
 In all recurring series whose scale of relation consists of 
 two parts, we must, in order to be able to compute the 
 successive terms, know the values of the first two terms, 
 which we have represented by Ai and Jl-y. 
 
 Since the values of a' and a", in (6), have not yet been 
 fixed, it is evident they may be so taken as to make the 
 first two terms of (6) agree with Ai and A^. This is effect- 
 ed by making 
 
 a'-fa"=.3.. 
 
 (7) 
 
 aV-|-a'V'=^2. 
 
 (8) 
 
 These equations immediately give 
 
 
 r'—r" ' 
 
 (9) 
 
 
 (10) 
 
SERIES. 279 
 
 The nth term, Jin-, of the recurring series (6) is 
 
 o'r'''-'-|-o"r"''-^ 
 
 Hence, if we substitute the values of a' and «", as given 
 by (9) and (10), we shall have 
 
 r' — ;•" r — r 
 
 (208.) By a similar train of reasoning, we might show 
 that a recurring series whose scale of relation consists of 
 three parts, is the sum of i\\xee geometrical recurring series, 
 having the same scale of relation. 
 
 1. The recurring series l-\-x-\-3x'-\-5x-'-\- &c., whose 
 scale of relation is x and 2x-, is the sum of the two follow- 
 ing g-eo7?ie^7?ca/ recurring series, each having the same scale 
 of relation: 
 
 3 '3 ^3^3 ~ 3 ' 
 
 1 1.1., 1 , , 1 4 s 
 
 3 3^3 3 ~ 3 
 
 1_|_ a;_^3a;^_|_5x3-f-lla;^ + &c. 
 
 2. The recurring series, l-j-.r-j-Sx^-f-l^^^'-f-'^c., whose 
 scale of relation is 2x and 3x^, is the sum of the two follow- 
 ing gfCome^Wca/ recwm?2o- series, each having the same scale 
 of relation : 
 
 1.3 , 9 . . 27 3 ,81 , , , 
 2"^2'' + 2 "^2" "^2"'" + 
 1 1 , 1 , 1 . , 1 , . 
 2~2'''^2^'~2 "^2 ^ ^^'•'• 
 1+ x-f-5x2+ 13x^-1- 41 x'-f-&c. 
 
280 
 
 3. The recurring series, l-{-x-\-2x'--\-3x^-\-6x'^-{-kc., 
 whose scale of relation is 2a:, a;'-, and — 2x^j is the sum of 
 the three following geometrical recurring series, each hav- 
 ing the same scale of relation : 
 
 -x4--x- — -x'^-\ — X* — &c. 
 6 ^6 6 ^6 
 
 1+ rr-f-2x--|-3r^+ 6 x^-f &c. 
 
 (209.) From what has been shown, it follows that we 
 may regard all geometrical series as recurring series, but 
 all recurring series are not geometrical series. When 
 a recurring series is not a geometrical series, it is the sum 
 of two or more geometrical series. Hence, a geometrical 
 series may be regarded as a particular case of a recurring 
 series, the recurring series being of a more general form. 
 
 (210.) We will now give some examples in which the 
 general term, J?,,, of a recurring series is required. 
 
 EXAMPLES. 
 
 1, Suppose the scale of relation of the series 
 1 + x -f 3a;- + 5r'+ Ux' -\- 21x^ -}- &c., 
 to be ^ = a; ; g = 2x^, what will be the 7ith term ? 
 
 In formula (A), the values of r' and r" are given by (4). 
 In this example they become 
 
 r'=2x; r"=^ — x. 
 
 From the given series we have 
 
 Ji\ = 1 : »/?2 == X. 
 
SERIES. 281 
 
 Substituting these values in formula (A), vrc find the 
 Ans. ^n= !(2a:)"-' + ^(— x)'^K 
 
 2. If the scale of relation of the series 
 
 1-l-x-f 5a:2-}-13x3 + 41x^+121x«+ &c., 
 is ^ = 2x ; q = 3a;"^, what will be the nth term ? 
 
 Ans. ^„=i(3x)"-i + i(— x)"-'. 
 
 3. The scale of relation of the series 
 
 1 + 2x -f 5x* + 13x3 _|_ 34a;4 ^ 89x^ + &c., 
 being j9 = 3x; q = — x^, what will be the nth term ? 
 
 (v/5 + i)( 3 + -y^y-' ^„_, 
 
 Ans. ^„ = <f ^"^^ 
 
 (^5-l)(3-^/5)"-^ ^„_,^ 
 
 Referring to the (question of the oak tree, under Chap. 
 XII, Higher Arithmetic, we see that if we call x = 1, the 
 above expression for Jl„ will give the number of branches 
 of the tree, at the end of n years, thus the number of 
 branches at the end of 20 years is 
 
 (1 + v/5)(3+ v/5)'^ _ (l-v/5)(3-v/5V^ 
 2-V5 2-V5 
 
 (211.) Having shown how to find the general term of 
 a recurring series, it is easy to find the sum of n terms by 
 the aid of formulas (C) and (D), Art. 203. 
 
 1. Find the sum of 7i terms of the recurring series 
 1-f-x-f 3x2-(-5x3-|-llx'-|-21xS4- &c., 
 whose scale of relation is 
 
 j) = x; q=2x^ . 
 Under the last Article, we have found the nth term to be 
 
282 
 
 Writing n — 1 for n, we find 
 
 A-i = |(2x)"-H^(-^)"-'- 
 
 Substituting these values of A and A-i? together with 
 the known values of p and q, in (C), Art. 203, we have 
 _2''+'{l+x)x^-{ {l—2x){—xY—3 
 '^" ~ d{2x^-i-x — l) ' 
 
 or, perhaps a simpler form is 
 
 (2»+^^2)x"+'-f-(2»+'dbl)x''— 3 
 ^"~~ 6x-^ + 3x — 3 
 
 In this last expression, the upper sign is to be used when 
 n is even. 
 
 2. Find the su-m of 7i terms of the recurring series 
 l-{-x-{- 5x-+ 13x^-j-41x^ -f &c., 
 
 whose scale of relation is 
 
 p =2x'^ q = 3r^. 
 
 In this example, the nih. term is 
 
 A=|(3xy-+l(-x)"-', 
 
 or which is the same thing, 
 
 A=l(3'-'=Fl)x«-'. 
 
 The upper sign must be used when n is even. Writing 
 n — 1 for 71, we find 
 
 The values of ./?„, ^„_i, /j, and 7, being used, cause for- 
 mula (C) to become 
 
 3(3"-'Tl)a:''+' + (3"drl)x''4-2j — 2 
 *""" 6x'^-|-4x — 2 
 
 Use the upper sign when n is even. 
 
CONTINUED FRACTIONS. 
 
 CHAPTER VIII. 
 
 CONTINUED FRACTIONS. 
 
 (212.) Suppose we have the following conditions: 
 -^=2/+^, (1) 
 
 ^.=2/. + g,, (2) 
 
 D-^=y+^, (3) 
 
 ■Us 
 
 ^3=3/3 + £, (4) 
 
 &c. &c. 
 
 In (1), for Di, write its value as given by (2), and it 
 
 becomes Jl = y- 
 
 X.2 
 
 yi-^~. In this expression, for Da, 
 write its value given by (3), and we have 
 
 ..+^-^ 
 
 y^'+D, 
 
284 
 
 CONTINUED FRACTIONS. 
 
 Now substituting for D3, its value given by (4), and we 
 obtain j3=y-| — ! 
 
 3/2- 
 
 X3 
 
 J/3- 
 
 X4 
 
 Di-\- &c. 
 
 (5) 
 
 (213.) Such expressions as the above value of ^, equa- 
 tion (5), are called continued fractions. 
 
 The expressions — , — , — , &c., of which — is the gene- 
 
 Vn 
 
 expressions — , 
 
 2/1 y-y 3/3' 
 
 ral term, we shall call partial fractions. The numerators 
 xi, x-2 X3, a:,i, we shall call partial numerators. 
 
 The denominators i/i, y^, 3/35 Vn-, in like manner, 
 
 we shall call partial denominators. 
 
 If we compute successively 1, 2, 3, 4, &c., terms of the 
 continued fraction (5) , by reducing them to the form of 
 common fractions, we shall find 
 
 91' 
 
 
 1 
 1 
 
 yyi+Ji 
 3/1 
 
 P3 
 9i' 
 
 H 
 
 y^_^^ ^my2_-h^^yr±y^ ^^3^ 
 
 Xi 
 
 3/i-f- 
 
 I ^3 _ yyiy9y3-\-Xiy2y:i-\-yxc^9-\-yyiX3-\-XiX 3^ pi 
 
 ^ yz ^1^23/3+ 3^22/3 +.yi-^8 94* 
 
 &c &c. 
 
 We shall hereafter represent these successive values, 
 which we shall call approximative fractions ., by the abridged 
 
 expressions ^, ^-, ^, &c., of which the general term is ^. 
 qi q-i 93 qn 
 
CONTINUED FRACTIONS. 286 
 
 By carefully examining the above approximative frac- 
 tions, we discover the following relations : 
 p3=p^y2-\-pix,, (6) '73= 922/0 +9ta^3, (8) 
 
 Pi = p:iy3-\-p23:3j (7) q\=q3y3-^q-2T3. (9) 
 
 By referring to our continued fraction (5) , we see that 
 
 — will change to—, if we substitute 3/3-I — for y^. Mak- 
 94 qs y\ 
 
 ing this substitution in (7) and (9), we find 
 
 ?3| y^-\ — - ) +i>2X3 v^{pm-\-pi^^) -\~P^^* 
 ^_ V yii ^ 
 
 93 1 ya -f ^ J + 9--i^3 yi{.q2y-3 + q-^z) + 93x4 
 .p-iy4-|-?3^4 
 
 94yi+!73J^< 
 
 Hence, ;j5 = 7>4y4+P3^4, (10) 90 = 904 + 932^4. (H) 
 
 And in the same way may pr, and 9,; be drawn from the 
 next two inferior values. Therefore the law is general and 
 may be expressed as follows : 
 
 Pn = Pn-OJn~\-\-Pn~^Xn~:^ (12) 
 
 • 9„=9„_jy,._,-|-9„-ox„_,. (13) 
 
 (214.) If we place our quantities in the following order : 
 
 y y\ y-2 ya y\ yn~i yn 
 
 I . y_. Vi . Pj. PJ.. PjlZl . Pjl.S^r 
 
 0' 1' q.' 93' 7.' 9«-i' qn' 
 
 X\ X X3 .T4 15 Xn X.-i-i 
 
 We may find the successive approximative fractions by 
 tke following 
 
286 
 
 CONTINUED FRACTIONS. 
 
 RULE. 
 
 Multiply the numerator of the last approximative frac- 
 tion by the partial denominator which stands over it, and 
 the numerator of the approximative fraction which precedes 
 this, by the partial numerator which stands under it; the 
 sum of these products, noticing the signs, is the numerator 
 of the next approximative fraction. In like manner we 
 must multiply the denominator of the last approximative 
 fraction by the partial d.enomAnator which is over it, and 
 the denominator of the approximative fraction which pre- 
 cedes this by the partial numerator which staiids under it; 
 the sum of these products is the denominator of the next 
 approximative fraction. 
 
 EXAMPLES. 
 
 1. Find, by the above rule, some of the approximative 
 fractions of the infinite continued fraction 
 
 9 — &c. 
 Our work, when executed agreeable to the above rule, will 
 be as follows : 
 
 a 
 
 3 
 
 5 
 
 
 7 
 
 9 
 
 
 1 
 
 a 
 
 3a— a' 15 
 
 a — 5a^ — a^ 
 
 105a— 35a-'— 10a 
 
 +a* 
 
 6' 
 
 i' 
 
 3 
 
 5 
 
 15 — a^ ' 
 
 105 — 10a2 
 
 
 — a2 
 
 — a' 
 
 — a 
 
 .' 
 
 — a' 
 
 — a' 
 
 [&c 
 
 2. 
 
 Find 
 
 some 
 
 of the approximate fractions of the 
 
 con- 
 
 tinued fraction 
 
 
 
 
 
CONTINUED FRACTIONS. 287 
 
 i+?i 
 
 3a 
 
 4a 
 
 14^" 
 
 I4-&C. 
 
 In this example, y, the integral part, is nothing, and our 
 work is as follows : 
 
 111 1 1 
 
 10a a a-f-3a- G-j-7a^ „ 
 
 0' T' r l+2a' 1+5^' 14-9a-f8a--*' ^' 
 a 2a 3a 4a 5a la 
 
 (215.) If, in our general expression (5), we suppose all 
 
 the partial denominators j/i, yo, 1/3, y„ to be positive, 
 
 and also 1 = xi = Xo == x^ = = a:„, we shall then 
 
 have 
 
 yr 
 
 y+'- 
 
 y^-\ (14) 
 
 1/4-}~&C. 
 
 This is the kind of continued fraction most commonly 
 employed. Any common vulgar fraction can be converted 
 into a continued fraction of the above form, by the method 
 explained in my Higher Arithmetic^ which is equivalent to 
 the following 
 
 RULE. 
 
 Divide the denominator by the numerator ; then divide this 
 divisor by the remainder^ and thus continue to divide the 
 preceding divisor by the last remainder, until there is no 
 remainder^ or until ice have obtained as many terms as rce 
 
288 
 
 CONTINUED FRACTIONS. 
 
 wish; then will these successive quotients be respectively the 
 
 values of 3/1, 1/2,1/3, yn- 
 
 Note. — In this rule we have supposed the vulgar fraction 
 to be less than a unit, and consequently the integral part 
 y = ; when the fraction is not less than 1, we may first 
 reduce it to a mixed number, and then proceed with the 
 fractional part agreeable to the above rule. 
 
 EXAMPLES. 
 
 1. Convert 4x411 4 into a continued fraction. 
 
 05 
 
 as Oi lo 
 
 OJ 
 
 CO CO 
 CO t^ 
 o< o 
 
 
 5^1 
 
 y 
 
 
 II 
 
 CO 
 
 
 Oi (N 
 
 
 
 
 kO 
 
 CO CO 
 
 
 O' 
 
 
 5»i 
 
 lO lO 
 
 
 
 il 
 
 00 CD 
 
 (N 
 
 Tt^cr. 
 
 lO 
 
 CD 10 
 
 
 ^ 
 
 
 
 
 
 
 co"© 
 
 CO 
 
 
 CO Tl< 
 
 03 
 
 
 r-l O) 
 
 1— 1 
 
 
 OJ 10 
 
 CO 
 
 
 CO^CO 
 
 
 
 
 
 
 lO 
 
 
 
 00 
 
 
 
 "* 
 
 
 
 CD 
 
 
 
 i-i 
 
 
 
 ^2 
 
 O CD 
 
 i-i 00 
 
 CO 
 
 CD 
 
 CO 
 
 03 l> 
 
 
 to c- 
 
 Ol 
 
 iO ^ 
 
 CD 
 
 /-^ 
 
 
 
 
 00 
 
 
 CO 
 
 
 <N 
 
 
 0< 
 
 
 O .1 
 
 (N I-I 
 
CONTINUED FRACTIONS. 289 
 
 Therefore, 41^7 5 = 
 
 o+ — J 
 
 '+— 1 
 
 8+-. 
 
 2. The fraction IHHMf IK expresses nearly the ratio 
 of the diameter of a circle to its circumference ; required to 
 expand it into a. continued fraction. 
 
 Proceeding as above, we find 
 1 
 
 7 + 
 
 1+- 
 
 292+&C. 
 (216.) Referring to (14), we see that Ayy^ since, in or- 
 der to obtain the true value of ^, something must be added 
 
 to y. Again, A < y-\ — , since, in order to obtain the true 
 
 2/1 
 value of A^ the denominator y\ must be increased, and con- 
 sequently y-\ will be diminished. We can show in the 
 
 same way that 
 
 Ayy^- A<:y^- ^ 
 
 3/1 + -, y-\ r- 
 
 2/2 , 1 
 
 3/3 
 
 37 
 
290 
 
 CONTINUED FRACTIONS. 
 
 (217.) Therefore., the value of A is always comprised be- 
 tween two consecutive approximative fractions. 
 
 (218. ) When \ == xi = x-, = x- = ^ a:„, equations 
 
 (12) and (13) become 
 
 Pn=Vn-Xyn--{-pn~; (15) 
 
 qn = 7„_i2/n-l +?„--. (16) 
 
 Multiplying (15) by </„_i, and (16) by p„_i, and then 
 taking the difference of the results, we find 
 
 p^qn-i. —p„-iqn = — {pn-iq,^-i—pn-2q>,-0- (l'^) 
 
 This shows that pnq„-i — Pv—iQn andp„_i9„_,. — ;>„_27,i_i 
 are equal in numerical value, but contrary in sign. When 
 n =2, equation (17) gives 
 
 Piq~ —ihq2 = — {piqo—pi,qi)- 
 
 We know that— =- and— ==- ; consequently, piqo — 
 <?i 1 go 
 
 poqi = — 1 ; therefore, p^qi — piq^z ~ 1, pnqo — p^qs^ — l? 
 
 p^q.t — p394 = 1 ; and so on for other similar expressions. 
 
 Hence, we always have 
 
 PnQn-l —Pn-\q„ = ±1- (l8) 
 
 The upper sign having place when n is even, and the lower 
 sign when n is odd. 
 
 If we take the difference of two consecutive approxima- 
 tive fractions we shall find 
 
 p«7«- 
 
 Pv-\qn 
 
 q„ qn-^ qnqn-i 
 
 By (18), we know that the numerator of the right-hand 
 member of this equation is = i 1 . Hence, 
 
 Pn Pn-\_ =bl 
 
 q» qn-i qnqn-\ 
 
 (19) 
 
CONTINUED FRACTIONS. 291 
 
 This shoics^ that the difference between any two consecu- 
 tive approximative fractions is equal to the reciprocal of 
 the product of their denominators. 
 
 We have already shown that the true value lies between 
 the values of any two consecutive approximative fractions, 
 
 and since 7„> qi^-x-, we have -^ > . Therefore 
 
 the difference between Lj^tL and Ji is less than . That 
 
 J5, the difercncc between the true value and any approxima- 
 tive fraction., is less than the reciprocal of the square of its 
 denominator. 
 
 dividing (15) by (IG) we find 
 Pn^]h:-^y^i±Pj^^ 
 7„ q„^iy„-i-\~q„^z' 
 
 If, in this ecjuation, for y _i, we substitute the complete 
 denominator, which we will represent by z, then will 
 
 "'— =^?. From the form of our continued fraction (14), 
 
 it is obvious that z will also be in the form of a continued 
 fraction. 
 
 Thus, c = ^_, + ^ 
 
 y.^'- 
 
 y..+1-f 
 
 This value of z being substituted for y,--. in (20), we 
 have ^=Plil-J-Jjlz:\ (21) 
 
 qn-\Z-\-qn-'2 
 
 Equation (21) immediately gives 
 
 ^-J^= ^"--'?"-^-P"-^?"-^ (22) 
 
 qn-j 9„_i(9,._iz4-7„-o) 
 
 qn-2 5„_2(9„_l2-j-<7„_2) 
 
292 CONTINUED TRACTIONS. 
 
 The condition of (18) causes these to become 
 
 A 
 
 A— 
 
 </„_;(ry„_iZ-f-(/„_o)' 
 
 Pn-- 
 
 (24) 
 (25) 
 
 g»-2 i7„-o(9„-iz+g„_o) 
 
 Now, by the nature of continued fractions, c>l, and 
 q„-\ > (7„_i:. Hence, the right-hand member of (25) is 
 greater than the right-hand member of (24). Considering 
 the numerical value without reference to the signs. This 
 shoios^ that each appro.iiinafe fraction is nearer the true value 
 than the preceding approximative fraction. 
 
 p. 
 
 If 
 
 ■> A^ conditions (24) and (25) will giv( 
 
 Pn-\ 
 
 qn-i 
 
 _Pn-2 
 
 ?«-l(7"-l-+?n-2) 
 
 7„_, q„^o{qn-lZ-{-qn~2) 
 
 If ?!!ii-< A, conditions (24) and (25) will give 
 
 qn-2 
 
 A=P-^' - . 
 
 qn-i 7„-i(7„_ir+g„_o) 
 
 A-P^' 
 
 (26) 
 (27) 
 
 )• 
 
 qn-2 qn-^iqn-iZ-^-q 
 Equations (26) and (27) give 
 
 ISP^^PjuillyA. 
 
 nqn-2 qn~x) 
 
 Equations (28) and (29) give 
 
 Hence., the successive arithmetical means of two consecu- 
 tive approximative fractions are alternately greater and less 
 than the true value. 
 
 (28) 
 
 (29) 
 
 (30) 
 (31) 
 
CONTINUED FRACTIONS, 293 
 
 If we take the product of (26) and (27), we find, after a 
 little reduction, 
 
 _ (p„_ig„_i- — p„_2?„-2)+(P»-l?»-2— ;j„-2g;,-l)zH-g 
 
 Now, since ^—^yjl, we have, by (18), 
 
 gK-2 
 
 {p,^iqn-2 — p>,~^qn-l)z-\-Z = 0. 
 
 Moreover, we have pn—\qn-\Z yp^-^q^-z. 
 Consequently, ^""'^"~' > S'. (33) 
 
 qn--2qn-l 
 
 Taking the product of (28) and (29), we find 
 
 >(34) 
 
 qn-zqn-i 
 
 {p„^^qn-lZ—pn-^n-z)-{-(pn-iqn-2— p„ -iq„-l)z-\-Z ^ 
 ■*" 9„_o9„_i(9„_iC-|-9h-2)- 
 
 Now, since ^^ < Ji, we have, by (IS), 
 
 {pn-iq„-2 — pn-^n-l)z— Z = 0. 
 
 And, as before, pn-\qn-\Z ypn—^n-z- 
 
 Consequently, ^""'^"'^ < A^. (35) 
 
 qn-iq>i-i 
 
 Equations (33) oTjd (35) show that the successive geomet- 
 rical means of two consecutive approximative fractions are 
 alternately greater and less than the true value. 
 
 (219.) All approximative fractions are always in their 
 lowest terms. For if not, let the numerator ;?„, and its 
 
294 CONTIXUED FRACTIONS. 
 
 denominator ^-n, of the approximative fraction—, have a corn- 
 s''* 
 mon divisor h. 
 
 Condition (18) shows, that if p,, and q^ are each divisible 
 by A, then its left-hand member must be divisible by A, and 
 consequently its right-hand member is also divisible by h ; 
 that is, ±1 is divisible by A, which is absurd ; consequently 
 it is absurd to suppose pn and §« to be divisible by h. 
 
 (220.) In the case of 1 =a:i=:a:2 =0:3 = = ar„, 
 
 the rule under (Art. 214) will require some modijfication in 
 order to appear in its simplest form. Thus, placing the 
 quantities as follows : 
 
 y 3/1 y-i y-i y^ y«-i yn 
 
 y pi Pi Pi Pn-l 
 
 ,f»,&c. 
 
 1 92 q:i Qi q^x q„ 
 
 We deduce the successive approximate values by this 
 
 RULE. 
 
 Multiply the mimerator of the last approxim,ative frac- 
 tion by the jio^rtial denominator standing over it^ and to the 
 product add the numerator of the preceding approximative 
 fraction^ and the sum will he the numerator of the next ap- 
 proximative fraction. In like manner multiply the denomi- 
 nator of the last approximative fraction by the partial de 
 nominator standing over itj and to the product add the de- 
 nominator of the preceding approximative fraction, and it 
 will give the denominator of the next approximative frac- 
 tion. 
 
 1. What are some of the approximative fractions of th«- 
 iccfinite continued fractioii 
 
CONTINI'ED FRACTIOyS. ' 295 
 
 3+— 1 
 
 1 2 7 30 157 972 
 1' 3' 10' 43' 225' 1393 
 
 2. What are the approximative fractions of the continued 
 
 1 
 
 fraction 
 
 3 + ^ 
 
 1 + i 
 
 
 , 1 
 
 1 7 8 23 100 523 623 1769 
 3' 22' 25' 72' 313' 1637' 1950' 5537" 
 This last value expresses accurately the true value of 
 the above continued fraction. Whenever the value of a 
 continued fraction is capable of being expressed rationally, 
 it must consist of a finite number of terms j but when the 
 value is irrational, the continued fraction will extend to 
 infinity. 
 
 (221.) Continued fractions may be employed for determi- 
 ning approximately the values of the square roots of surds. 
 
296 
 
 CONTINUED FRACTIONS. 
 
 Operating upon -v/19, we obtain the successive values : 
 ' v/19— 4 3 ^3 ^'^Da 
 
 m = 
 
 3 v/19 + 2 , , v/19 — 3 , 1 
 
 = ■ — =1H — ^=iJc>-\ . 
 
 v/19— 2 5 ^5 ^~~D3 
 
 „ 5 v/19 + 3 _,v/19-3 ,1 
 
 „ 2 v/19 + 3 v/19— 2 , 1 
 
 J. 5 vl9 + 2_ v/19-4 1 
 
 3 v/19+4_^ , v/19-4 _^, , 1 
 A- -^^^^^= - 1 8 + ^ -3'a+^^- 
 
 ' v/19 — 4 3 ^3 ^'^Ds 
 &c. 
 
 &c. 
 
 Collecting the results, we have 
 y = 4, 2/1 = 2, y3=:l, 2/3=3, 3/4= 1, 2/5 = 2, 3/9 = 8, 
 y7 = 2, &c. 
 
 v^29 + 4 _ v/194.2 v/19 + 3 
 i^i— g , JJi— , U3— , V4 — 
 
 v/19 + 3 ^ _ v/19+2 J. _ v/19+4 ^ _ v/19+4 
 5~~'^'"" 3""'^"- T"'^' 3~- 
 
 The values of Dj, Do, D3, &c., of which the general 
 term is D„, arc complete denominators of their correspond- 
 ing partial fractions. 
 
 The values of yi, 2/2? 3/35 &c., which we have already call- 
 led partial denominators^ are the greatest integral parts of 
 
CONTINUED FRACTIONS. 297 
 
 their corresponding complete denominators Dj, D2, D^, &c. 
 
 In this example, we see that Dt= Di = - — -^ , and 
 
 y7 = yi = 2, hence the operation must begin to repeat at 
 this point, and the partial denominators as well as the com- 
 plete denominators will recur in an infinite number of 
 periods. 
 
 (222.) Suppose we wish the value of the surd ^/5. 
 If a- is the greatest square contained in 5, with the re- 
 mainder 6, we shall obviously have 
 
 ' (36) 
 
 J. ^ _ 1 ^ vAB-U ^ y/^-f-g ■ 
 
 ^ VB — a B—a" h 
 
 The form of the general value of D,i will be 
 
 D.= ^^±^\ (37) 
 
 If n-\-\ is written for 71, this becomes 
 
 D„^a^ ^^+-^V (38) 
 
 Now, by carefully inspecting ^the operations just per- 
 formed for finding the value of v/19, (Art. 221), we draw 
 this relation : 
 
 ^« = 2/H-^. (39) 
 
 Substituting in (39) the values of D„, A.+i) given by 
 (37) and (38), we find 
 
 38 
 
298 
 
 CONTINUED FRACTIONS. 
 
 This cleared of fractions, becomes 
 
 B-\-{Mn-\-Mn+l) VB-j-MnMn+l ) 
 
 = ynJrWB + ynJrnMn+l-\-jynJyn+l. ^ 
 
 Equating the irrational parts, as well as the rational parts 
 (Art. 116), we find 
 
 Mn-{-Mn+l = ynJK, (41) 
 
 B-\-MnMn+l = ynJVnMn+l + JynJVn+1. (42) 
 
 These equations readily give 
 
 M„+i = ynMn—M„. 
 
 B — J^n+l 
 
 M„+l= 
 
 JVn 
 
 (43) 
 (44) 
 
 If in (44) we substitute for MnJf-\ its value given by (43) , 
 we find 
 
 _B-M£ 
 
 Equation (44) gives 
 
 j^, _B-MUi 
 
 yn'-J^n + ^ynMn. (45) 
 
 JVn 
 
 In (46) writing n — 1 for tj, we get 
 This causes (45) to become 
 
 JV;+i = K„^t—yn^jr+2ynMn 
 
 (46) 
 
 (47) 
 
 (48) 
 1, J\r„, and 
 
 Equations (43) and (48) show, that when JV, 
 Mn are whole numbers, then will J^n+i and jlf„+i be whole 
 numbers. When n=: 1 , we have by (36) JV„_i= 1 , JV^=6, 
 and Mn = a- 
 
 Hence, JV^ and Mn are whole numbers for all values ofn. 
 
 (223.) If in (21) we substitute 
 change to v/J?, and we shall have 
 
 ^;_i 
 
 for c, ^ will 
 
CONTINUED FRACTIONS. 299 
 
 VB = . (49) 
 
 Clearing this of fractions and reducing, we have 
 
 q„^iB+qn-i^I«-i VB+q»-^^^„-l ^/5 =pn-i VB 
 +;j„_,JW„_i+p,_,JV;_i. 
 
 Equating the rational quantities, as well as the irrational, 
 we have 
 
 Pr^^Mn-,-\-pn-^'n~l = Qn-lB . (50) 
 
 qn-lMn-l + g;,-oJV„_i = p,^i. (5 1 ) 
 
 From (50) and (51) we readily deduce 
 
 p„^iqn-2—q„-^Pn-2 
 
 j^^_^_qn-iqn--.B -pn-.,p^, ^ ^^3^ 
 
 p„^iqn-2 qn-lpn-1 
 
 It is readily seen, by reference to (18), (33), and (35), 
 that the numerators and the common denominator of (52) 
 and (53) always have like signs. 
 
 Consequently^ J^a and Mn are positive for all values ofn. 
 
 Equation (47) shows that Mn<iVB; that is, J\U cannot 
 exceed a. Equation (43), by transposition, becomes ynJ^n 
 = Mn+i -\-Mn ; which shows that J^n as well as y„ cannot 
 exceed 2a. 
 
 JVow, since the continued fraction, which expresses v/B, 
 must extend to infinity, and since Mn as well as JV„ are posi- 
 tive integers less than 2a, it follows that the values of Mn 
 
 and JVn in Dn = — — ^— rnust recur in periods. 
 
300 
 
 CONTINUED FRACTIONS. 
 
 Suppose the number of terms in a period to be w, so that 
 y„+i = 2/1, Mn+i = Ml, JSTn+i = JV\. (54) 
 
 If z denote the complete denominator, corresponding to 
 the partial denominator 2/„, we must have 
 
 z = y« — a-f^/5. (55) 
 
 Hence, 
 
 ^j5 ^ PnZ-^Pn-l _Pn {yn — a) +Pn VB+P n-l ^^g. 
 
 q,,Z + q,i-i qniVn — a) -j- ?» VB-\-q„^i ' 
 
 Proceeding with (56) as was done with (49) , and we 
 find 
 
 PniVn — a) +Pn-l = Bq„. (57) 
 
 gn(2/n — a) + qn-l = Pn- (58) 
 
 Equation (58) gives 
 
 Pjl = .y^-a-{.t=}. 
 
 (59) 
 
 Now, since — — cannot be a whole number, it follows 
 5/1 
 
 that Vn — a is the greatest integral part of ^ ; but, since the 
 
 process begins to repeat at this point, the greatest integer of 
 
 ^ is a; therefore, y„ — a^a, hence, y,i^2a. 
 
 So that, whenever we obtain a partial denominator which 
 is equal to twice the greatest square root of B, the process 
 must begin to repeat. 
 
 Equation (58) gives, when 2a is substituted for y^,, 
 qn-i=^Pn — o,<l>*j which, divided by q„, becomes 
 
 qn qn 
 
 =P-^-a. 
 
 (60) 
 
CONTINUED FRACTIONS. 301 
 
 Inverting both members of (16), it becomes 
 1 1 
 
 (61) 
 
 qn yn-iqn-l + qn-2 
 
 Multiplying the numerator of the left-hand member of 
 (61) by 5,1—1, and dividing the denominator of the right- 
 hand member by the same quantity, we obtain 
 
 q.-i 
 
 
 9'^ 
 
 qn~i 
 
 
 (62) 
 
 Writing 71 — 1 for 
 
 n, in (62), and it 
 
 gives 
 
 
 
 9„_o _ 
 
 1 
 
 
 
 
 7/„_o-f-(/,._3 
 
 
 
 
 qn-2, 
 
 
 
 which substituted in 
 
 (62) gives 
 
 
 
 
 9n-l _ 
 
 1 
 
 
 
 
 "■ V. 
 
 , 1 
 
 
 
 ' -+:: J 
 
 (63) 
 
 Again 
 
 , in (62), writing n — 2 for n 
 
 we get 
 
 
 
 qj^_ 
 
 qn-2 
 
 1 
 
 y„_3-t-9„_4 
 
 q'n-a, which substituted in (63), 
 
 we get 
 
 qn-, _ 
 
 1 
 
 
 
 
 3/n-l 
 
 , 1 
 
 
 
 . 1 ' 
 
 
 
 yn-2-t- 
 
 1 
 
 
 
 
 !/n-3 + 
 
 qn-4 
 
 (64) 
 
302 CONTINUED FRACTIONS, 
 
 Continuing this process we discover, that the value of 
 
 i2zL is expressed by a continued fraction, less than a unit, 
 
 qn 
 of which the partial denominators are the same as those of 
 the continued fractions arising from ./Bj tal^ in a reverse 
 order. 
 
 From this, we see that if the partial denominators are 
 symmetrical, that is, of the following form : 
 
 yi>3/2,J/3, y3, 3/25 2/15 (65) 
 
 then will i^=-, or q„^i = j^,,^ and conversely, if 
 
 9n qn 
 qnr~i=pni then will the partial denominators be symmet- 
 rical as given by (65.) 
 
 Now, (60) shows, that ^^ is the same as the value of 
 
 ^ in the expansion of v//j, after a is subtracted. 
 
 Hence it follows, that, if we neglect the integral part 
 a = yj the partia/ fJcnominators of the continued fraction 
 arising from t.K ualue of ^/B, will recur in symmetrical 
 periods. 
 
 (224.) We will now extract the square root of some surds 
 by the method of continued fractions. 
 
 From equations (43) and (44), we readily find Jl/„-t-i, 
 
 JV*„^.i, when 2/„ is known. Then by condition ^ — ^ 
 
 = yn+i + &c., we readily deduce i/n+i- Continuing the 
 process in this way we find in succession all the different 
 partial denominators, of which the general term is y„. 
 
 Observing the above law of derivation, we have in the 
 case of v/19, the following successive operations : 
 
CONTINUED FRACTIONS. 303 
 
 ^W:"=,„+&c; M.y,^M..=M.^,, 5^'=^.+.. 
 ^E±^=4 + &c; IX 4-0 = 4; li=i^=3. 
 ^=2 + &c;3x2-4 = 2;l^^=5. 
 
 ^^2=l + &c;5xl-2 = 3;i^' = 2. 
 
 ^+l=3 + &o; 2x3-3 = 3; ^^^^=5. 
 
 :^3=l+&c; 5 Xl-3=2; 1^^'=3. 
 &c., &c., &c. 
 
 These operations are all so simple that most of the work 
 can be performed mentally. Consequently, the conversion 
 of the square root of a surd into an infinite repeating con- 
 tinued fraction, is a very simple matter. 
 
 If jB=2S, we shall have by proceeding as we have 
 already done for B =: 19, the following periods of partial 
 denominators : 
 
 !/) 2/15 2/2, 2/3, 2/45 
 
 5; 3, 2, 3, 10; 3, 2, 3, 10; 3, 2, 3, 10; &c. 
 
 and our continued fraction becomes 
 
 ^28 = 5+-— 
 
 3 + i-, 
 . '+— 1 
 
 3+'- 
 2+-, 
 
 3+ 
 
 10 + &C. 
 
304 
 
 CONTINUED FRACTIONS. 
 
 If we compute some of the approximate fractions, by 
 Rule, under Art. 220, we shall find 
 
 5, 3, 
 1 5 
 0' 1 
 
 37 
 
 7 
 
 10, 
 127 
 24 
 
 3, 
 1307 
 
 247 
 
 &c. 
 
 v/28>5j ^/28<|; ^/28>^; ^28 <^ ; v/28>^^; 
 
 and so on for the successive values 
 
 from the square root of 28 by a quantity less than 
 
 This last value differs 
 1 
 (247)^' 
 
 In the same way we find, for the square root of 31, the 
 following partial denominators, the first terra being always 
 the integral part : 
 
 5 ; 1, 1, 3, 5, 3, 1, 1, 10 ; &c., the approximate fractions are 
 
 1 5 6 11 39 206 657 863 1520 16063 . 
 _._._. . . . . . . . • 5j,c 
 
 ' 1 ' 1 ' 2 ' 7 ' 37 ' 118 ' 155 ' 273 ' 2885 ' 
 The square root of 44 gives the partial denominators 
 
 6 ; 1, 1, 1, 2, 1, 1, 1, 12; &c., the approximate frac- 
 tions are 
 
 1 6 7 13 20 53 73 126 199 2514 , 
 ' 1 ' 1 ' 2 ' 3 ' 8 ' 11 ' 19 ' 30 ' 379 ' " 
 The square root of 45 gives 
 
 6; 1, 2, 2,2, 1, 12; &c., 
 1 6 7 20 47 n_4 161 2046 
 ' 1 ' 1 ' "3 ' 7" ' 17 ' 24 ' 305 ' 
 
 For the square root of 53 we have 
 
 7; 3, 1, 1, 3, 14; &c., 
 q 
 
 &c. 
 
 1 7 22 29 51 182 2599 
 
 ' 1 ' 3 ' 4 ' 7 ' 25 ' 357 
 
 (225.) If we suppose 5 to equal the following infinite 
 continued fraction, we have 
 
CONTINUED FRACTIONS. 306 
 
 1 
 
 2a-f-- 
 
 ^2a+&c. (66) 
 
 Transposing a, and then inverting both members of (66), 
 we have 
 
 1 =2a.' ' 
 
 2a-\ 
 
 2a' 
 
 2a + &c. (67) 
 
 Adding a to both members of (66), it becomes 
 
 s-^a = 2a-\-^ 
 
 2aA-- 
 
 2a-^^ 
 
 2a + &c. (68) 
 
 Equating the left-hand members of (68) and (67), we 
 
 have s-\-a ^= j clearing of fractions, s- — a^^l ; 
 
 5 — a 
 
 .- . s= v/a=^+l, and 
 
 2a-|-i- 
 
 2a-' ^ 
 
 2a + &c. (69) 
 
 If we make a =^ 1 in (69), we find 
 
 2+— 
 
 ^■^2-f&c. no) 
 
 If we make a:=2, we have 
 39 
 
306 
 
 CONTINUED FRACTIONS. 
 
 4+- 
 
 4 + 
 
 4+&C. 
 
 (71) 
 
 Again, suppose 
 
 = a+- 
 
 2-f- 
 
 2a- 
 
 2+; 
 
 2a4-&c. (72) 
 
 Transposing a and inverting both members, we have 
 
 2a-j — 
 
 2+- 
 
 2a-f-; 
 
 2-f-&c. (73) 
 
 Transposing 2 and inverting, we have 
 
 l_2.-|-2a ^2_^1 
 
 2a +- 
 
 2-f-&c. (74) 
 
 Adding a to both members of (72), and it becomes 
 
 ^-|-a=2a-| 
 
 2+- 
 
 2a -h 
 
 2+&C. 
 
 (75) 
 
 Equating the left-hand members of (75) and (74), we 
 have 
 
 s-{-a = 
 
 l—2s-\-2a 
 Clearing this of fractions, and reducing, we have 
 
 (76) 
 
2+1 
 
 CONTINUED FRACTIONS. 307 
 
 1 
 
 2a+— 
 2+; 
 
 2a+&c. (77) 
 If we take a = 2, in (77), we have 
 
 ^6 = 2+L_ 
 
 '+— 1 
 
 2 + 
 
 4+&C. 
 Making a=13, we have 
 
 2+- 
 
 v/l82 = 13+ J 
 
 26+^ 
 
 2+; 
 
 26-J-&C. 
 
 (226.) A continued fraction, and consequently any com- 
 mon fraction, can be converted into a series as follows: 
 Equation (18) gives 
 
 72 qi qxqi 
 
 P3 ya _ — 1 
 
 93 92 9293 ' 
 
 Pi_Pji_ J_ 
 
 94 93 9394* 
 
 Pn P„-l_(— 1)" 
 
 9„ 7„_i 9„_,9„ 
 
308 
 
 Hence, by addition, 
 
 CONTINUED FRACTIONS. 
 
 ?2=^ + J-. 
 qn qi qiqz 
 
 qzqa qaqi 
 
 qn-iqa 
 
 The terms of this series continually decrease, and are al- 
 ternately positive and negative ; consequently the error 
 committed by taking n terms of the series is less than the 
 (n + l)thterm. 
 
LOGARITHMS. 309 
 
 CHAPTER IX. 
 
 LOGARITHMS. 
 
 (227.) Logarithms are numbers, by the aid o-f which 
 many arithmetical operations are greatly simplified. 
 
 In the following relations : 
 
 (A) 
 
 a— 6, 
 
 (1) 
 
 ay = c, 
 
 (2) 
 
 a' = d, 
 
 (3) 
 
 &c., 
 
 &c. 
 
 X, y and z are respectively the logarithms of 6, c, and d. 
 
 (228.) The assumed root a is called the base of the sys- 
 tem of logarithms. 
 
 (229.) If in (1), of equations (A), we make x^ 0, we 
 shall have a*'=:6= 1, for all values of a, therefore the 
 logarithm of 1 is always 0. 
 
 (230.) If in (1), we suppose the base to be negative, we 
 shall have ( — a)*=6. If b is positive, then x must be 
 even, if b is negative, then x must be odd ; hence we can 
 not represent all values of b by the expression ( — a)'. 
 Therefore the base of every system of logarithms must 
 be positive. 
 
310 
 
 LOGARITHMS. 
 
 (231.) If in (1), wfc suppose b to be negative, we shall 
 have 0^= — b. Now, since a is always positive, the ex- 
 pression a^ is positive for all values of x either positive or 
 negative. 
 Therefore, the logarithm of a negative quantity is impossible. 
 
 (232.) Each different base must produce a different sys- 
 tem of logarithms ; the logarithms in common use have 10 
 for their base. 
 
 So that we have 
 10°= 1; 101=10; 102=100; 10'= 1000; &c. 
 
 Hence, we have 
 
 log. 1 = 0, 
 
 log. 10=1, 
 
 log. 100 = 2, 
 
 log. 1000=3, 
 
 
 log. 
 
 log. 
 
 10000 = 4, 
 &c. 
 
 10000 
 &c.. 
 
 4, 
 
 (233.) If we take the product of equations (1) and (2) 
 of group (A), we shall have 
 
 aT+'j— 5c, (4) 
 
 from which we discover that, the logarithm of the product 
 of two quantities is equal to the s%tm of their logarithms. 
 
 And in general, the logarithm of a number consisting of 
 any number of factors is equal to the sum of the logarithms 
 of all its factors. 
 
 (234.) It also follows from the above, that n times the 
 logarithm of any number is equal to the logarithm of its 
 nth power. 
 
LOGARITHMS. 311 
 
 (235.) If we divide equation (1) by (2), of group (A), 
 we shall find 
 
 a-^=-, (5) 
 
 c 
 
 from which we see that, the difference of the logarithms of 
 
 any two quantities is equal to the logarithm of their quotient. 
 
 (236.) We have just shown that the logarithm of a 
 number raised to the wth power, is equal to n times the 
 logarithm of the number. Conversely, the logarithm of 
 the wth root of a number, is equal to the nth fart of the 
 logarithm of the number. 
 
 (237.) We will now show how the numerical values of 
 logarithms may be found. 
 
 If X is the logarithm of JV for the base a we shall have 
 this condition : 
 
 a'=jY. (6) 
 
 If we assume 
 
 we shall have 
 
 (l-|-m)*=l + n. (8) 
 
 Involving both members of this to the yth power, we shall 
 have 
 
 (l+^'0'^=(l+^y- (9) 
 
 By the Binomial Theorem, we find 
 
 , , . xy(xy — 1) „ , xyixy — l)(xy — 2) „ 
 
 . \-[-xym-\--^^-^ -■ m^-\- ^^ ^ , '^ ;^ .m^-\- e^c, 
 
 ^ ^ ~ 1.2 ' 1.2.3 ' 
 
 Equating these expanded values, rejecting the units of 
 both expressions, we have, after dividing through by y. 
 
312 
 
 LOGAUITHMS. 
 
 ( , xy—\ , , {xy—\) (j-V — 2) , , „ } 
 a:J,n_|_..-^. m-+\J- ^L '. ,a^J^ &c. ^ = 
 
 ' 2 "^ 2.3 ~ 
 
 This must be true for all values of y. 
 
 When 2/= 0, it becomes 
 x[m — Im'^-^-liiv' — |7n'-f-&c.} = n — \n^-\-\n^ — -]7i^-j-&c. 
 
 Hence, 
 a:=z: W.JV= log. (14-w)= n-— !-_±_ — i_-J (10) 
 
 Re-substitutins: a — 1 for m, and we have 
 
 log. (1 + n): 
 
 n — ln--\-\n^ — i^^+ &c. 
 
 (11) 
 
 If we assume 
 
 1 
 
 M 
 
 (a-l)-Ka-ir+K«-l)'— K«-l)' + &c 
 we shall have 
 
 loc. {\-\-n) = M \n—^y-+'^n^—\n'+\n^—kc.]. (B) 
 If the base be so chosen as to render M= 1, then for- 
 mula (B) will become 
 
 log. (1+n) 
 
 Ln- + in3 
 
 &c. (C) 
 
 (238.) The logarithms obtained by formula (C) are 
 called hyperbolic or JVapierean, whilst the common loga- 
 rithms given by formula (B), are called Briggean. 
 
 Lord Napier, or Neper, is supposed to have first con- 
 structed logarithms. The logarithms in common use, were 
 first calculated by Mr. Briggs. 
 
 (239.) We shall hereafter denote the Napierean loga- 
 rithms by the abbreviation JV* log., whilst the common or 
 
LOGARITHMS, 313 
 
 Briggean logarithms will be represented simply by log. 
 
 Hence formula (C) will become 
 
 JVlog. (l-j-n)=w — i7i--f i;i3_ ,^^,_^^^6_&P ^Q,^ 
 
 (240.) By comparing formulas (B) and (C) we discover 
 this relation 
 
 ./J/XJVlog. (l-fn) = log. (1+n). (12) 
 
 Therefore, 
 
 *-^Mog.(l+,0 ^^> 
 
 ■^= („-i)-k«-i)Vk°-i)'-&c. ' '^ "^"""^ ""^ 
 
 modulus of the system of logarithms whose base is a. 
 
 From (12) we see that, the logarithms of any j^articular 
 system is equal to the Kapierean logarithm multiplied by 
 the modulus of that particular system. 
 
 (241. )We wall now proceed to calculate some Napierean 
 logarithms. 
 
 Resuming formula (C), which is 
 JVlog. (1+n) = 71 — ^?!3_^ -in3— i7i4 + ]n5 — •&c. 
 we have, when n is made negative, 
 
 JVl0g.(l — 7l)= — 71 — In2— i7l3— |7l* — &C. (1) 
 
 Subtracting (1) from (C), we have 
 
 JVlog.(l-f7,)-JVlog.(l-7.)=JVlog.l±^ ^2) 
 
 ^■1 
 
 = 2(7l-|-l7l3+J7lS-fl7l^+i7l»-f-&C.) 
 
 If we assume n = - — — , we shall find -2!^ = -^X_ ,an(l 
 2/> + l' \—n p ' 
 
 (2) will become 
 
 JVlog.^^=2M- + L__^._J -f&c.^3) 
 
 ^ P hp+1^3(2p+l)3^5(27>+l)'^^ r^ 
 
 Or, which is the same thing, 
 
 40 
 
314 
 
 LOGARITHMS. 
 
 + &C. 
 
 JVlog.O;-fl) = 
 
 If we take p = lj formula (E) becomes 
 
 I (E) 
 
 3^ 
 
 0.66666666 -^ 1: 
 0.07407407^ 3: 
 0.00823045 -r 5: 
 0.00091449-r 7: 
 0.00010161-i- 9: 
 0.00001129^11: 
 0.00000125-^13: 
 
 : 0.66666666 
 : 0.02469136 
 : 0.00164609 
 : 0.00013064 
 : 0.00001129 
 :0.00000103 
 : 0.00000010 
 
 0.00000014- 
 0.00000001 
 
 15 = 0.00000001 
 
 0.69314718 = JVlog. 2. 
 Take p =4, in formula (E), and we get 
 
 11 
 
 JVlog.5 = JVlog. 4+2^^ + 3^ 
 
 + 
 
 5.9 
 
 ^+7:9^+^" 
 
 But JVlog.5 = JVlog. 10 — JVlog.2; 
 
 also, JVlog. 4=2 JVlog. 2. 
 
 Hence, substituting these values of JVlog. 5 and JVlog. 
 4, in the above expression, and we get, after transposing, 
 
 JVlog. 10 = 3^1og.2+2Jl+J<^+±, + ^,+ &c.| 
 
 Executing the calculation, for the sum of the series, as in 
 the above example, omitting the ciphers on the left, we ob- 
 tain the following : 
 
LOGARITHMS. 
 
 316 
 
 0.22222222 -M =0.22222222 
 2469136 
 274348-^3 = 91449 
 
 30483 
 3387-^5 = 677 
 
 376 
 42-^7 = 6 
 
 5 
 
 0.22314354 = sum of series. 
 3 JVlog.2 = 2.07944154 
 
 2.30258508 = JVlog. 10. 
 
 We are now prepared to find the modulus of the Briggean 
 system. Since the base of the Briggean system is 10, and 
 the logarithm of the base of any system is 1, we have log. 
 10 = 1; formula (D) shows, that the common logarithm 
 of any number divided by the Jfapierean logarithm is equal 
 to the modulus of the common system. 
 
 Hence, 
 
 = 0.43429448. 
 
 JVlog. 10 2.30258508 
 
 This value, when carried to 35 decimal places, is 
 
 Jkr= 0.43429448190325182765112891891660508. 
 
 We will now proceed to calculate common logarithms. 
 
 Since all numbers are either primes, or composed of a 
 certain number of prime factors, and since the logarithm 
 of any number is equal to the sum of the logarithms of 
 all its factors, it follows that it will be necessary only to 
 calculate the logarithms of prime numbers. 
 
316 
 
 LOGARITHMS. 
 
 By equation (12), Art. 240, we see that the Napierean 
 logarithm multiplied by JW, gives the common logarithm. 
 
 Hence, 
 
 log. 2 = JVlog. 2XM= 
 
 0.69314718 X 0.43429448 = 0.30103000. 
 
 The logarithm of 10 we know to be 1, therefore the 
 
 log. 5 
 
 log. —=1 — log. 2: 
 
 : 0.69897000. 
 
 Formula (E), when adapted to common logarithms, be- 
 comes 
 
 log. (p + l)= -) 
 
 ^o.-^+^-^l,-^ + 3l2il)3- 
 
 5(2;, + l)^ 
 
 &c. 
 
 (F) 
 
 log-(;'+l) = log.p-f 
 
 0.86858896 5 -i—-f-—l-—H ! -f &c. I 
 
 Take p = 2 in (F) , and we get 
 log. 3 = log. 2+0.86858896 1 l+^^+_l- + A_ +&e. j 
 
 5 
 
 0.86858896 
 
 
 25 
 
 0.17371779-Hl .-= 
 
 0.17371779 
 
 25 
 
 694871-^3 = 
 
 231624 
 
 25 
 
 27795^5 = 
 
 5559 
 
 25 
 
 1112-^7 = 
 
 159 
 
 25 
 
 44^9 = 
 2 
 
 5 
 
 0.17609 126 = sum of series 
 log. 2 = 0.30103000 
 
 0.47712126= log. 3. 
 
LOGARITHMS. ST 
 
 Ifj in (F), we make p = 49, we get 
 log. 50 = 
 
 log. 49 + 0.86858896ji+3^3+^Lp+&c. j 
 
 And since 
 
 log. 50 = log. lO-flog. 5j and log. 49 = 2 log. 7, 
 we have by substitution and transposition, 
 2 log. 7 = 
 log. 10+log. 5-0.86858896 j i+3-^3+5-^.+&c 
 
 Calculating the series, we find 
 
 99 0.86858896 
 (99)2 ^ 9301 877362 -j- 1 = 0.00877362 
 
 89-^3 = 29 
 
 0.00877391 = sum of 
 
 log. 5=0.69897000 
 log. 10=1. 
 
 1.69897000 
 0.00877391 
 
 1.69019609 = 2 log. 7, 
 0.84509804 = log. 7. 
 
 We might have calculated the log. 7 by substituting 6 for 
 p in (F), but the operation would have been more lengthy 
 than the above. 
 
 The next prime in order is 11 ; to find its log. we make, 
 in equation (F), p = 99, observing that log. 100 = 2, also, 
 that 
 
318 
 
 LOGARITHMS. 
 
 log.99 = log. 9+log. ll=2log. 3+log. 11, 
 we thus obtain 
 2 =2 log. 3+log. 11+0.86858896| ^+ ^^^^ &c. | 
 
 Or, by transposing, it becomes 
 log.ll=2-2log.3-0.86858896|i^+^^3 + &c.| 
 
 199 
 39C01 
 
 0.86858896 
 
 436477-1-1 =0.00436477 
 11-^3= 4 
 
 0.00436481 = sum of series. 
 2 log. 3 =r 0.95424252 
 
 0.95860733 
 
 2.00000000 
 0.95860733 
 
 3+ &C. 
 
 1.04139267 = log. 11. 
 
 To find the log. ofthe next prime 13, we assume _p = 1000 
 in equation (F), and obtain 
 
 log. 1001 = 
 
 log. 1000+0.86858896! 
 
 Now, since 
 1001 =7X11X13, log. 1001=log.7-j-log. 11+log. 13 
 
 Hence, 
 log. 13 = 
 3-log.7-log. 11+0.86858896 jA_ + 3--J^^+&c. J 
 
 2001 ~ 3(2001) 
 
2001 
 
 LOGARITHMS. 319 
 
 0.8685S896 
 
 43407 = sum of series. 
 
 log. 7 = 0.84509804 
 log. 11 = 1.04139267 
 
 3.00043407 1.88649071 
 
 1.88649071 
 
 1.11394336= log. 13. 
 
 We might proceed in this way until we should have cal- 
 culated the logarithms of all the prime numbers within the 
 limits of the tables. 
 
 If, in formula (F), we substitute q- — 1 for p^ it will be- 
 come 
 
 log. f = log. (,"- l)+2-Af l^^ + 3j5^,+ &c. j 
 
 Now, since log. (f = 2log. q, 
 and log. (5=— 1) = log. (<?+l)+]og. (g — 1), 
 we have 
 log. (g+l)^21og.<7 
 
 -log.(,-l)-0.86858896J^-^^+3^3+^-j 
 
 When ?>13, we have this very simple formula : 
 
 0. 86858896 , 
 log.(7+l)=2log.(/-log.(9 — 1) 2^irY-- (^) 
 
 This formula will be true to 8 places of decimals. 
 
 Having already obtained the logarithms of all numbers 
 as far as 13, we may now make use of formula (G) for all 
 numbers exceeding 13, and thus shorten the labor. 
 
 (242.) We have already (Art. 237) said, that the base a 
 of the Napierean system of logarithms satisfies the following 
 equation : 
 
320 
 
 LOGARITHMS. 
 
 {a-l)-i{a-lfi-l{a-iy-^{a-iy + kc. = l. (1) 
 From example 3, page 260, we see that if we have 
 
 (y-1) 
 
 (3/-l)^ (2/-1? , (3/-iy 
 
 -|-&C.: 
 
 (2) 
 
 + &C. 
 
 (3) 
 
 then will 
 
 2 ' 2.3 ' 2.3^4 
 Equation (2) will agree with (1) when y=a, and j;=r 1. 
 
 x^ x^ 
 3/ = l+a;+- + — + ; 
 
 Making these changes in (3), we find 
 
 1.1.1 I 1 , 1 , 1 
 a= 1 + 1+2 + 2:3 + 2:3:4 +2:3X5 
 
 This series may be summed as follows : 
 
 1 
 1 
 
 0.5 
 
 0.16666666 
 
 4166666 
 
 833333 
 
 138888 
 
 19841 
 
 2480 
 
 276 
 
 28 
 
 2 
 
 (4) 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 
 2.71828180 = base of Napierean logarithms. 
 This value, when extended to 35 decimals, is found to be 
 e = 2.71828182845904523536028747135266249. 
 
 EXPONENTIAL THEOREM. 
 
 (243.) This theorem makes known the law of the develop- 
 ment of a^ according to the ascending powers of x. 
 
LOGARITHMS. 321 
 
 To determine this law, we will assume 
 
 a- = l-f.^x + 7.V+rx' + Dx^ + &c., (1) 
 
 both members of which become 1, when x = 0. 
 Changing x into y, in (1), and we have 
 
 a^=l + .% + 7?.y-^+Cr + Dy+&c. (2) 
 
 Subtracting (2) from (1), and actually dividing the right- 
 hand member by x — y, we obtain 
 
 ^ y _^D{x^-^3^y-i-xy''-\-y^)-\-kcA ^^ 
 Writing x — y for x, in (]), and it becomes 
 a-'^ = 1 +Jl{x — y)+B{x - y) -f C{x - yY+kc. (4) 
 Transposing the 1, and multiplying by a", we get 
 ay(a=^-y-l)=ay[Jl{x-y)i-B{x—yy+C{x-yy-^kc.] (5) 
 Dividing (5) by x — y, after replacing its left-hand mem- 
 ber by its equivalent value a' — «'', we find 
 
 ^—^ = ay[^+B{x-y)-{.C{x-yy-]-D{x-yy-^kc.]{e) 
 X — y 
 Equating the right-hand members of (3) and (6), we 
 
 have 
 
 r^-^B{x-^y)-^C{x'-\-xy-^f) ) ) 
 
 ) -^D{x'-\-xhj-j-xf-\-y')-\-kc.S [ (7) 
 
 C ==a^[JJ+B{x-y)-\-Cix-yY+D{x-yy-^kc.] ) 
 This is true for all values of x and y. 
 
 ^_^2Bx4-3Cx=-f4Dx3+&c. = a^.^. (8) 
 
 For a', substituting its value, equation (1) , we find 
 
 A~\-2Bx-\-3Cx'-\-iDx^-\-kc. ) /gx 
 
 = A-^A'x-{-^Bx''-{-^Cx^-\-&.c. S 
 
 Equating the coefficients of the like powers of x, (Art 
 183), we find 
 
 41 
 
322 
 
 LOGARITHMS, 
 
 2B = A-, 
 3C=^w\ Therefon 
 
 \/l = .i, 
 ^-2:3' 
 
 &c., &c. 
 Hence, (1) becomes 
 
 D = 
 
 ,fi' 
 
 2.3.4' 
 &c. &c. 
 
 A-x~ 
 
 A'x' 
 
 It now remains to find the value of Ji. 
 For this purpose, put l-|-6^fl, and we have a=^=: 
 (l_j_6")2:j Avhich, by the Binomial Theorem, becomes 
 
 (1 + 6)- = 
 
 xb . x{x — l)b^ . x{x — }){x — 2)b ' 
 ~^ T"^ 1.2 "^ 1.2.3 
 
 +&c. 
 
 (11) 
 
 Performing the multiplications indicated, we find the 
 coefficient of the first power of x to be 
 
 1 2^3 4^ ' 
 or, re-substituting a — 1 for 6, it becomes 
 
 Therefore, 
 
 ^=(a — 1) — 
 
 {a-iy {a-iy _ ( g-iy 
 
 ft ' 1" O A 
 
 + &c. (12) 
 
 If in formula (C'), we put a — 1 for 7J, we shall find 
 
 jv,„,.„=(„_j)_(i_-zI)'+(i:=:ll'-(^Vc, 
 
LOGARITHMS. 
 
 323 
 
 Hence, 
 
 
 j3=JVlog. .'/. 
 This value of A substituted in (10), gives 
 
 (13) 
 
 ar^l+jVlog. a.a:+l ^_^_ + ^ ^^_ 
 
 -+ (A) 
 
 When 
 
 
 a= e = 2.7182818 &c., 
 then 
 
 • 
 
 ./Ylog. a=./Vlog. e= 1, 
 and (A) becomes 
 
 
 e^=l+x+- + _+iL.+ &c. (B) 
 
 APPLICATION OF LOGARITHMS. 
 
 (244.) By the aid of a table of logarithms, we can easily 
 perform the following operations : 
 
 1 . Find the value of ^-^ — —^ — by logarithms. 
 
 Recollecting (Art. 233) that the logarithm of the product 
 of several factors is equal to the sum of their respective 
 Jogarithms ; and (Art. 235) the logarithm of the quotient 
 of one quantity divided by another is equal to the logarithm 
 of the dividend diminished by the logarithm of the divisor, 
 we find for the logarithm of our expression 
 
 log. — '^l^i^ =log. 3.75+log. 1.06 — log. 365. 
 
 By the tables we have 
 
 log. 3.75 = 0.5740313 
 log. 1.06 = 0.0253059 
 
 0.5993372 
 log. 365=2.5622929 
 
 log. 0.01089 = 2.0370443. 
 
324 LOGARITHMS. 
 
 Therefore, the above expression is nearly equal to 
 0.01089 
 
 2. Finil the 11th root of 11, that is, the value of the '^11. 
 Taking the logarithm of this expression, we find 
 
 locr. iVn = TVof log. 11 = -'-of 1.0413927 = 0.0946721 
 
 = log. 1.24357 &c. 
 Therefore, ^ 11 = 1.24357. 
 
 3. What is the value of ^" ^ Y ' '? 
 
 V6 
 
 5 X log. 8-f- ?f log. 7 - 1 log. G = 4.51545 4-0.2816993 
 
 — 0.1556302 = 4.6415191 =log.43794.53. 
 Therefore, our expression is equivalent to 43804.53. 
 
 (245.) The above exaniples will show the great advantage 
 of logarithms in abridging arithmetical labor. In the higher 
 parts of analysis, the use of logarithms is indispensable. It 
 would not be difficult to propose questions, which by loga- 
 rithms might be wrought in a few moments, but if wrought 
 by arithmetical rules, would require years. The following 
 example will illustrate the above remark. 
 
 How many figures will be required to express 9^^ 1 w 
 The exponent of the above expression is 
 
 90 = 387420489 . • . 999 = 9^ « ' * 2 4 s 9 
 
 Putting it into logarithms, we have 
 
 log_ 9 3 87. 2 04 89^ 387420489 X log. 9 = 
 
 387420489x0.954242509439= 369693099.63 &c. 
 
 Hence, the number answering to this logarithm must con- 
 sist of 369693100 figures. This number, if printed, would 
 fill upwards of 256 volumes of 400 pages each, allowing 60 
 lines to a page, and 60 figures to a line. 
 
LOGARITHMS. 325 
 
 EXPONENTIAL EQUATIONS. 
 
 (246.) An exponentiiil equation is one where the unknown 
 quantity enters as an exponent. 
 
 Thus, a^=h; x^ = c; &c., 
 
 are exponential equations. 
 
 (247.) When the equation is of the form a'^ = b, we find, 
 by taking the logarithm of both members, xX log. a=log. 6. 
 
 Therefore, a::^:^ j-Sli'. 
 
 log. a 
 
 (248.) When the exponential is of this form a;*=c, we 
 must find the value of x by the following double position 
 
 RULE. 
 
 Find by trial two numbers as near the value of x as possi- 
 blcy and substitute them successively for x; then, as the dif- 
 ference of the results is to the diference of the two assumed 
 numbers, so is the difference of the true result, and either 
 of the former, to the diference of the true 7iumber and the 
 supposed one belonging to the result last used-, this differ- 
 "ence, therefore, being added to the supposed number, or sub- 
 tracted from it, according as it is too little or too great, 
 will give the true value nearly. 
 
 ^ind if this near value be substituted for x, as also the 
 nearest of the first assumed numbers, unless a number still 
 nearer be found, and the above operations be repeated, we 
 shall obtain a still nearer value of x; and in this way we 
 may continually approximate to the true value of x. 
 
 EXAMPLES. 
 
 ]. Given x^=^ 100, to find an approximate value of z. 
 The above equation, when put into logarithms, becomes 
 
326 LOGARITHMS. 
 
 xXlog. a- = log. 100 = 2. (1) 
 
 By a few trials we find the value of x to fall between 3 
 and 4. If we substitute, in succession, 3 and 4 in (1) , we 
 shall find 
 
 3 X log. 3 = 1.4313639 
 
 4 X log. 4 = 2.4082400 
 
 0.9768761 = diff. of results. 
 0.9768761 : 1 : : 0.4082400 : 0.418. 
 Hence, 4— 0.418 = 3.582 = x nearly. 
 
 Upon trial, this value is found to be rather too small, whilst 
 3.6 is rather too great ; therefore, substituting each of these 
 in succession in (1), we find, 
 
 3.582 X log. 3.582 = 1.9848779 
 3.6 X log. 3.6=2.0026890 
 
 0.0178111 =difF. of results. 
 0.0178111 : 0.018 : : 0.0026890 : 0.002717. 
 Hence, 
 
 3.6 — 0.002717 = 3.597283=0; nearly. 
 
 2. Given x" =z 5, to find an approximate value of x. 
 
 Ans. x=2.1293. 
 
 3. Given a:* = 2000, to find an approximate value of a: 
 
 Ans. x = 4.8278. 
 
 4. Given X^' = 100, to find an approximate value of x. 
 
 Ans. rr = 2.2127. 
 
 COMPOUND INTEREST AND ANNUITIES BY LOGARITHMS. 
 
 (249.) Interest is money paid by the borrower for the use 
 of the money borrowed. 
 
LOGARITHMS. 327 
 
 It is estimated at a certain rate per cent, per annum; that 
 is, a certain number of dollars for the use of $100 for one 
 year. 
 
 The sum upon which the interest is computed is called 
 the principal. 
 
 The principal, when increased by the interest, is called the 
 amount. 
 
 When the interest of a given principal is paid at the end 
 of each year, it is called simple interest ; but when the in- 
 terest due, at the end of each year, goes to increase the prin- 
 cipal, it is called compound interest. 
 
 The present worth, at compound interest, of a given debt, 
 due at some future time, is such a sura as, being put out 
 at compound interest, will, in the given time, amount to 
 the debt. 
 
 Jin annuity is a fixed sum which is paid periodically, for 
 a certain length of time. 
 
 (250.) In our calculations we shall use the following no- 
 tation : 
 
 ^ = the principal. 
 r = the interest of $1 for'one year. 
 ii = $l-{-r = the amount of $1 for one year. 
 a = the amount of the given principal. 
 j3 = an annuity. 
 
 a' =the amount of a given annuity. 
 P = the present worth of a given annuity. 
 n =the time in years. 
 
 Since $l-|-r= 7? is the amount of $1 for one year, it 
 follows, that the amount of a given principal, /), will in the 
 same time be pR, and this being considered as a new prin- 
 cipal, will in the next year amount to pR X R ^=^pR^: which. 
 
328 
 
 LOGARITHMS. 
 
 in turn, will the next year amount to pR^xR =pR^; and 
 so on. 
 
 Hence, 
 
 pR =z amount for 1 year. 
 jj7{* = amount for 2 years. 
 pR^ = amount for 3 years. 
 pjR4 _. amount for 4 years. 
 
 pR'^ = amount for n years. 
 Therefore, we have this relation, 
 
 w^hich, in logarithms, becomes 
 
 log. a = log, 7J-|-7i log. 7?. (1) 
 
 (251.) When an annuity is left unpaid for n years, it is 
 obvious that the annuity due at the end of the first year, must 
 be on interest n — 1 years, and must therefore amount to 
 AR"^^ ; the annuity due at the end of the second year will 
 be on interest n — 2 years and will therefore amount to 
 JlR'^'^j and so on, hence, the amount of the annuity at the 
 end of n years will be 
 
 J?(/J»-4-il»-2+ -R+1). 
 
 The geometrical progression within the parenthesis being 
 summed, we have, after substituting r for R — 1, 
 
 .'=^(^). (.) 
 
 We have said that the present worth of a debt is such a 
 sum as being put out at interest, will, in the given time, 
 amount to the debt, hence we have 
 
 PR^ 
 
 i'^} 
 
 (2') 
 
LOGARITHMS. 329 
 
 from which we find 
 
 P=tiL^, (3) 
 
 When the annuity is continued forever, the value of n 
 becomes infinite, making this substitution in (3), we find 
 
 The amount of $1 at compound interest for n years at r 
 per cent. J is 
 
 (l + r)». (5) 
 
 The amount of $1 at simple interest for n years at r per 
 cent., is 
 
 l-\-nr. (6) 
 
 Expression (5), when expanded by the binomial theorem, 
 becomes. 
 
 r 
 
 (l+r)"= l + nr+ -^__^-^ + ^ ^ -V'+&c. 
 
 When ri= 1, this expression becomes 
 
 l-\-nr. 
 When n>l, this expression is >1-|- nr. 
 When 7i<l, it is <l-}-;2r. 
 
 Hence, the compound interest computed by formula (1), 
 is equal to the simple interest when the time is one year j 
 it is greater than the simple interest, when the time is greater 
 than one year, and less than the simple interest, when the 
 time is less than one year. 
 
 42 
 
330 LOGARITHMS. 
 
 1. How much will $875 amount to in 12 years, at 6 per 
 cent, compound interest ? 
 
 In this example, we have 
 
 p = S15', n=12; R= 1.06; 
 and we are required to find a. 
 
 Substituting these values in equation (1), we have 
 
 log. a=log. 875 -f 12 log. 1.06. 
 By actually consulting a table of logarithms, we find 
 
 log. 875 = 2.9420081 
 12 log. 1.06=0.3036708 
 
 log. a = 3.2456789. 
 Therefore, a = $1760.672. 
 
 2. What principle will, in 10 years, at 5 per cent., amount 
 to $1000? 
 
 By transposition, equation (1) becomes 
 
 log. p = log. a — 71 log. R. 
 Substituting for a, n and R their given values, we have 
 \og.p=z\og. 1000— 10 log. 1.05, 
 .-. ,og.p = 3 — 0.2118930 =2.7881070. 
 And, 2)= $613,913. 
 
 3. At what rate per cent, will $100 in 16 years amount 
 to $160? 
 
 Fquation (1) gives 
 
 log./i='.2SiJ?-!£iiP, 
 
 n 
 which, in this example, becomes 
 
LOGARITHMS 331 
 
 „ :2.2041200 — 2 ^^,„^.^. 
 log. R= ; =0.0127575. 
 
 .-. il= 1.02981. 
 Therefore, the per cent, is 2.981, or nearly 3 per cent. 
 
 4. In how many years will $460 at 7 per cent., amount 
 to $1000 ? 
 
 Again, equation (1) gives 
 
 log. a — log. p 
 
 which, in this example, becomes 
 3 — 2.6627578 
 
 ?l: 
 
 11.477 years, nearly. 
 
 0.0293838 
 
 5. What is the amount of an annuity of $200, which 
 has remained unpaid 14 years, at 6 per cent., compound 
 interest ? 
 
 Equation (2), when put into logarithms, becomes 
 log. a'=log. ^-}-^og- (^" — 1) — log. r. 
 In the present example 
 
 r = 0.06 ; i? = 1.06 ; .^ = 200 ; n= 14. 
 
 log. R" = 71 log. R = 0.3542826. 
 
 .-. 72" = 2.2609 and /{"— 1 = 1.2609. 
 
 Hence, 
 
 log. a'= 2.3010300-1-0.1006806 —2.7781513, 
 and 
 
 log. a'= 3.6235593. 
 Therefore, 
 
 a'=$4203 nearly. 
 
 6. What is the present worth of the above annuity ? 
 Equation (3) gives 
 
 log. P=: log. a' — nlog. R. 
 
332 
 
 LOGARITHMS. 
 
 In this particular case, we have 
 
 log. P = 3.6235593 — 0.3542826 = 3.2692767. 
 and P =$1858.988. 
 
 7. What is the present worth of an annuity of $100, to 
 continue forever, at 7 per cent 1 
 
 A 
 r 
 
 By equation (4), which is P ^ — , we find 
 „ $100 
 
 0.07 
 
 = $1428.571. 
 
 8. A debt, due at the present time, amounting to $1200, 
 is to be discharged in seven yearly and equal payments. 
 What is the amount of one of these payments, if the inter- 
 est is calculated at 4 per cent 1 
 
 In this example, we have given the present worth of an 
 annuity, the time of its continuance and the rate of interest, 
 to find the annuity. 
 
 Equation (2'), by a slight reduction, becomes 
 
 jR"— 1' 
 which, in logarithms, is 
 
 log. ^ = log. P-f log. r-f-n log. i2 — log. (/?»--l). 
 If we take 
 
 P = $1200; r = 0.04; 21=1.04; 7i = 7, 
 
 we shall find 
 
 .^ = $199,931. 
 
 9. In what time will a given principal, at compound inte- 
 rest, amount to m times the principle 1 
 
 Under example 4, wc have the formula 
 
 ^og- a — lo g.p 
 
 **- log. iJ— ' 
 
LOGARITHMS. 333 
 
 To make tins agree with the present case, we must, for a, 
 write mp, by which means it becomes 
 
 log. m 
 
 (252.) When the interest, instead of being added to the 
 principal at the end of each year, is added at any other 
 regular period, as half yearly, quarterly, &c., n must be 
 considered as standing for the number of those periods, and 
 r will be the interest for one of those periods. 
 
 10. What is the amount of $100, for three years, at 6 
 per cent, per annum, when the interest is added at the end 
 of every 6 months l 
 
 Equation (1), when adapted to the present example, 
 becomes 
 
 log. a = log. 100 + 6 log. 1.03, 
 from which we fmd 
 
 a =$119,405 
 
 11. If the interest of $1, for the crth part of a year, is 
 - , what will be the amount of $1 for ?i years, when a* = 00 1 
 
 X 
 
 The formula for the amount will, in this case, be 
 
 -;)■ 
 
 Expanding the right-hand member by the Binomial Theo- 
 rem, we find 
 
 • , r , nx{nx—V) r^ , nx(nx—l)(nx—2) r^ , „ 
 
 a=l-t-nx.— J \— — '. ^-\ 5: il i^ ,-h&c- 
 
 X 1.2 x^ 1.2.3 x»^ 
 
 When X = CO, this becomes 
 
 ^ ^1.2^1.2.3^1.2.3.4^ 
 
334 
 
 LOGARITHMS. 
 
 Comparing this with formula (B), Art. 243, we have 
 
 a = e"'. 
 Using the common logarithms, we have 
 log. a =/)rX 0.4342944819. 
 
 This formula gives the logarithm of the amount of $1 for 
 n years at r per cent. 
 
 If we call n = 1 , r = 0.07, we shall find 1 .0725 nearly, for 
 the amount. That is, the instantaneous compound interest 
 of $1 for 1 year, at 7 per cent, per annum^ is 7 j cents nearly. 
 Hence, however often a person adds the interest to the prin- 
 cipal, to form a new principal, he cannot make more than 
 1{ per cent., when the rate is 7 per cent, per annum. If 
 we take the rate per cent, per annum, at 6|^, and compute 
 the instantaneous compound interest on $1 for 1 year, it 
 will be just the same as the simple interest of $1 for the same 
 time at 7 per cent. 
 
 (253.) Before closing this chapter, we will show how for- 
 mulas 17, 18, 19, and 20 of Geometrical Progression were 
 found. 
 
 By taking the logarithm of both members of No. 1, as 
 given in the table under Art. 178, we have 
 
 log. I =log. a-\-{n — l)log. r. 
 
 This gives 
 
 n — i=!£Hiizzl^ii? 
 
 log. r ' 
 
 lop;. / — log. a . , 
 
 or, n=-li- ^-+1, 
 
 log. r 
 
 which agrees with No. 17. 
 
 No. 5 is readily put in the following form : 
 
 (i-\-{r — l)s — ar^. 
 
LOGARITHMS. 336 
 
 Taking the logarithms, we have 
 
 log- [« + ('"— 1)5] = log. a-f-nlog.r, 
 from which we readily get 
 
 „^ Iog-[fl + (^ — 1)^]— log-" 
 log. r ' 
 
 which agrees with No. 18. 
 
 No. 12 may take the following form : 
 
 a[s — aY~'^ = l{s — /)"-', 
 which, in logarithms, is 
 
 log. a+(7i — l)log. (5— a) = log. /•+■(" — l)log.(5 — O5 
 which gives 
 
 log./ — log. a , 
 
 ^-log.(,_a)-log.U-/)"^ ' 
 which agrees with No. 19. 
 
 Again, No. 16 may be written as follows : 
 ,.(.,_/^,— 1 __.„.«-! _}_/ = 0, 
 which is readily reduced to 
 
 [r/—(r—l »"-! = /. 
 Taking the logarithms, we have 
 
 log. [rl — (r — l)5]+(?i — 1) log. r^log. I. 
 From this we find 
 
 _ log. / — log, [rl — (r — 1 )s] 
 
 1 +^' 
 
 log.r 
 
 which is the same as No. 20. 
 
336 
 
 GENEBAL PROPERTIES OF EQUATIONS. 
 
 CHAPTER I. 
 
 GENERAL PROPERTIES OF EQUA- 
 TIONS. 
 
 (254.) Any number or quantity which, when substituted 
 for the unknown quantity in an equation, satisfies the equa- 
 tion, is called a root of that equation. 
 
 If the general algebraic equation, 
 a;'»+j3ix'^^+^.>c"-2 .... +A,,-,x+An = 0, (1) 
 
 IS satisfied by making a:=ai, then a^ is a root of equa- 
 tion (1) . 
 
 Substituting a^ for a; in (1), we get 
 aV+Aar'^+'^acr'- • • • -}- A-id -|-^n = 0. (2) 
 
 Subtracting (2) from (1), we have 
 
 ^„. x(x — aO=0. ( ^^ 
 
 We know that each of the expressions 
 x» — ay, 
 a:"-* — ay-', 
 x"-= — ap=, 
 
 x--ai, 
 
GENERAL PROPERTIES OF EQUATIONS. 337 
 
 is divisible by a: — Oi ; consequently, the left-hand member 
 of (3) is divisible by x — a,. 
 
 Equation (3) does not ditfer from (1), since (3) was de- 
 rived from (1) by subtracting from it equation (2), which 
 is equal to 0. Therefore, equation (1) is also divisible by 
 X — Ci ; hence the following property : 
 
 (255.) If ai is a root of the general algebraic equation 
 
 z"+J?,x"-'+^2a;»-2 -\-Jn-ix-\-Jln = 0, 
 
 then its left-hand memher will be divisible by x — aj. 
 As an example, suppose 3 is a root of the equation 
 
 a:3 _ 7x2 _f 36 = 0. 
 Now, by the above property this equation must be divisi- 
 ble by x — 3. 
 
 Actually performing the division, we have 
 
 x3_7x-+36 
 x3 — 3x2 
 
 X — 3 divisor. 
 
 X- — 4.T — 12 quotient. 
 
 — 4x2 
 
 — 4x2-1- 12x 
 
 — 12x+ 36 
 
 — 12x-h36 
 
 0. 
 (256.) If we divide our general equation 
 
 x»+^iX"-> + ^.^'-2....-f-^„_,x+A = 0, (1) 
 by X — flfi, we shall obtain for a quotient, a new equation 
 of one degree less than equation (1), which may be repre- 
 sented as follows : 
 
 X»-'+5,X»-2-f-B2X"-3 .... -|-5„_oX + J?„_i = 0. (2) 
 
 This equation must also have a root, which we will rep- 
 resent by cg. Again, dividing (2) by x — a^, we shall obtain 
 43 
 
338 GENERAL PROPERTIES OF EQUATIONS. 
 
 a new equation one degree less than (2), and consequently 
 two degrees less than (1). Let this new equation be rep- 
 resented by 
 arn-s-f Cix"-3 -f Cox'-' + C„_sx4- C„_o = 0. (3) 
 
 If fl; is a root of equation (3), we can divide it by x — 03, 
 we shall thus find a new equation of three degrees less than 
 equation (1). If we continue in this way, we shall, after n 
 divisions, obtain an equation whose degree = ; therefore, 
 equation (1), is composed of n factors. 
 
 X — flj ; X — fl.j ; X — 03; &c. 
 
 Hence, we have the following property : 
 
 (257.) If til, a-2, Oiiy a„, denote the n roots of our 
 
 general equation of the nth degree^ then this eqtiation will 
 take the following form : 
 
 {x — ai){x — ai){x — ai) {x — a;,_i)(x — c„) = 0. 
 
 This equation is verified by making either of the n fac- 
 tors = ; that is, by making x = Ci, or x = a^, or x = a^, 
 &c., from which we infer, that every equation of the nth 
 degree, has n roots. 
 
 (258.) It does not however follow, that all the roots Oj, 
 02} 035 "4, &c., are different, since two or more of them may 
 be equal, but still, their number must be 7i, since there are 
 n factors. 
 
 (259.) If all the roots C], oo, 03, an are negative, 
 
 then each factor of the equation 
 
 (x+a,)(x+rto)(x-fa3) (a:+a„-i)(a:-f fln) = 0, 
 
 will be positive, consequently each term of its equivalent 
 value 
 
 x»-|-^ia:'-' + ^,x'^= _|_^^_^3,_|_^_ _0 
 
 •will be positive. 
 
GENERAL PROPERTIES OF EQUATIONS. 339 
 
 If the roots are all positive, tlien will the terras of 
 
 (x — ai){x — a.2){x — 03) {x — a,.-i)(x — a„) = 0, 
 
 when expanded, be alternately positive and negative. 
 
 (260.) HmcCy if the terms of any equatioii are neither 
 all positive^ nor alternately positive and negative^ that equa- 
 tion must contain both positive and negative roots. 
 
 (261.) Reasoning after this manner, Harriot has shown 
 
 That every equation whose roots are possible, has as many 
 changes of signs from -{-to — , or from — to -\-,as there 
 are positive roots; and as many continuations of the same 
 signs from -f- ^o -f-j or from — to — , as there are nega- 
 tive roots. 
 
 (262.) If, as we have already supposed, the n roots of 
 an e , nation of the 7ith degree be denoted by ^i, a^, Qa, . . . . a,,, 
 we can put the equation under the following form : 
 (x — a^){x — a.i){x — 03)(a: — a^) (x — a„) = 0. (!) 
 
 Let us suppose «;>«.. ; a,>03; ajya,i; and so of the rest. 
 
 If a quantity b greater than «i be substituted for x in 
 (1), the result will be positive, since all the factors w-ill 
 then be positive. 
 
 If a quantity c less than ci, but greater than Qo, be sub- 
 stituted for a-, the factors will be all positive except one, 
 and consequently the result will be negative. 
 
 If a quantity d less than ag, but greater than a.), be sub- 
 stituted for X, all the factors except two will be positive ; 
 and since two negative factors produce a positive product, 
 the result must be positive. 
 
 By following out this plan of reasoning, we deduce the 
 following property : 
 
340 GENERAL PKOPCUTIES OF EQUATIONS. 
 
 (263.) if tu-0 qucnitilics he successively substituted for x 
 in any equation, and give results ajfected with different 
 SIGNS, there must be an odd number if roots heticeen these 
 quantities. 
 
 But if the tico quantities when substituted for x give 
 results affected with the same signs, there must he either 
 no roof, or else an even number of roots between these quan- 
 tities. 
 
 EXAMPLES. 
 
 1. Find the first figure of one of the roots of the equation 
 
 x'-\-l.bx^-^0.3x — 46 = 0. 
 If we substitute 3 for x, the result will be — 4. 6, a nega- 
 tive quantity. If we substitute 4 for x, the result will be 
 43.2, a positive quantity. Therefore, the first figure of the 
 root sought must be 3. 
 
 2. Find the first figure of one of the roots of the equation 
 
 xi-j. 3r'+2x-^-f 6x— 148 = 0. 
 Putting 2 for x, the result is — 88, and putting 3 for z, 
 we get 50, .*. the first figure of the root sought is 2. 
 
 3. Find the first figure of one of the roots of the equation 
 
 a^— 17x"+54x— 350 = 0. 
 
 In this example, the two consecutive numbers between 
 which there is a root, are 10 and 20, therefore, the first 
 figure of the root sought is 1 in the ten's place. 
 
 (264.) By actual multiplication, we find 
 
 (x — ai)(x — a.) = x2 ^Vx-f-cifl.a, 
 
 fll ^ +^102 i 
 
 (x — a))(x — ao)(x — a,i)=r' — a.. > x-+^i"^ / ^ — fliOaflsj 
 — 03 7 H-OoCa^ 
 
GENERAL PROPERTIES OF EQUATIONS. 341 
 
 {x—ai){x—a.>){x—a-3){x—m)=x' _C3 / ^' 
 
 — 04 ) 
 
 -j-OiflaX 
 
 
 -i-aias) 
 
 OiOoO^ -\ 
 
 + aia4( 
 
 > a- + 01020304, 
 
 + 02^3^ 
 
 — 010304 i 
 
 -j-fl-iOiX 
 
 — 020304 ^ 
 
 + 0304/ 
 
 
 , &c., 
 
 &c., 
 
 &c. 
 
 By carefully examining the above results, we discover 
 the following properties : 
 
 (265.) The coefficient of x in the first term is always 1. 
 
 The coefficicjU of the second term^ is the sum of all the 
 roots with their signs changed. 
 
 The coefficient of the third tenrf, is the sum of all the 
 products of the roots taken two at a time. 
 
 The coefficient of the fourth term, is the sum of all the 
 products of the roots with their signs changed, taken three 
 at a time. 
 
 And so on for the succeeding terms, until we reach the 
 last term, which is independarit of x, and is equal to the 
 continued product of all the roots, ivifh their signs changed. 
 
 (266.) The general form of an imaginary or impossible 
 root of an equation is o-f- ^ — b. 
 
 The only factor which will render o-f- ■/— 6 rational, is 
 a — ^—h. 
 
 We have just seen, that the last term of our general 
 equation 
 
342 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 is composed of the continued product of all its roots. 
 
 Hence, if a-j- ^ — 6 is a root of this equation, then also 
 will a — ^ — 6 be a root, unless Jl,^ is imaginary. 
 
 In the same way, if a' -f ^ — b' is a root, then will ai — v^ — h' 
 be a root, and so for other imaginary roots. From this we 
 infer the following properties : 
 
 (267.) Every equation has an even number of impossible 
 rootsy or else none at all . 
 
 Jin equation of an even degree may have all its roots im- 
 possible; but if they are not all impossible, two of them at 
 least are possible. 
 
 If all the roots of an equation are impossible, then what- 
 ever values are substituted for x in that equation, the results 
 will always be affected loith the same signs. 
 
 An equation of an odd degree has at least one real root. 
 
 (268.) If we divide both members of the identical equa- 
 tion 
 
 (x — a])(x- 
 by x", we shall obtain 
 
 2X"-\...^,,^,x+A=? 
 fl,.)(x — a.) (x— a„) ) 
 
 - , j?i , ^3 •/?«— 1 , Jin 
 XX- X"-* X" 
 
 (-?)('-'7)('-?)--(-?)- 
 
 Taking the lognritiims of both members, we find 
 
GENERAL PROPERTIES OF EQUATIONS. 
 
 343 
 
 + loo-. 
 
 + Iog.(l-f)V(A) 
 
 If we actually take the logarithm of the left-hand mem- 
 ber of (A), by formula (C), Art. 237, where 
 
 X X2 X"-^ X^ 
 
 is put for 7J, we shall obtain 
 
 By taking the logarithms of the terms of the right-hand 
 member of (A), we get 
 
 — (oi-f-ao+as «-.)r 
 
 -^(«l+«^+«^3 «.t)^/ 
 
 -^(«?+«.^+«? <)-2 
 
 &c., &c. 
 
fli-f-aa+fla- 
 
 ...a„ = — ^,, 
 
 a'.+al+al. 
 
 ...a==^=-2A>, 
 
 a',+al+al. 
 
 . . .G,=j = — yi?+3^j^o— 3^3, 
 
 a\+al+al. 
 
 ...< = 
 
 344 GENERAL PROPERTIES OF EQUATIONS. 
 
 By equating the coefficients of the like powers of x, in 
 (B) and (C), we find the following interesting properties : 
 
 (D) 
 
 ^5-4^-:^, -|-4A^3+2^S— 4^4, ' 
 &c., &c. 
 
 (269.) These relations make known the sum of the 7?ith 
 powers of all the roots of an equation in terms of its coeffi- 
 cients. 
 
 (270.) If we suppose the general equation is deprived of 
 its second term, or which amounts to the same thing, if we 
 suppose ^1 = 0. the above results of (D) will become 
 
 a,-\-a,-{-a^....an=0, ^ 
 
 c?+a^+ai....a-;=-2^2, / 
 
 aJ + a3 + a^...a3z=-3^3, )> (E) 
 
 &c. 
 
 — 3^3, \ 
 
 &c. ) 
 
 TRANSFORMATIONS OF EQUATIONS. 
 
 (271.) We will resume our general equation 
 a;»+^ix"-'+^oa:«-2 -f A_ix+A = 0. (1) 
 
 If in this equation we suppose x = u-\-x', m being a new 
 unknown quantity, and x' an indeterminate quantity, we 
 shall have 
 
 {u-\-x'Y-\-M^+^'Y-'+Mu+x'Y-' I /2^ 
 
GENERAL PROPERTIES OF EQUATIONS. ^ 345 
 
 which, when expamled by the Binomial Theorem^ becomes 
 
 
 
 Ax'"-'/ 
 +^W > = 0. (3) 
 
 + ■ 
 
 Now, since x' is wholly arbitrary, we are able to give it 
 such a value as to satisfy this condition 7?x'-f-^i = 0; 
 
 n 
 
 which is done by making x' = ^. 
 
 n 
 
 This value of x' substituted in (3), will give an equation 
 of the following form : 
 
 W''4-jBotf"-2_^5oM"-3 Bn-iU-\-Bn = 0^ (4) 
 
 which -is deprived of its second term. 
 
 (272.) Hence, to cause the second term of an equation to 
 disappear y we must replace the unknown by a new unknown 
 augmented by the coefficient of the second term with its 
 sign changed, ajid divided by the number denoting the degree 
 of the equation. 
 
 • EXAMPLES. 
 
 1. Transform the quadratic equation 
 
 into a new equation wanting its second term. 
 
 A' 
 Assume x^w — , and it will become 
 
 thisj when reduced, becomes 
 
 44 
 
346 , GENERAL PROPERTIES OF EQUATIONS. 
 
 -•^-(f-^i=='' 
 
 or, by transposing, 
 4 
 
 
 and, . = ._:|^ = -|l±V^fZ7, 
 
 The same result as was obtained by the direct solution 
 of the above equation under Art. 151, formula (D). 
 
 2. Transform the cubic equation 
 
 a;-' -j- ^,x» + Ji^x + Jl^ = 0, 
 into a new equation, wanting its second term. 
 
 AssuminjT x = m — 
 
 ^1 
 
 we get 
 
 which, when expanded and reduced, gives 
 
 'here 
 
 5o = . 
 
 -}-^„ 
 
 We might proceed in this way for the transformation of 
 equations of higher degrees, but its easy to see that this 
 method would be very lengthy and complicated for such 
 equations, we shall therefore seek some law by which these 
 transformations can be made with less labor. 
 
 (273.) If in the general equation 
 
 x» -f- ^ix**-' 4- ^2X— '-^ 4- An-xX -\-An^Q = X. 
 
GENERAL PROPERTIES OF EQUATIONS. 
 
 347 
 
 we substitute x' -\-u for a;, and imitate the operations of 
 Art. 271, we shall have 
 
 +.^,x'»-2 + (n — 2)^2x'"-3 1 
 
 
 +n. 
 
 n — 1 
 
 
 = 0. 
 
 w'+...+w« 
 
 If in the above transformation, we put X' for the coefficient 
 of M°5 or which is the same thing, for the sum of the terras in- 
 dependent of tc. Also, put X" for the coefficient of w, and 
 
 X'" . X"" 
 
 — - for the coefficient of m^, — — - for the coefficient of w^, 
 ^ 2i.o 
 
 and so on, we shall have 
 
 X= x" + ^ia;'-i + Aia:'-^ ^„_ix-f A, 
 
 r=a;"'-f-^:x'"-' + A2'»-=^ A_ix'+A, 
 
 J^"=nx"»-»-f (n — l)^ix'"~^+(n— 2)^2a:"'-l . . . A-i, 
 
 :X"'= 7i(7i— l)x"»-2-f- (n — 1) (n — 2).^,x"'-^+ 
 
 Jt""=;2(n— l)(n— 2)x"'-3+(n— l)(n— 2)(n— S).'?,!'"-'.. 
 &c., &c. 
 
348 GENERAL PROPERTIES OF EQUATIONS. 
 
 If we examine the above expressions, we shall discover 
 the following law : 
 
 X' is derived from the general equation X, by simply 
 changing x into x'. 
 
 X" is derived from X' by multiplying each of the terms 
 of X' by the exponent of x' in that term, and diminishing 
 this exponent by a unit. 
 
 X'" is derived from X" in the same manner as X" was 
 derived from X'. 
 
 And, in general, a coefficient of any rank, in the above 
 transformed equation, is formed by means of the preceding, 
 by multiplying each term of the preceding by its exponent, 
 and dividing the product by the number of coefficients which 
 precedes the terms sought, and diminishing the exponent by 
 a unit. 
 
 (274.) The polynomial X" is called the^rs^ derivedpoly- 
 nomial of X'. 
 
 The polynomial of X'" is called the second derivedpoly- 
 nomial of X' ; and so on for the succeeding polynomials. 
 
 (275.) We will add a few examples to illustrate the above 
 law. 
 
 1. Transform the equation 
 
 X ' — 12x3 -|- 17x2 — 9x-}- 7 = 0, 
 into an equation wanting its second term. 
 
 12 
 
 By Art. 272, we must substitute w+— =m-}-3, or 3 -[- w 
 
 for x; this transformed by Art. 273, will be of this form : 
 X'JrX"u^'^u'^^'^.u^+u* = 0. 
 
 Now, by the above law, we find 
 
GENERAL rUOrEUTlES OF EQUATIONS. 349 
 
 X' = (3)'— 12(3)''+17(3)2— 9(3)'+7 = — 110, 
 :^"=4(3)^' — 36(3)-+34(3)'— 9 =—123, 
 
 -2-=6(3)-'-36(3y+17 =- 37, 
 
 |^==4(3).-12 = 0. . 
 
 Hence, our transformed equation is 
 
 w' — 37m3 — 123;/. — 110 = 0. 
 
 2. Transform 
 
 x^ — I0a:^+7a;3+4x — 9 = 0, 
 into a new equation wanting its second term. 
 
 Proceeding as above, we find 
 
 X' = {2Y— 10(2;-' +7(2)^+4(2)^ — 9 =— 73, 
 J^" = 5(2)^ — 40(2)'+21(2)--|-4 =—152, 
 
 ^=10(2)^ — 60(2)H21(2)^ =—118, 
 
 ^^'=10(2)-^ -40(2)' + 7 =- 33, 
 
 Y'"" 
 
 Hence, our transformed equation is 
 
 u' — 33m'* — 1 18?^^ — 152m — 73 = 0. 
 
 3. Transform 
 
 3x-^+15x^+25x — 3 = 0, 
 into an equation wanting its second term. 
 
 Dividing each term by 3, in order to make it agree with 
 the general equation, we get 
 
 ar^_j_5ar2-J-±:x — 1=0. 
 
350 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 Now, in order to make the second term disappear, we 
 5 
 must, by Art. 272, substitute — ^-\-u for x. 
 o 
 
 Hence, 
 
 -- (-1) +^ (-B +f H)- 
 
 25 
 3 
 
 _ 152 
 
 "" 27' 
 
 = 0, 
 
 = 0. 
 
 Hence, the transformed sought, is 
 , 152 
 
 27 
 
 = 0. 
 
 In this example, the third term vanished at the same time 
 as the second. 
 
 4. Transform 
 
 4x3 — 5x'^-f-7a:— 9 = 0, 
 
 into a new equation, of which the roots shall exceed by a 
 unit, each of the corresponding roots of the given equation. 
 
 We must assume u = x-j-1 or x=u — 1, which gives 
 
 X'=4: (—1)3 — 5 (— l)-'-j-7(— 1)' — 9 = — 25, 
 X"=12(— IV— 10(— l)'-f-7 = 29, 
 
 X'" 
 
 l^=12(-l)'-5 
 
 = —17, 
 
 2.3 
 
 = 4. 
 
 Hence, the transformed equation is 
 
 4^3— 17tt-'+29u — 25 — 0. 
 
(2) 
 
 GENERAL PROPERTIES OF EQUATIONS. 351 
 
 (276.) The derived polynomials possess some remarkable 
 properties, which we will develop. 
 
 x"+A,x^~'+A.^"-^+ + A_;X+^, = 0, (1) 
 
 have uij 02, a3, a„_i, a„, for its 7i roots, we shall 
 
 then have by Art. 257, the identical equation 
 
 X- + Ax^' + + A-isc 4- A = 
 
 (x— ai){x — ao) {x — c„_i) {x — a^). 
 
 In (2) change x into x -\- v, and it will become 
 
 {x-\-uY-{-A{x-huY~' ...... ..l.^i{x-\-u)+.(l„ = ) ,3. 
 
 [u-^{x-a,)\[u-\-ix-a,)] [u -^ {x - a,.)]. \ ^ ^ 
 
 The left-hand member of (3), by Art. 273, is 
 
 X + X'u-^^^c^-i- w. (4) 
 
 If we should actually perform the multiplication of the 
 factors of the right-hand member, we should find, by pay- 
 ing attention to the properties under Art. 265, that the part 
 independent of u is equal to 
 
 (x — ai){x — aj){x — a,;) {x — a„_i)(a; — a„). (5) 
 
 The coefficient of u will equal the sum of the products 
 
 of all the terms x — r/j, a: — ao, x — 03, taken n — 1 
 
 at a time. 
 
 The coefficient of u- will equal the sum of the products 
 of the same terms taken n — 2 at a time. 
 
 Hence, by equating the coefficients of the like powers 
 of u, in the two numbers of (3), we have 
 
352 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 X:= {x—ai){x — ao)(x — as) . . . .(x — a„_i)(x — a,,) 
 
 x—ai X — a-2 X — 03 x — a,j 
 
 X" = X _^ X |_ 
 
 (x— ai)(x— Co) (x— fli)(x— 03) 
 
 &c. 
 
 (x— a^i)(a:-a„) 
 &c. 
 
 (A) 
 
 EQUATIONS HAVING EQUAL ROOTS. 
 
 (277.) Let X denote the first member of the equation 
 
 x^+^iX^-'-f^ax"-' + A-ia:4- A = 0, (1) 
 
 and suppose m factors equal to x — a, 7n' factors equal to 
 X — b, m" factors equal to x — c, &c. ; also, that it contains 
 the simple factors x—p,x—q,x—r, &c., then we shall have 
 
 x= {x-ar{x-bnx-cr . . . . ? 
 
 {x-p){x-q)ix-r)...,= 0. S 
 
 Calling X' the first derived of X, we shall, by (A), Art. 
 260, have 
 
 x=i^+t:^+"^+....^+-^+^+..(3) 
 
 x—a X — b x — c X — p X — q x — r 
 
 Hence, the greatest common divisor of Xand X' is 
 
 D = {x — a)'^\x—b)""-\x-c) "-'. (4) 
 
 (278.) From this wc conclude, that when the equation 
 X = has no equal roots, then the polynomials have no 
 common measure. 
 
 (279.) If the greatest common divisor D, equation (4), is 
 of the first degree, and equal to x — ^ = 0, we conclude that 
 equationX = has two roots equal to k. And in general, if 
 
GENERAL PROrEllTIES CV EQUATTONS. 353 
 
 it is of the form {x — h)" = 0, then the equation has n-f-1 
 roots equal to h. 
 
 When it is of the form >r--]-»/':/i.r-|-yio = 0, we must find 
 the two vahies of a by quadraties, which we will suppose to 
 be Ic and k', so that the eijuation will have two roots ^/c, 
 and two more =k'. 
 
 EXAxMPLES. 
 
 1. Has the equation 
 
 2x^ — 12x='+l9a;-— 6a:+9 = 
 any equal roots, and if so, what are they 1. 
 
 X = 2a- « — 12j:^+ ] 9x- — 6x+9, 
 X'=8x3 — 3Cx'-+3Sx —6. 
 Now, by the method of Art. 50, we find the greatest com- 
 mon measure of X and A'' to be 
 
 D=zx-3. 
 Therefore, the above equation has two roots equal to 3. 
 Dividing its first member by 
 
 {x — 3Y = x^ — 6x-i-9, . 
 we find 
 
 2x- — 12a:'+ 1 9.r- — 6x+9 
 2x= — 12r'+lSx- 
 
 6a:-|-9. 
 
 '2a;-H-l. 
 
 X-' — 6i--f 9 
 x3_ 6x4-9 
 
 0. 
 The two roots of 2x--|-l ^0 arex^^zb"^ — I. 
 Hence, the four roots of the above equation are 
 
 3,3, + v/=T, _v/=T. 
 2. Find the equal roots of 
 
 x^ — 2x4+ 3x'' _ 7j;:_|_8x — 3 =: 0, 
 if it has any. 
 
 45 
 
354 
 
 GENERAL PROPERTIES OF EQrATIO.No. 
 
 X-- 
 
 X'-- 
 
 a-5_2^'_)_ 
 
 + 8X-3, 
 
 ~ — Sx'-\-9--'—Ux -f-8. 
 
 Seeking the greatest common divisor of X and X', we find 
 D = x^ — 2x-{- I ={x — ly, 
 
 hence, there are 3 roots equal to 1. 
 If we divide the value of X by 
 
 {x—lY = x^ — 3x--\-3x—l 
 we shall obtain the quotient .r* -f- x-|-3. 
 
 The two roots ofx- -|-3^H-3 = 0, are x ■■ 
 hence the five roots of 
 
 x^ — 2x4 -f 3a;:i _ 7a;2 _|_ Sa: — 
 
 :±i^— 11 
 
 1, ], i,-^4-;v/: 
 
 11. 
 
 3 = 0, 
 ■^v^— 11. 
 
 3. Find the roots of 
 
 x-i + 5x^ 4- 6.T^ — 6x^ — 15x3 — 3x2 -f 8x 4- 4 = 0. 
 
 Proceeding as in the last example, we find 
 X= x^+ 5x«4- 6x^— 6x^—15x3 — 3x=+8x + 4, 
 X' = 7x« + 30x^ 4- ^0^' — 24x3 _ 45^,0 _ g^ _|. g^ 
 D— x-« + 3x3 -j- X-— 3x — 2. 
 
 Isow, since the greatest common divisor D, surpasses the 
 second degree, we cannot immediately resolve it. 
 
 If we apply the same process to D, as we have done to 
 X, we shall find 
 
 D= x' 4-3x3-}- 3,2 _ 3a, _ 2, 
 D' = 4x^ 4- 92;2 -f 2x — 3 = first derived of D, 
 D" = X -j- 1 = greatest common divisor of D and D'. 
 Hence, D has two roots equal to — 1 . Dividing it by 
 (^+l? = ^^+2x+l, 
 we obtain the quotient x'--\-x — 2, 
 which equated to zero, gives x= 1, or x = — 2. 
 
GENERAL PROPERTIES OF EQUATIONS. 355 
 
 Therefore, D= {x-]-iy{x — l){x ^2), 
 
 and consequently, A'= (x + ^Yi^ — l)-(-^" -\- ~) ' 
 and the equation has three roots, ^=^ — 1 ; two roots, =; 1 ; 
 
 : — 2. 
 
 RECURRING EQUATIONS. 
 
 (280.) v/i recurring equation is one which remains the 
 same when - is substituted for ar. 
 
 X 
 
 All recurring equations are of this form : 
 
 z" + ^ix"-' + .V"—' + Ji'iX- + ^ix + 1 = 0, (1) 
 
 where the coefficients of the terms equi-distant from the 
 extremes are equal, because, if for x we substitute -, the 
 above equation will become 
 
 i+r^'- +i?i ^^^j^i=o, (2) 
 
 which, when cleared of fractions by multiplying by x", 
 becomes 
 
 l-\-A,x-^Jl^-^ 4-^ox"---}-^iX''-'+x''= 0, (3) 
 
 which is just the same as equation (1), only the terms are 
 taken in a reverse order. 
 
 From the above definition of a recurring equation, we 
 
 know, that if ax is one of the roots, then W'ill — also be 
 
 a root of this equation. 
 
 Hence, recurring equations are sometimes called recipro- 
 cal equations. 
 
 (281 ) A recurring equation of an odd degree can in 
 general be represented by 
 
35G 
 
 GENERAL PllOrElll lES OF EQ^ATIO^-S. 
 
 a:--+i^i,j'"db. ?.a-^-'*-'d= . . . .±J2rx'-±:.hx±l =0. (4) 
 Now, if the corresponding coefficients have the same sign, 
 a:=— Iwill satisfy (4), but if the corresponding coeffi- 
 cients haA'e contrary signs, then x^=l will satisfy (4). 
 
 (282.) Heiice^ — 1 or -\-\ is ahcays one root of a recur 
 ring equation of an odd degree; consequently ^ by Jlrt. 255 
 we know that a recurring equation of an odd degree is divisi- 
 ble by x-f- l^orhy x — 1 ; and the quotient will be a recurring 
 equation of one degree loioer, and consequently of an even 
 degree. 
 
 (283.) The general form of a recurring equation of an 
 even degree is 
 x2»-|-^iz3"-'-)-^2.r«-2_j_ ^ ^ __j_^,a;2_^^^^j^l ^ 0. (5) 
 
 This divided by a;", becomes 
 
 a:"+^,x"^'-f^„a;~-+....-f 4^.4- "^L^-L^O, (6) 
 
 which becomes, by bringing the terms of equal coefficients 
 together, 
 
 ^"+x-.+-^'(^ -'+i;!=,)+-^-^(-'-Hj;^,)+&c.=0. (7) 
 
 If we expand I .r-| — ;, ) X I .r-(-- j) we shall obtain this 
 identical equation, 
 (^'•+1) X (.+^) =."+.+-ip,+.»-.+J^. (8) 
 
 By transposing, we have 
 
 where z = a:-|- -. 
 
GENERAL PROPERTIES OF EQUATIONS. 357 
 
 If in formula (9) we suppose successively 
 
 '' = 1,2,:^,4,5, , 
 
 we shall find 
 
 x^-\- 
 
 (A) 
 
 These values of x-f-- ; x^ 4- -^, : x^ -4- —.: &c., in terms 
 ' X x~ x^ 
 
 of z, being substituted in the general recurring equation of 
 
 an even degree, will give an equation in terms of z of but 
 
 half that degree. 
 
 (284.) From Art. 281, we know that a recurring equation 
 of the degree 2n-[-l, can be immediately reduced to a re- 
 curring equation of the degree 2w,by dividing by x-f-l, or 
 x — 1. Consequently a recurring equation of the degree 
 2?i-f-l can be reduced to an equation of the 7ith degree. 
 
 Suppose, for example, we wish to find the five roots of 
 the recurring equation 
 
 x'^—]\x'-\-llx^-\-\1x^—nx-{-l = 0. (1) 
 
 Since this is a recurring equation of an odd degree, and 
 the corresponding coefficients have the same sign, it follows, 
 by Art. 281, that one of its roots is — 1. Dividing this 
 equation by x-J-1, we obtain for a quotient this new recur- 
 ring equation of the fourth degree. 
 
 x'—\2x^-]-2dx'-—12x-\-l=0. (2) 
 
]5S 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 Dividing this by x\ and reducing the result to the form 
 of (9), Art. 283, we have 
 
 ,.+ i_,2(,+_l) + 29 = 0. 
 
 (3) 
 
 Substituting, in (3), for x^ -\ — ^, a; -j- -? their values in 
 
 terms of 2, as given by group (A), Art. 283, we obtain 
 s2_ 2— 122 + 29=^0, (4) 
 
 or z2 — 12z + 27 = 0. (5) 
 
 Equation (5), solved by the usual rule for quadratics, 
 gives 
 
 z = 9, or c = 3. 
 
 Taking the first value of r, we have 
 1 
 
 X 
 
 9x = 
 
 Solving (7) by quadratics, we find 
 9 1 „^ 
 
 Taking the second value of z, we have 
 
 z = x-\ — = 3, or X- — 3x=: — 1. 
 
 Equation (9) gives 
 
 3 1 ^ 
 
 2^2^^'- 
 
 (6) 
 (7) 
 
 (8) 
 
 (9) 
 (10) 
 
 Therefore, the five roots of the proposed equation are 
 9-f-sr77 9— v/77 3-|-v/5 3 — s/5 
 ' 2 ' 2 ' ~~2 ' 
 
 h 
 
 If the numerator and denominator of the third root be 
 each multiplied by 9-}- v/ 77, and the numerator and de- 
 nominator of the fifth root, be each multiplied by 3-j- >/ 5, 
 the roots will assume the followincr form : 
 
GENERAL PROPERTIES OF EQT'ATIONS. 359 
 
 9-f-^/77 2 34-V/5 
 
 2 
 
 2 '9-|-v/77' 2 '3-f-v/5' 
 which shows that the third root is the reciprocal of the 
 secondj and the fifth is the reciprocal of tjie fourth. 
 
 BINOMIAL EQUATIONS. 
 
 (285.) Binomial equations are of this form : 
 
 in which, if we substitute ax for y, and divide the result by 
 a", we shall obtain 
 
 for the general form of binomial equations. 
 
 (286.) If n is even, the equation cr" -j- 1 ^ ; or, 
 x'* = — 1, gives for x the impossible expression ^ — Ij 
 hence all the roots are imaginary. But the equation 
 X" — 1 = ; or X" = 1, gives a;=yi=-|-], or — 1; 
 so that the equation has two real roots, and n — 2 imaginary 
 roots. 
 
 (287.) If n is odd, the equation x'-f- 1^=0 ; orx"= — 1; 
 gives X = V — 1 = — 1 ; so that there is one real root and 
 n — 1 imaginary roots. But the equation x" — 1=0; or 
 x"= 1, gives x= V l = -j- 1, so that, as before, we have 
 one real root and n — 1 imaginary roots. 
 
 (288.) If a is one of the imaginary roots of the binomial 
 equation^ 
 then will x "= 1 , 
 
 («■)" = «'— 1, 
 {a^)" = a''-"= 1-=1, 
 (a3)"=a3'»= 13=1, 
 
360 GENERAL PilOPEUTIKS OF E^UATIOKS. 
 
 So that a', a-j a', a', &-C , satisfy the equations = 1, 
 when substituted for x. These quantities are therefore roots 
 of the above equations. 
 
 Hence^ if a is one of the imnginar§ roots of the equation 
 rc'*^!, then any j^ower of a, will also be an imaginary 
 root. 
 
 From this it follows, that the roots a;"= 1, may be rep- 
 resented under an infinite variety of forms, each term in 
 the following series being a root. 
 
 1, fl, a-, a3, """'j^ 
 
 a", a"+^, a"r2j «^"~S( 
 
 a2", 02"+', a'^"r2^ a3— 1, > (A) 
 
 (289.) When n is a prime number, the roots of the 
 equation x"=^ 1, are all contained in either of the expres- 
 sions (A), for in each of these series of roots all the n terms 
 will be different. But when n is a composite number, the 
 roots of the equation are not all contained in cither of the 
 series (A), for some of them will be the same root under 
 different forms, for suppose 7i = 7; X 7, and let 5'>jo, then 
 the first series of (A) is the same as 
 
 1, «, a', o ", fl?', aP+', a?'+-, a\ a'^', a^-'. 
 
 Now, since 
 
 therefore the terms 1, «'', and a*?, are each equal to 1, and 
 consequently, each must be the same root under different 
 forms. 
 
 (290.) Suppose we have x^=\^ where p = a prime. 
 If wc put xP==y; then yf= 1. 
 
GENERAL PROPERTIES OF EQUATIONS. 361 
 
 Now, suppose I is a root of i!/'':i^ 1, it will follow from 
 Art. 288, that the p roots of 3/''= 1 will be denoted by 
 
 1, 5, h-, P, bP-K 
 
 Hence, by substitution, we have 
 
 -X^-1=:0, (!)• 
 
 xP—b=0, (2) 
 
 ,._3,= ^-^-^^=0, (3)1 ^^, 
 
 The ]) roots of the first equation x^ — 1=0, have already 
 been found to be 
 
 1, h,b-, b\ bP-\ 
 
 If we make x = c V Z), the second equation of (B) will 
 become 
 
 xP—b = {zr'—\)xh = Q; 
 
 therefore the roots of x^ — 6 = 0, are equal to the roots of 
 zi'— 1 = multiplied by V h. 
 
 Hence, the p roots of (2) are 
 
 V b, bV h, h~ V b, bP-^V b. 
 
 Again, if we make x = r V b'-, the third equation will 
 become 
 
 xP—b-= {zP— 1 ) X 6-= ; 
 therefore, the roots of xP — b-= 0, are equal to the roots 
 of r''— 1 = 0, multiplied by V b\ 
 Hence, the p roots of (3) are 
 
 V &-, h V ft'S b- Vb--, ft''"' V 6-. 
 
 Proceeding in this way m;iy find the following for the pp 
 
 roots of xPP= 1. 
 
 46 
 
362 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 hb,b', bP~\ 
 
 V6, h Vb, 6- Vb, bP-^ V&, 
 
 V&S b V&-, b^' Vb% bP-' V6S 
 
 (C) 
 
 V bP-\ bVbP-', b- VbP-\ b^^VbP-K 
 
 (291.) Again, suppose we have x^^ — 1 = 0, where j9 and 
 q are both primes. 
 
 If we put xP = y, we shall have yi — 1=0. 
 Let the q roots of this equation be 
 
 1, a, a-, a^, a'-', 
 
 or which, by Art 288, is the same as 
 
 1, aP, aP, cv'P a^t-^^P^ 
 
 then by substitution, we find 
 
 a;P— 1=0, (]) 
 
 , a;P — aP = 0, (2) 
 
 l^p — «->=0, (3)' 
 
 xP—y 
 
 I 
 
 (D) 
 
 xP — «(«-')/^=0. {q) 
 
 We will denote the values of x in a-'' — 1 = 0, by 
 
 1,&, 6-', 6', IP-K 
 
 If we make x = or,, equation (2), of (D), will ber,ome 
 xP — aP = {zi — \)bP = Q; 
 
 therefore the roots of a^ — uP = 0, arc equal to the roots of 
 zP — 1 = multiplied by a. And in a similar way we dis- 
 cover that the roots of xp — q'-p = 0, are equal to the roots 
 of o-P — 1 =: multiplied by a-, and so for the other equa- 
 tions of (D). 
 
 Hence, the pq roots of x^? — 1 = 0, are 
 
GENERAL PROPERTIES OF EQUATIONS. 363 
 
 l,^i^i^ ^^-s 
 
 a, a6, ab'j ab^, afc^"', 
 
 a2, 0=6, a-6-, a%', a'^bP-\ 
 
 (E) 
 
 a'-', a? -'6, ai-'b'^ai-^b-', o'-'ftP^'. 
 
 As a particular case, suppose we wish the 15 roots of the 
 equation x^^ — 1 = 0, or x^-^—1 =0. 
 
 In this case, p = 3, and 5=5; we must therefore seek 
 the roots of x^ — 1 = 0, and x^ — 1 == 0. 
 
 We know, by Art. 287, that a- 1= 1 will satisfy each of 
 the above equations ; hence they are, by Art. 255, both di- 
 visible by X — 1. If we effect the division, we shall have 
 
 a;2-|-x-]_l=0, and x*-\-x^-{-x--j-x-{-l = 0, 
 
 for the results; the first of these, x"-|-x-|-l = 0, being 
 solved by quadratics, gives 
 
 a:= — i + iN/"^, or X = — 1 — iv/^^. 
 
 The other equation, x^-j-x^-j--"^""!"^"!"! =0) is a recur- 
 ring equation. Dividing it by x^, ^ve have 
 
 x=-^x+I+l+?^==o, 
 
 X X* 
 
 (..+ij+(.+y+i=o. 
 
 Substituting for x'--|-— , and a;-j--, their values in terms 
 
 of c, Art. 28", we find 
 
 z-' + z-l = 0. 
 This, solved by quadratics, gives 
 
 z = -i + iv/5,orz = — i — iv/5. 
 
261 GENERAL PROPERTIES OF EQUATIONS. 
 
 Taking the first value of ;:, we have 
 z =x+i = — i+ 1. v/5, or x- — (^ v/5 — i)x = — 1, 
 
 X 
 
 which, solved by quadratics, gives 
 
 :c = J [v/5 — 1+ >/— 10 — 2x75], 
 x= i[v/5 — 1— v/_lO — 2v75]. 
 Taking the second value of z, we have 
 
 z ^ x-\-- = — 
 
 v/5,orx^+(iv/5 + i)a: = — 1, 
 
 which, solved by quadratics, gives 
 
 a: = — i[v/5 + l—N/— 10+2^5], 
 or a;=3 — i[^5 + lH-v/— 10-h2v/5]. 
 
 In this case weVave for the three roots of x^ — 1 = 0, 
 the following : 
 
 1 = 1, 
 
 62 = — ^ — I ^/— 3. 
 
 We have for the five roots of x^ — 1 = 0, the following : 
 
 1 = 1, 
 
 a = l[^/5 — 1 4-v/— 10 — 2v/ 5], 
 
 a« = - i [v/5 + 1— v/-10 + 2"75], 
 
 a» == — I [v/5 + l + v/— 10 + 2s/5], 
 
 a' = ^v/ 5— 1— v/— 10— "275], 
 
 Consequently, the fifteen roots of a:"' — 1 = 0, are 
 
 1 = 1, 
 
 a= ;[v/5 — l+v^— 10 — 2v' 5], 
 a-^ = — 1 [,/5 + 1 — y — I0-f-' 275], 
 
 a^= — i[V5 + 1 +v/^To~+T7^], 
 
 a^= 1 [^5 _l_v/_ 10^^75], 
 
GENERAL DlOrilUTIES OF ECiUATIONS. 
 
 365 
 
 6=_. [l_v^_3j, 
 6a=— 1 [1— v/— 3JX[V5 — l+v^- 
 ba^= i [1 — n/^^3Jx[v5 + 1 — ^Z- 
 ba'= i[l — V^^Jx[^/5 + 1 +v/:_i0_|_2v5], 
 
 62o=— 1 [1 4-v/— 3jx[v/5 
 
 6V = Ul + ^'' --3 1 X [ v^O + 1 — ^/_10+2^75], 
 
 1 [1 ^ v/Zrr3] X [ ,/5 + 1 + ^/ — 10 + 2^ 5], 
 - - [1 -|.\/ir3]xrv/5 — 1 — x/— 10— 2^75]. 
 
 10— 2v/oJ, 
 T0-f2v/5], 
 
 1 _v/— 10 — 2v/5], 
 
 1 4_x/_iO — 2v/5j, 
 
 
 3]x[^/5- 
 
 If we extract the roots imlicated, to 7 places of decimals, 
 
 and reduce the results to their simplest forms, we shall 
 have 
 
 1= 1, • _ 
 
 a= 0.3090170 + 0.95105G5^/— 1, (3) 
 
 a2 :r^ _ 0.8090170-1-0.5877853 V—l, (6) 
 
 a' = — 0.8090170 — 0.5S77S53\/—1, (6') 
 
 0^= 0.3090170 — 0.95105G5v/—l, (3') 
 
 6 = — 0.5000000+0.8660254%/^, (5) 
 
 ah — — 0.9781476 — 0.20791 17 %/—l, (7') 
 
 a-b = — 0.1045285 — 0.9945219%/^, (4') 
 
 a^b= 0.9135454 — 0.40673G6v/^, (1') 
 
 a^b= 0.6691306 — 0.7431448 n/^, (2') 
 
 62 = _ 0.5000000 - 0.8C602"4x/— 1, (5') 
 
 ab-= 0.6691306 + 0.7431448 n/—1, (2) 
 
 a"-b'-= 0.9135454-1-0.4067366 v/^, (1) 
 
 fl-ife^ = — 0.1045285-1-0.9945219%/^, (4) 
 
 a'f^ = — 0.97814764-0.20791 17 v/—l, (7) 
 
 These imaginary roots are each of this form. 
 And in all cases, 
 
366 
 
 GENERAL PROPEKTIES OF EQUATIONS, 
 
 For a complete and lull discussion of the Binomial Equa- 
 tion^ ac" — 1 = 0, when n is a prime, the reader is referred 
 to the 5th part. Vol. II, of Legendre's Theone des JVom- 
 hres, 3d edition^ where he will find collected and demon- 
 strated the many beautiful theorems on this subject, which 
 were first published by M. Gauss^ in his Disquisitiones 
 Arithmeticcs. 
 
 (292.) Before closing this subject, it may not be amiss 
 to apprise the student, that the solution of binomial equa- 
 tions are most readily found by the aid of Trigonometrical 
 formula. 
 
 GENERAL SOLUTION OF AN EQUATION OF THE THIRD 
 DEGREE 
 
 (293.) We have seen. Art. 272, that an equation of the 
 third degree may be put under this form : 
 
 x^^A^x-{-Jl^=.0. (1) 
 
 If we assume x=^y-\-z^ (2) 
 
 we shall find x' = {y ^ zf = y^ -\- z^ -^'iyz^y -[• z) 
 
 
 :^-2>yz.x — y^-z^ = 
 
 0. (3) 
 
 If we equat 
 we shall find 
 
 e the coefficients of (3) 
 
 A, = — 3yz, 
 A = — y^ — z\ 
 
 with those of (1), 
 
 (4) 
 (5) 
 
 Which give 
 
 
 (6) ' 
 
 
 y^^z^ = — A,. 
 
 (7) 
 
 Cubing (6), 
 
 we obtain 
 
 ^ 0-7 
 
 (8) 
 
 Squaring (7), we get / + 2/z'-f c«= ^^ (9) 
 Subtracting four times (8) from (9), and wc have 
 
GENERAL PROPERTIES OF EQUATIONS. 367 
 
 Extracting the square root of (10), we find 
 
 2,3_,3_y^^3+4^?. (11) 
 
 By adding and subtracting half of (11) to and from the half 
 of (7), we find 
 
 2+^^ 4 +27' 
 2 ^ 4 ^ 27' 
 
 (12) 
 
 (13) 
 
 Hence, 
 x = y-\-z = 
 
 If we assume 
 
 
 (14) 
 
 2 *^ 4 ' 27 
 the above value of x will become 
 
 x=^m-\-n. (15) 
 
 Now, to obtain the other two roots, we will depress the 
 equation 
 
 by dividing it by r — {ni-f-n). See Art. 255. 
 
368 GENERAL PROPEr.TIES OF EQUATIONS. 
 
 OPERATION. 
 
 \x''-\-{m-{-n)x-\- (m + n)2-|-^i. 
 
 (m-^n)x''-\-^iX 
 {m-\-n)x^ — {m-^nyx 
 
 [{m-\-ny-\-Jl,]x-\-^2 
 [{m-\-ny-\-Ai]x—{m-\-ny—{m-\-n)Ai 
 
 {m-\-?iy-\-{m-\-7i)^i+^.. 
 
 As it regards this remainder, we see that since m-^n is a 
 root of equation (l)5it will be satisfied by substituting m-j-w 
 for X- making this substitution in (1), we find 
 (m + n)3 -(- (m -f- 7^)^i -f- /ia = 0, 
 which proves our remainder to vanish. 
 
 Hence, the true value of the depressed equation is 
 
 x''-\-{m-\-n)x-\-{vi-\-7iy-\-Jli = 0. (16) 
 
 This, solved by quadratics, gives 
 
 — {m-{-7,)± v/— 3 (7/1 + ny — 4^1 
 
 (17) 
 
 So that equations (15) and (17) give the three roots of 
 equation (1 ). 
 
 The two roots contained in (17) may bo found from (15), 
 as follows : Comparing equation (14) with (12) and (13), 
 we find 3/» = m', z^ = n'' ; therefore, by Art. 288, we have 
 
 y = am, > and c = o;?, > 
 
 y = a'm, } z — «^72, } 
 
 where 1, a, a', are the three cube roots of 1 ; that is, 
 
GENERAL PRCPERTILIS OF KqrATIONS. 369 
 
 « = — ! 4- i v/:zi3 . a-= — ! — : ^— 3. 
 
 See example under Art. l: '. 
 
 The only \vay in which we can combine the above six 
 values of y and c, so that at the same time their product 
 
 shall equal — '-—, equation (6), is as follows : 
 
 x=:m + n, giving the root of equation (15), "^ 
 
 x = a7n-\-a^n. ) ... „ ,,^, \ as.^ 
 
 >e[ivino; tlie roots 01 equation (1/), ( ^ ■' 
 
 The roots given by (17) may be simplified as follows : 
 
 Since y = m and c = ?7, we have yz --^ mn. Comparing 
 this with (6), we find ^^i = — 3m«, this value of c^^i, sub- 
 stituted in (17), gives 
 
 m -\- n , 7)1 — 71 / — r- "^ 
 ^ = --^ + -^—^-37 
 
 , C (19) 
 
 X = ■ V J. 1 
 
 Collecting in one point of view, the roots of the equa- 
 tion r' -(- jTiX + JI2 = 0, we have 
 
 x=zzm-f-n, (1)" 
 
 . = -'^ + iLp^=:^, (2)1 
 
 (B) 
 
 wnere m and 71 are given by equations (14). 
 
 We will now see what conditions must be fulfilled, in 
 order that one or all of the roots may be real. 
 
 47 
 
370 GENERAL PROPERTIES OF EQUATIONS. 
 
 CASE 1. 
 
 In this case, the values of m and 7i are real, and each 
 equal to \/ — ', and the values of x, given by (B), are 
 
 ^V-i\ (1) 
 
 -V=f. (3) 
 
 CASE II. 
 
 --(tr+(fr>«- 
 
 In this case, the values of m and n are both real, and 
 unequal. Hence, the first root as given by equation (B), 
 is real, whilst the other two are imaginary. 
 
 CASE III. 
 ...(-.)% (-J^o. 
 
 In this case, since I -^ 1 is positive for all values of j?2. 
 
 it follows that ./3i<0. This is called the irreducible case, 
 since m and n are both imaginary. 
 
GENERAL PROPERTIES OF EQUATIONS. :'71 
 
 Nevertheless, we can prove, that in this irreducible case, 
 all the roots are real. For, 
 
 Then we shall have 
 
 (C) 
 
 ^- =(/>+? ^—l)^ 
 
 See questions 13 and 14, Art. 191, which give the ex- 
 panded form of m and n as follows : 
 
 (p-9^-1)*- 
 
 J J _-' 2 —1 2 5 — - 
 
 Therefore, we find 
 
 2 ='' +5:6'' '-*"=■ 
 
 m— !j ( 1 -I 2.5 -i , , ) , — 
 -2-=J3P 9-3^P V + &c.|^_l. 
 
 Hence, the three values of t, as given by (B) , become 
 
372 
 
 GENERAL PRO'.EllTIES OF EQIATIONS. 
 
 
 \l''''''''-i-'h'''''''''+^'-\'^^'S 
 
 
 l.r. 
 
 (B") 
 
 ) I, 1 -J 2.5 -I , , 
 
 where Uie values of y; and 7 are given by (C). 
 
 And since p and q are real quantities, it follows that the 
 three rooU; as given by (B") are real. 
 
 GENERAL SOLUTION OF AN EQUATION OF THE FOURTH 
 DEGREE. 
 
 (294.) Let the equation of the fourth degree be put under 
 this form : 
 
 x'4-./J,a;^-f,?,r-f.,^, = 0. (1) 
 
 If we assume s = y-\-z-\-v, (2) 
 
 we shall find x- ■=i/'-\-z--\-u--^2{yz-\-yu-\-zu), 
 or x^ - {y-^Jrz'-\.u')= 2{yz+yu-^zu). (3) 
 
 By squaring (3), Ave find 
 
 X*- 2(r'-fz-^-f-wV-H(?/H~'^+'''')'= I /4X 
 
 ^y'z'-\-y■^u^+z'u^)-{-Syzu{y-j-z+u). S ^ ' 
 
 Replacing y-^z-\-u by x, in (4), and transposing, we find 
 x* — 2{r-{-z-+n^ x^ — 8yzu.x } _ 
 
 Now, in order that (5) and (1) may become identical, w< 
 must have 
 
GENERAL PROPERTIES OF EQUATIONS. 373 
 
 Ji.. = —Syzu, V (A) 
 
 From these conditions, v:e immediately deduce 
 
 yv-^+yV+c=i.= =:^^IZ_!:i\ ^ (B) 
 
 , o o ^; 
 ^ 64 
 
 Now, by Art. 265, we know that the sum of the three 
 roots of a cubic equation with their signs changed, is equal 
 to the coefficient of the second term of that equation ; and 
 the sum of their products taken two and two, is equal to the 
 coefficient of the third term ; also the continued product 
 of the three roots is equal to the absolute term. 
 
 Hence, the values of y^ ~"'5 and w-, will correspond with 
 the three roots of this equation : 
 
 ~ 2 ~ 16 64 ^ ' 
 
 If we suppose t = -. equation (6) will become 
 4 
 
 s' -{-2.1 16-^ + ( ^ 1 — 4^3)5 — j21=0. 
 
 If we denote the three roots of this equation, as found by 
 
 method explained under Art. 293, by s', s", a'", we shall have 
 
 u= ztW^'"- ) 
 Now, in order to find x, a root of (1), we must add the 
 values of y, c, and «, observing tliat their signs are so taken 
 
374 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 that their continued product may be afifected with a contrary 
 sign with A,^ so as to satisfy the second condition of (A). 
 
 CASE I. 
 
 WlienJi<i<Q. 
 The four values will be as follows : 
 
 x=-\-l^s'—\ Vs" — ^ Vs"',{ 
 x = —Ws'-i-Ws"-Ws"': 
 x = —Ws' — Ws"-\-^Vs"'. 
 
 (D) 
 
 CASE II. 
 
 When ^2> 0. 
 The four values will be as follows : 
 
 x = —l s/i' — i Vs" — ^ Vs'", 
 x= — lVs'-\-Ws"-\-^s^s"', 
 x = -\-W^''—Ws"-\-Ws"', 
 
 X = -j- ^ V^*' ~l~ I '^^" i y/s'". 
 
 (D') 
 
 The method of solving a cubic equation as given under 
 Art. 293, is generally supposed to have originated with 
 Cardan^ an Italian analyst of the 16th century j it is there- 
 fore frequently referred to as Cardan's Method. Montucla, 
 in his Hisloire des Mathcmatiques^ seems to have proved 
 that it was also discovered about the same time, independ- 
 ently of each other, by Scipio Ferreus and JVicolas 
 Tartalea. 
 
 The above method for equations of the fourth degree, 
 which is a close imitation of the method for cubic equations, 
 was first given by Euler, a distinguished analyst. 
 
 As yet, analysts have not been able to obtain the general 
 solution of equations beyond the fourth degree. 
 
general properties of equations. 375 
 
 Sturm's theorem 
 
 (295.) Let X=0 be an algebraic equation having real 
 coefficients ; we will suppose, also, that it has no equal roots. 
 Call A'l its first derived polynomial^ found by the method 
 of Art. 273. 
 
 Apply to X and Ai the method of finding the greatest 
 common measure, as explained under Art. 50, with this con- 
 dition, always to change the sign of the remainder at each 
 operation^ and to use this remainder, thus modified, for a di- 
 visor in the next operation. 
 
 Designate, moreover, by An, As, A4, Ar, the succes- 
 sive remainders, taken with contrary signs. 
 If we denote the successive quotients by 
 
 ?t5 92, ?3J qr^\, 
 
 we shall have the followino- relations : 
 
 A=A,7, — Ao, (1) 
 
 Ai = Ao<7, — A., (2) 
 
 A2 -^ A35 — A4, (3) 
 
 A.-o==X-i«7._i — A.. (r-1) 
 
 (A) 
 
 We shall necessarily have Xr independent of a*, and dif- 
 ferent from zero, (Art. 278.) 
 
 After having obtained the functions A, Ai, Ao, Ar, 
 
 suppose we substitute in them for x, two numbers p and q 
 of any signs whatever, p being < q. 
 
 The substitution of p will give results either positive or 
 negative ; if we only take account of the signs, and write 
 them, one after another in a line, they will give a certain 
 number of variations and permanences. 
 
376 GENERAL PROPERTIES OF EQUATIONS. 
 
 The substitution of q will in like manner give a succes- 
 sion of signs, of a certain number of variations and perma- 
 nences. 
 
 Now, the Theorem of Sturm consists in this : 
 
 The difference between the number of variations given 
 by the first series of signs, and the number of variations given 
 by the second series of signs, will express exactly the number 
 of real roots of the proposed equation, which are comprised 
 between p and q. 
 
 (296.) We shall now proceed to demonstrate this beau- 
 tiful theorem, 
 
 I. Consider the function A' in particular, and suppose a, 
 is a real root of A = 0. If we substitute a[ -j-w for x, in X, wr 
 shall obtain. Art. 273, a result of this form : 
 
 ^+^'u-{.'^u^ + f^tc^ w"; (1) 
 
 where ^ is w'hat X becomes when ai is put for x, and 
 
 v4', ^", ^"', are the successive derived polynomials 
 
 of J?, found by the method of Art. 273. 
 
 Now, by hypothesis, ai is a root of AT ^ 0, therefore j? ^ 0, 
 
 and the preceding expression becomes 
 
 (an am \ 
 
 ^^'+2''+l:3^'+ -^''r ^^^ 
 
 We can always take u sufficiently small to cause the quan- 
 tity within the parenthesis of (2) to have the same sign as 
 its first term A' . 
 
 II. If, in the functions A, Ai, A2, Ar, ive substi- 
 tute any quaiitity a for x, it cannot happen that two conse- 
 cutive functions shall vanish at the same time. 
 
 For take any three consecutive functions as An-i, An, X.+i- 
 Then, by conditions (A), w^e have 
 
 X„., = Xnq.-Xn+U (1) 
 
GENERAL PROPERTIES OF EQUATIONS. 377 
 
 Now, if we arc able to have at tlie same time 
 
 X->=0, (2) 
 
 X. = 0, (3) 
 
 we must also, by comlition (1), have 
 
 AVi = 0. (4) 
 
 Since the relation (1) is general, it must be true when we 
 write n-\-l for 7i; hence we have 
 
 Xn = X,-^i9«+i — Xt+J. (5 ) 
 
 In (5), substituting the values of X„, X,+i, as given by 
 (3) and (4), and we obtain 
 
 a;+2 = 0. (6) 
 
 By continuing this process, we should finally find 
 
 a; = o, (7) 
 
 which is absurd, since we have already shown that Xf can- 
 not equal zero. 
 
 III. The relation X„-i=X„qn — Xn-\.i, shows that if a 
 function X^ becomes by the substitution of x = a, the two 
 functions X„_i, X„_|-i, between which it is 'placed^ have ne- 
 cessarily contrary signs for x = a. 
 
 (297.) Designating by k a quantity positive or negative, 
 but less than each of the the real roots of the equations, 
 Z =0, 
 
 . X, = 0X (B) 
 
 X-i-0, 
 
 Conceive that the value of a: is made to increase continu- 
 ously from X = kj and that its successive values are substi- 
 tuted in the functions X, Xi, Xq, Xr. Now, so long 
 
 as the increasing values of x are less than each of the roots 
 of equations (B), the signs, arising from their substitution in 
 48 
 
378 GENERAL PROPERTIES OF EQUATIONS. 
 
 the functions of X, Xi, X^, X, will occur in the same 
 
 order ; for, in order that the number of the variations and per- 
 manences of signs should change, it is necessary that some 
 one of the above functions, as X„, should have passed through 
 the stage in which X,^ = 0, which cannot have happened, . 
 since x is supposed less than the least value which can satisfy 
 either of the equations (B). 
 
 (298.) We will now suppose that x has reached a value 
 x = a, which causes one of the intermediate functions X], 
 
 Xo, Xs, X-i, to vanish, without causing x to vanish. 
 
 We will also suppose X^ to be the one which vanishes when 
 x = a; then by II, under Art. 296, we know that X,-i, 
 Xn-^], cannot vanish, and by III, under the same Art., we 
 also know that X,_i and A',4-1 must have contrary signs. 
 Now, if we consider the sign of the vanishing term Xn to be 
 either plus or minus, the three consecutive functions Xi-ij 
 X„, X.-i-i, can produce only these two combinations of signs, 
 
 or _ ± 4- S 
 
 Either of which gives one variation and one permanence. 
 
 We know, by Art 297, that the signs of X„_i, X„-|-i, will 
 not be changed from x=k to x = (i, and since we are able 
 to take u as small as we please, it follows that they will not 
 be changed from a: = ato x = a-\-ti. 
 
 Hence\, the hypothesis x = a, introduced in the series of 
 
 functions X, X|, Xo, , ca7i produce neither a gain or 
 
 loss in the number of variations. 
 
 (299.) We will now suppose that x = Oi causes X to va- 
 nish. Let [/and l]\ represent the values of X and Xi, when 
 
 Represent, as in Art. 273, by A^ A' ^ A", , the va- 
 
GENERAL PROPERTIES OF EQUATIONS. 379 
 
 Ities of A' and its successive derived functions, when a:= a. 
 
 In the same way, represent by ^5,, ..^/, ^i", , the 
 
 ■values of X^ and its successive derived functions. 
 By Art. 273, we shall have 
 
 Since Ci is a root of A'=0, we must have Jl = 0. 
 Again, the values A' and ./5, each represents the value of Xi 
 when a, is put for a-, and since the equation A:=:0 is sup- 
 posed not to have any equal roots, A' or its equal cannot 
 vanish, therefore (C) becomes 
 /I" 
 
 2 
 
 (D) 
 
 01 which the right-hand members will have the same signs 
 as their first terms Jl'u, Jl', if we take u sufficiently small. 
 
 Hence^ when u is positive^ U and U^ will have the same 
 si on. 
 
 When u is negative, U and C/j will have contrary signs. 
 
 From which it follows that the functions X and Aj will 
 give a variation for x = ai — m, and a permanence for 
 x = a, -(- w. 
 
 Consequently, in the passage of the continuously increas- 
 ing values of xfrom x = ai — u to x = ai-\- u a variation 
 will be changed into a permanence. 
 
 The same results would have place, if the value x = ai, 
 which causes A to vanish, should at the same time cause 
 
 some one or more of the functions Ai, A2, A3, to 
 
 vanish. (Art. 298). 
 
 Now, commencing with z = Ci-j-w, if we suppose the 
 
380 GENERAL PROPERTIES OF EQUATIONS. 
 
 value of X to increase continuously, the number of varia- 
 tions in the series of signs ^vill remain the same, although 
 the order of the succession of the signs may be changed 
 until we reach another value, ar= 02, which causes X to 
 vanish, and which is therefore a root of X=^0 ; in which 
 case a second variation must be changed into a permanence ; 
 and so on. 
 
 HencCy the number of variations lost when x increases 
 from X = k to X = k' , must be equal to the number of real 
 roots of X=^ 0, comprised between k and k'. 
 
 APPLICATION OF STURM'S THEOREM. 
 
 (300.) Before passing to the application of this theorem, 
 we shall do well to pay attention to the following principles : 
 
 I. In obtaining the functions X, Xj, Xo, A',, we are, 
 
 by Art. 53, at liberty to introduce or suppress any numeri- 
 cal factor, provided that it is positive ; but it is necessary 
 to pay particular attention to the signs, and make only the 
 changes mentioned under Art. 295, as the peculiarities of 
 this theorem depend principally upon this change of the 
 signs of X, Xi, X-, . . . . X;.. 
 
 II. If we simply wish to know the total number of real 
 roots, without fixing in any manner their limits, we need 
 only substitute in the first terms of X, X), X^, . . . .X^, the 
 values — CO and -\- cc. 
 
 EXAMPLES. 
 
 1 . How many real roots has the equation 8x^ — 6.r — 1= ? 
 The first derived of Sx^ — 6a; — 1 is 24a:- — G, or sup- 
 pressing the positive numerical factor 6, it becomes 4x- — 1. 
 
 Now, applying to 8x^ — 6a: — 1 and 4a;- — 1 the method 
 of finding the greatest common divisor, we obtain — 4a; — 1 
 for the first remainder, changing its signs it becomes 4a;+l, 
 
GENERAL PKOPERTIES OF EQUATIONS. 381 
 
 continuing the operation ^vith 4a;'- — 1 and 4z-f ^j ""^'^ ^"^^ 
 
 — 3 for the remainder, hence we have 
 
 X=Sx' — 6x — l,\ 
 X, = 4x^-1, ( 
 
 A',=:4xH-l, ( ^^^ 
 
 ^3 = 3. ) 
 
 Now, if for X in the above functions we substitute — co, 
 the signs of the results will be 1 \- giving 3 variations. 
 
 If we substitute + co, they will be + + + + giving 
 variations. 
 Hence, 
 
 If in the same functions (A), we substitute the three 
 consecutive values x= — l,a:=:0, x = l, we shall find that 
 
 for x^ — 1 the signs are 1 \- giving 3 variations, 
 
 " 0- = " (- + " 1 " 
 
 "■ x=l " + 4. 4_ -f " " 
 
 Hence, two roots lie between — 1 and ; and one root 
 between and 1. 
 
 If we substitute x = — 1 5 "^ve shall find -f- ± (- giving 
 
 2 variations. 
 
 T^herefore, one of the negative roots lies between — 1 
 and — ^, and the other between — ^ and 0. 
 
 2. How many real roots has x^ — 5x--}- 8x — 1 z= ? 
 In this example, we find 
 
 A'=x3 — 5x-^-f-8x — 1, 
 X=3r*— lOx + S, 
 X^ = 2x — 31, 
 Xi = — 2295. 
 
 When X = — 00, we find j , giving 2 variationg, 
 
 « x = -foo, « + + +-, « 1 " 
 
382 GENERAL PROPERTIES OF EQUATIONS. 
 
 Therefore, the above equation has but one real root, and 
 consequently, it must have two imaginary roots. 
 
 3. How many real roots has x* — 2x^ — 7ar^-{- lOx-j- 
 lO = ? 
 
 In this example, we find 
 
 A" = a;^ — 2a;^ — 7ar^ -f- lOx -f 1 0, 
 Xi = 2x^ — 3x^ — lx-\-5, 
 X.2= 17x- — 23a; — 45, 
 X3= 152a: — 305, 
 X4 = 524785. 
 
 When X = — 00, we find H i [-, giving 4 variations. 
 
 « x=+oo, " + + + ^-_f, " 1. " 
 
 Consequently, the roots are all real. 
 We also find 
 
 -|- -(- — \- giving 2 variations, 
 
 + + 
 
 + + 
 
 4- + + + + 
 -+ +- + 
 +- + 
 
 " x = -3 -f - +- + 
 
 From which we see that the equation has two positive 
 roots between 2 and 3 ; one negative root between and 
 — 1 ; and one negative root between — 2 and — 3. 
 
 4. How many real roots has 2x* — 13x'4-10x — 19=0? 
 Here we find 
 
 X = 2x' — 13x2 _j_ lOx _ 19, 
 Xy = 4x» - 13x-f 5, 
 ^2= 13x'^ — 15X+38. 
 It is not necessary to calculate Xn and X4, since the 
 two roots of ^2= 13x2 — l5x-|-38 :=0, are imaginary, for 
 (15)^<4 X 13 X 38. See Art. 149, Formula (B). 
 
 hen 
 
 X = 
 
 
 
 
 X = 
 
 1 
 
 
 x = 
 
 2 
 
 
 x = 
 
 3 
 
 
 X ::=: - 
 
 -1 
 
 
 X = - 
 
 -2 
 
 
 2 
 
 
 
 2 
 
 
 
 
 
 
 
 3 
 
 
 
 3 
 
 
 
 4 
 
 
GENERAL PROPERTIES OF EQUATIONS, 383 
 
 Using only the values A'', Xi, Xo, we have 
 
 ■whenx = — go -\- — + giving 2 variations, 
 
 « a; = +oo 4- + -I- " " 
 
 Therefore, the two remaining roots are real. 
 
 5. How many real roots has a:^ — 36x^+72x2 — 37a:-j-72 = 01 
 Here we find 
 
 X =a:^ — 36x^-1-72x2 — 37a; -f 7 2, 
 
 Xi = 5x' — 108x^ + 144x — 37, 
 
 Xo= 18x3 — 54x--f-37x— 90, 
 
 X-.i = 1319x^ — 2487x — 684, 
 
 Z4 = — 2960933x-f- 34935426, 
 
 Xs = - 
 whenx = — 00 we have i pH gi'^'i'^g^ variations, 
 
 " X=:-f-00 " + + + H " 1 " 
 
 Hence, the proposed equation has three real roots and two 
 imaginary ones. 
 
 . 6. How many real roots has x^-{-M,x-^^2'==-0 1 
 In this example, we find 
 
 X =x^'-(-.^,x-{-^,, 
 
 Xi = 3x--(-^:, 
 
 X, = — 2.;^ix — 3A', 
 
 X.,=-- — 4^?— 27^=. 
 
 CASE I. 
 
 WAen— 4^J— 27^=>0. 
 
 Now, since — 27^P, is negative for all values of .//■_>, it is 
 necessary that ^,<0, in order to fulfil the above condition. 
 Consequently, 
 
 when x^ — x we have 1 \- giving 3 variat-oro. 
 
 " x= + oo " -I- 4- -^ 4. " " 
 
384 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 Therefore, when -4^?— 27^^ v>o or 4^?4-27^§<0, then 
 the three roots are real. See Case III, page 370. 
 
 CASE II. 
 
 When —^J^, —21^l<0. 
 
 This condition can be fulfilled for values of ^i either posi- 
 tive or negative, so that 
 
 when x = — oo we have f- db — giving 2 variations. 
 
 " a; = -f-oo " -f--f-=F— " 1 " 
 
 Therefore, when — 4^? — 27^-:<0, or 4^?+27^p0, 
 then there will be but one real root, and consequently two 
 imaginary roots. See Case II, page 370. 
 
 CASE III. 
 
 When — 4.^3 _ 27^52 =.0. 
 
 In this case, we know, by Art. 279, that there are two 
 equal roots which will be given by 2^ia;-|-3^2= 0. Hence, 
 
 one of the equal roots is a: = — -—^, and the other root muK 
 
 2^i' 
 
 SJlo 
 
 bex = — — . See Case I, page 370. 
 
 7. How many real roots has x^-{-jiiX--\-^2X-\-^3 = 1 
 
 T^ere we find 
 
 X =x*+Aa:^ + .V4-^3, 
 
 ^i=4a:»+2^iX+A., 
 
 Xz=z — 2j?,x- — 3^20: — 4^.,, 
 
 Jr4= 16^3(^? —^Ji■^)^-Al{^A\—\UJ],A..-\-'ilAl^ 
 
GENERAL PROFF.KTIES OF EQUATIONS. 385 
 
 CASE I. 
 Whe7i .'2i<0, S.^fJh— 2./^? —9^5 >0, and 
 
 Then, 
 when x = — ex, we find -\ ( [-, giving 4 variations. 
 
 CASE II. 
 
 When 16^i3(^5= — 4.^3)K^::(4^?? — 144^1^3+27^^). 
 
 Then, 
 when r= — oo, we find -j 1 , giving 3 variations. 
 
 " x = -^oo, " +-j , " 1 « 
 
 Therefore, in this case, there must be two real roots, and 
 consequently, two imaginary roots. 
 
 When neither of these conditions are fulfilled, all the 
 roots are imaginary, 
 
 GENERAL METHOD OF ELIMINATION AMONG EQUATIONS 
 ABOVE THE FIRST DEGREE. 
 
 (301.) Suppose we have two equations, each containing 
 X and y, represented by 
 
 X=0, (1) 
 
 ^,=0, (2) 
 
 Now, if we seek the greatest common measure of the 
 polynomials .Y and Xi, by the method of Art. 50, we shall 
 
 have 
 
 X=X,q-^r, (3) 
 
 49 
 
386 GENEUAI. PROPEKTIKS OF EQUATK .<;.. 
 
 where q is tlie nuolicnt of X divicled by X-i, and r is the 
 remainder. Now, since by (^l)Tind (2), Jt and X\ are each 
 zero, it follows thai r as given by (3), must also be zero. 
 
 (302.) FrGiu which we conclude^ thai if ire operate upon 
 the polynomials X fmd X], hy the method for finding the 
 greatest commuiv measure^ we shall have the successive re- 
 mainders each equal to zero. 
 
 If wc arrange the j.olynoinials with reference to either 
 of the letters^ before operating vpon them., we shall iilti- 
 mately find a reni(i)ider independent of that letter.) when 
 the polynomiiils have iw common measure.^ which remainder 
 being 'put eqind to zero, will give an equation containing 
 hut one nnknoirn quiintity. 
 
 When the two polynomials have a common measure., it 
 must be put equal to zero., if it contains both the unknown 
 quantities., then divide both polynomials by it., and proceed 
 with the results as in the first case. 
 
 Note. — In the operation of finding the greatest common 
 measure of two polynomials, it frequently becomes neces- 
 sary to suppress factors, as well as to introduce new factors. 
 When this is done, we must carefully examine whether 
 such ffictors are able to effect the final result. If no 
 factors other ihan numerical, are either suppressed or 
 introduced, then the above method is rigidly correct, but 
 in other cases, the rule would require some modification. 
 
 EXAMPLES. 
 
 ]. Obtain from the two equations 
 
 •^=+-r.y+!/'^-i = o, (1) 
 
 x'-j-y'=0, (2) 
 
 a single equation in terms of y. 
 
GENERAL PROPERTIES OF EQIATIONS. 387 
 
 Proceeding by the method of finding the greatest common 
 measure, Art. 50, we have for the 
 
 FIRST OPERATION. 
 
 ^ ~ry lar-j-yx-j-i/* — 1 
 
 — yx- — y-x — y'^-\-y 
 
 'x — y. 
 
 x-j-^T/' — y = first remainder. 
 
 Again, dividing X2-\-yx-^y- — 1 by this remainder, we 
 find for the 
 
 SECOND OPERATION. 
 X^J^yxJ^-y — 1 I X-|-2^3 _ y 
 
 x-'J^{2y'-y\r 
 
 —{2y^-2y)x-^f—\ 
 
 — (2r-2.v)x-47y«+6y^— 27/- 
 
 4,1/''— 6j^^-|-3i/- — 1 = second rem. 
 
 Putting this rema nder, which is independent of x, equal 
 to zero^ we have for the equation sought : 
 
 ^.v'-6i/' + 3r-l=0. (3) 
 
 If we were required to find, from the above two equations, 
 one single equation in terms of x, we observe, that all that 
 would be necessary would be to change y into x, in equation 
 (3), since x and y can be changed the one for the other in 
 equations (1) and (2) without affecting their form. 
 
 2. Obtain an equation independent of y from the two 
 equations 
 
3S8 
 
 GENKSAL PROPKRTiES OF EQCATiONS, 
 
 H22/H32/-4)r+y— 1 ) 
 
 (2) 
 
 Proceeding with these equations agreeably to the above 
 method, we find for the first remainder the following: 
 
 Repeating the process, we find for the second remainder, 
 the followin g: 
 
 which being put equal to zero, gives for the equation sought, 
 
 3. Obtain an equation independent of y from the two 
 equations 
 
 3x»y ' + (3x- - 3x)y' — (2x-^^ x)f j ^ ^ 
 
 + (x • — 2x--' +2x — 3 )2/+x ' — X— 2 S 
 3x2^3 _ 2xy'2 — (2x2 _ x)y-^x^^x — 3=0. (2) 
 Ans- X" — X — 2=0. 
 
 (1) 
 
 (303.) When we have three equations involving three 
 unknown quantities. We must first eliminate one of the un- 
 knowns by combining either of the equations with the other 
 two ; we shall thus obtain two new equations involving 
 only two unknown quantities, which, as we have just shown, 
 will give a final equation involving but one unknown quan- 
 tity. 
 
 1. Obtain an equation containing only x, from the thret 
 equations 
 
GENERAL PROPERTIES OF EQUATIONS. 389 
 
 x-\-y^ — a=0. (1) 
 
 y+z'-h=0. (2) 
 
 ^^^_c=0. (3) 
 
 Eliminating z between equations (2) and (3) we have the 
 following 
 
 OPERATION, 
 
 z^-\-y — 6 z-\-x- — c 
 
 2^-(-(x- — c)z 
 
 z — {x'-c). 
 
 —{x^—c)z-i-y—b 
 —{x"-—c)z-x*-{-2cx-—c^ 
 
 x^ — 2cx'^-\-y-\- c^ — 6 = remainder. 
 
 Putting this remainder equal to zero, we have 
 
 x'—2cx~+y-^c'—b = 0. (4) 
 
 Now, eliminating y between equations (1) and (4), we 
 have the following 
 
 OPERATION. 
 
 yi^x — a \y^x'~-2cx''-\-c- — b 
 
 j;J-j_(a;i — 2cx--{-c- — b)y 
 
 1 j/ — (x' — 2cx^ -\- c- — b). 
 
 —{x^ — 2cx--\-c'^ — b)y-\-x—a 
 
 — (x' — 2cx--(- c- - b)y — {x* — 2cx= -}- c^ — 6)= 
 
 {x* — 2cx--\-c''' — b)--\-x — az= remainder. 
 Expanding this remainder, and then equating it with 
 zero, we have 
 
 x« — 4cx«-f-(6c-2— 26)x' — (4^ — 4/)c)x^)_ 
 
 -|_x_(_c'-f-6-^— 26c- — aS~ ' 
 By simply permutating these quantities, (Art. 85), we 
 have 
 
 /_4ay«-f-(6a=— 2cy— (4a^— 4cay>^ 
 
 _^3/_|_a. + c2 — 2ca- — 6S ' ^ ^ 
 
390 
 
 GENERAL PROPERTIES OF EQUATIONS. 
 
 _|_;_j_/,4_(_f,2_2a63 — c! 
 
 0. (7) 
 
 2. In a similar manner, find three equations each contain- 
 ing but one unknown quantity, from the three equations 
 a:2_^^.y_.a = 0, (1) 
 
 r-{-yz-b = 0, (2) 
 
 ::2_{_z2;— c = 0. (3) 
 
 thirst, eliminating z between (2) and (3), we find 
 
 y^ — xy' — {2b-\-c)y'-\-bxy-^b' = 0. (4) 
 
 Secondly, eliminating y between (1) and (4), we find 
 2x«— (7o-f-36-|-c)x« + (9a-^-f-5a6 + 2crc-f ^•^)a:^; 
 — (5a3 -f- 2a'6 -f a2c)x-^-l- a^ : 
 By permutating these letters, we find 
 
 0.(5) 
 
 22' 
 
 — (56-^-f-26-^c-f 6'a)?/ 
 (7c+3a+fe)c« + {9c^-\-5ca -f 2cb + a^)z' 
 
 — {oc'-\-2c-a -f c^b)z-'-\-c 
 If a = 1€, 6 = 17, and c = 18. 
 Then will the eight sets of values be 
 
 rx=:iz 4.173281, 
 1. < y ^ ± 4.287098, 
 (z = ^ 0.330363. 
 rx = ± 2.525516, 
 ■2.}y=zt 2.969156, 
 (z= ± 3.240579. 
 rx = ± 0.418924, 
 
 3. ) y = i 3.912240, 
 (z= ± 4.048877. 
 r X = ^ 4.003756, 
 
 4. ) 7/ = dr 0.007100, 
 ( c = =F 4.245971. 
 
 5:S=o,(6) 
 i(=o.(7) 
 
GENERAL PKOPmiTlES CF EQUATIONS. 3!>1 
 
 3.' Find three Li[uations each containing but one unknown 
 quantity, from the three equations 
 
 .• + 1/c - a = 0, (i) 
 
 J--}- zx — b = 0, (2) 
 
 r + a-2/ - c = 0. (3) 
 
 Operating as in the preceding examples, we find the 
 
 following results : 
 
 x^ — ax* — 2x'-\-{2a + bc)x' — {b'^-\-c-—l)x-\-bc — a = 0j 
 3/S— 6y' — 2y3+(26+a0y-— (C-+0-— l)y+cr — i^.O, 
 z^—cz' — 2z'-{-:^c+ab)z''—^n'+h'—l)z-{-ab — c = 0. 
 
 4. Find three equations each containing but one unknown 
 quantity, from the three equations 
 
 ar-]-yz — a = 0, (1)' 
 
 y' + zx- I . 0, (2) 
 
 z- + ay — c = 0. (3) 
 
 '8x^ — 20aa;«-f (I8a2 — 26c)a:' ? 
 
 + {babe — 7a'' — b^ — c')x'-+ (a- — bcY 
 
 = 0, 
 
 ^^g ,8]/« — 20&y«+ (186-- + 2ca) ^ 
 
 4- {bbca — lb^ - c3 — a3)3/^4-(ft2_ ca)2 
 
 82« — 20cz'' + (l8c'— 2flZ;) c' > 
 
 + (5ca6 — 7c^ — a^ — 6-^)c-+(c-^ — a6y^S~~ ' 
 
 (304.) When there are foui equations, we must first 
 reduce the number to three by eliminating any one of the 
 unknown quantities, and then proceed as above. From 
 what has already been done, it will not be diflficult to know 
 how to proceed for a greater number of equations, but it is 
 obvious that in many cases this general method must be 
 very long and laborious, still it is valuable on account of 
 the certainty of the result. 
 
392 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 CHAPTER XL 
 
 NUMERICAL SOLUTION OF CUBIC EQUA- 
 TIONS, AND EQUATIONS OF 
 SUPERIOR DEGREES. 
 
 (305.) Let ^ix5-j-^2x'+^3X = A (1) 
 
 be any cubic equation, and suppose that two consecutive 
 numbers in either of the series 
 
 1, 2, 3, 4, &c. 
 10, 20, 30, 40, &c. 
 
 0.1, 0.2, 0.3, 0.4, &c. 
 0.01, 0.02, 0.03, 0.04, &c. 
 &c. &c. 
 
 (A) 
 
 are found such, that by substituting the first for x in equa- 
 tion (1), the result shall be less than ./?4,and by substituting 
 the second, the result shall be greater than Ji^ ; then, by Art. 
 263, the fust of these numbers, omitting the cyphers if it 
 have any, will be the first figure of one of the roots. Let 
 this figure be denoted by n, and the other succeeding figures 
 of the same root by r?, 7-3, r4, &c., respectively. That is, 
 7-1, ra, ra, Ta-, &c., are the local values of the successive fig- 
 ures of the root. 
 
 If for X, in equation (1), we substitute its first figure n, 
 we shall have Air\-\-Air\ +^3ri = ^4. (2) 
 
 a 
 
 Therefore, r, = .5-^^-3^ (3) 
 
NUMEKICAL SOLLTION OF IIIGHEU EQUATIONS. 393 
 
 If we put 3/ for the excess of the true root above its first 
 figure, we shall have a- ^ ri-f-y ; this value being substi- 
 tuted in (1), we get 
 
 A,f-{- A'.y'^ + A'.,y -^ B = A„ 
 
 or A,y'-\-A'2f-\-A',y = A'„ (4) 
 
 where A' , = A+^An, (1) ) 
 
 ^';, = ^3-f-2^erj + 3Ar^, (2) [ (B) 
 
 Equation (4) is in all respects similar to the original equa- 
 tion (1); therefore, repeating the above process upon this 
 equation, we shall obtain 
 
 where ro is the first figure of the root of equation (4), or the 
 second figure of the root of equation (1). Putting z for the 
 sum of all the remaining figures, we have y =?-^,-|-z; this 
 value substituted in (4), gives 
 
 A',z-\-A',r^_=A'^, 
 
 A'zz''-{-2A'.r,z-\-AUrl = A'^y^, 
 
 A^z'-\-3AirsZ'-\- 3Airlz-f- Air^= A^f, 
 
 A,z^-^ A",z'-\- A",z-^ B' = A\, 
 
 or A^z'-\-A",z--}-A"3Z = A"4, (6) 
 
 where ^", = j^'a-fS^.r.,, (1) 
 
 A", = A'3-\-2A',r-2-^3A,rl, (2)} (B') 
 
 A^" = A\ — A',ro — A'2rl —A,rl (3) 
 
 Here, again, equation (6) is similar to equations (4) and (1) 
 50 
 
394 NUMERICAL SOLUTION" OF HIGHER EQUATIONS. 
 
 We might now proceed to find the first figure of the 
 root of equation (6), the value of which must be such, that 
 we shall have 
 
 (306.) Now, by observing the formation of the coeffi- 
 cients ./3'j, .^'2, in equation (5), and recollecting that 7-1 
 being the first figure of the root, must be greater than rg, it 
 will appear obvious that ^^'y must constitute the largest 
 portion of ^'3 + J^'grg-f- .iir^, which is the denominator 
 of the value u as given by (5), and if nis already known, 
 then (2), of (B), will make known J^'g, which maybe used 
 as a trial divisor for finding ro, the second figure of the 
 root ; the same may be observed of the succeeding divisors, 
 and it is obvious that these trial divisors j3"3, ^'"3, &c., 
 will continually approach nearer the true divisors. 
 
 (307.) If we multiply the first coefficient by r,, and add 
 the product to the second coefficient, we shall find 
 
 A<i+Air.. (8) 
 
 Multiplying expression (8) by r,, and adding the product 
 to the third coefficient, we have 
 
 ^3H-Avr, + ^ir?. (9) 
 
 Multiplying expression (9) by rj, and subtracting the 
 product from A\^ we have 
 
 ^.1 — -^ari — ^2^7 — ArJ. (10) 
 
 Again, multiplying the first coefficient by ri, and adding 
 the product to expression (8), we have 
 
 ^2+2^ir,. (11) 
 
 Multiplying expression (11) by 7-j, and adding the pro- 
 duct to expression (9), we have 
 
 ^3+2Ari + 3^ir?. (12) 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 395 
 
 Again, multiplying the first coefficient by ?• , and adding 
 the product to expression (11), we have 
 
 A+3^iri. (13) 
 
 Expressions (13), (12), and (10), are the values of the 
 coefficients ^5'o, ^^'j and .^'i respectively, of equation (4), 
 as given by (B). 
 
 (308.) From the above method of operation, we dis- 
 cover that the root of a cubic equation having numerical 
 coefficients, can be found by the following 
 
 RULE. 
 
 I. Having found the first figure of the root, multijdy it 
 into the first coefficient^ and add the iproduct to the second 
 coefficient^ which sum, multiply by the same figure and add 
 the product to the third coefficient^ imiltiply this last sum by 
 the same figure and subtract the product from the term 
 which constitutes the right-hand member of the equation; 
 the remainder we shall call the first dividend. 
 
 Again, multiply the first coefficient by the same figure, 
 and add the product to the lust number under the second 
 coefficient, which sum must be multiplied by the same figure 
 and the product added to the last number under the third 
 coefficient, the result we shall call the first trial divisor. 
 
 Again, multiply the first coefficient by this same figure, 
 and add the product to the last number under the second 
 coefficient. 
 
 II. Find the second figure of the root by dividing the 
 FIRST dividend by the first trial divisor, proceed with 
 this second figure precisely as was donewiththe first figure, 
 observing to keep the work so that units shall stand under 
 units, tens under tens, ^c, ^c. 
 
 Note. — The above rule bears a close resemblance to 
 the rule for extracting the cube root of a polynomial, as 
 given under Art. 106. 
 
39f5 
 
 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 EXAMPLES. 
 
 1. Find a root of the cubic equation 3a:'-j-2x'-j~'^^^='^^- 
 
 OPERATION. 
 
 <U 4) fc. H 
 
 3 2 
 
 14 
 
 20 
 
 215 
 
 230 
 
 245 
 
 ^171 
 
 2492 
 
 2513 
 
 25151 
 
 25172 
 
 25193 
 
 251957 
 
 4 75(2.57 7 9 &c.= a;. 
 
 20 40 
 
 48 =: Ist trial divisor. — 
 
 5875 35 = first dividend. 
 
 7025 = 2d trial divisor. 29375 
 
 719797 
 
 73724 1 = 3d trial divisor. 5625 = 2d dividend. 
 73900157 503S579 
 
 74076361 =4th trial divisor. 
 
 7409903713 586421 = 3d dividend. 
 
 517301099 
 
 691 19901 =4th dividend. 
 66689133417 
 
 2430767583. 
 
 Find one of the roots of the equation 
 lx--^x^ — Ux=lG15. 
 
 1 
 
 43 
 
 85 
 127 
 1284 
 1298 
 1312 
 13162 
 13204 
 13246 
 132509 
 132558 
 132607 
 1326112 
 
 14 
 244 
 
 754 
 
 77968 
 
 80564 
 
 8135372 
 
 8214596 
 
 822387163 
 
 823315069 
 
 82339463572 
 
 1675(6.2676&c.= x. 
 
 1464 
 
 211 
 155936 
 
 55064 
 48812232 
 
 6251768 
 5756710141 
 
 495057859 
 4940367S1432 
 
 1021077568. 
 
NUMERICAL SOLUTION CU HIGHER EQUATIONS. 
 
 39-: 
 
 Find a i 
 
 oot of the equation 
 
 Zx' 
 
 -2a:--f-a-=3. 
 
 — 2 
 
 1 
 
 
 3(1.1417&c.=:x 
 
 1 
 
 2 
 
 
 2 
 
 4 
 
 6 
 
 
 — 
 
 7 
 
 673 
 
 
 1 
 
 73 
 
 749 
 
 
 673 
 
 76 
 
 78108 
 
 
 
 79 
 
 81364 
 
 
 327 
 
 802 
 
 S144663 
 
 
 312432 
 
 814 
 826 
 
 SI 52929 
 
 
 
 815871877 
 
 
 1456S 
 
 8263 
 
 
 
 8144663 
 
 8266 
 
 
 
 
 8269 
 
 
 
 6423337 
 
 82711 
 
 
 
 5711103139 
 
 712233861. 
 
 (309.) The above roots are all positive. We will now 
 give a couple of examples when the value of a: is negative. 
 The operation will remain the same if we observe the alge- 
 braic rule for the signs. 
 
 4. Find a root of the equation 5x^ — 6x--f-3.r= — 85. 
 
 — 6 
 
 3 
 
 — 86(— 2.16139&C. 
 
 — 16 
 
 35 
 
 — 70 
 
 — 26 
 
 87 
 
 
 — 3'; 
 
 9065 
 
 — 15 
 
 — .365 
 
 9435 
 
 — 9065 
 
 -370 
 -375 
 
 966180 
 989040 
 
 
 -5935 
 
 — 3780 
 
 98942405 
 
 — 5797080 
 
 — 3810 
 
 — 3840 
 
 989S0SI5 
 9S99233995 
 
 
 — 137920 
 
 — 38405 
 
 9900386535 
 
 — 98942405 
 
 - 38410 
 
 — 38415 
 
 990073231455 
 
 
 — 38977.595 
 
 -384165 
 
 
 — 29697701985 
 
 — 384180 
 
 
 
 — 384195 
 
 
 -9279893015 
 
 -3841995 
 
 
 — 8910659083095 
 
 36923393J905, 
 
398 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 5. Find a root of the equation x^-{-x'^-{-'70x=^ — 300. 
 
 1 
 
 70 
 
 — 300(— 3.73S79&C 
 
 —2 
 
 76 
 
 — 228 
 
 —5 
 
 91 
 
 
 —8 
 
 9709 
 
 — 72 
 
 —87 
 
 10367 
 
 -67963 
 
 —94 
 
 — 101 
 
 1039739 
 1042787 
 
 
 — 4037 
 
 — 1013 
 
 1043602S4 
 
 — 3119217 
 
 — 1016 
 
 — 1019 
 
 104441932 
 10444908229 
 
 
 — 917783 
 
 — 10198 
 
 10445623307 
 
 — 834882272 
 
 — 10206 
 — 10214 
 
 1 04457 1525''71 
 
 
 IVIIC/I lKj^%J^t I 
 
 — 82900728 
 
 — 102147 
 
 
 — 73114357603 
 
 — 102154 
 
 — 102161 
 
 
 
 
 — 9786370397 
 
 — 1021619 
 
 
 — 9401143727439 
 
 .385226600561. 
 
 (310.) When the second or first power of a: is wanting, 
 we must consider its coefficient as being d=0. 
 
 6. Find a root of the equation x^ — ]2x = 28. 
 
 ±0 
 
 — 12 
 
 28(4.30213&c. = a;. 
 
 4 
 
 4 
 
 16 
 
 8 
 
 36 
 
 — 
 
 12 
 
 3969, 
 
 12 
 
 123 
 
 4347 
 
 11907 
 
 126 
 
 43495804 
 
 
 129 
 
 43521612 
 
 93 
 
 12902 
 
 4352290261 
 
 86991608 
 
 12904 
 12906 
 
 4352419323 
 435245804199 
 
 
 6008392 
 
 129061 
 
 
 4352290261 
 
 129062 
 
 
 
 
 129063 
 
 
 1656101739 
 
 1290633 
 
 
 1305737412597 
 
 350364326403. 
 
NUMERICAL SOLUTION OF HIGHE-R EQUATIONS. 
 
 599 
 
 7. Find a root of the equation 2x^-\-3x^ = 850. 
 
 3 
 
 17 
 31 
 
 ±0 
 119 
 336 
 
 850(7.05025&c.= x 
 833 
 
 45 
 
 3382550 
 
 17 
 
 4510 
 4520 
 
 3405150 
 34052406008 
 
 16912750 
 
 4530 
 
 34053312024 
 
 87250 
 
 453004 
 453008 
 
 3405353853050 
 
 6S104812016 
 
 453012 
 
 
 19145187984 
 
 4530130 
 
 
 17026769265250 
 
 211841"<718750. 
 
 (311.) In all the preceding examples, the first figure of 
 the root has been in the units' place ; \\c will now add two 
 examples in which the first figure is in the tens' place. 
 
 8. Find a root of the equation 3a:^ — 7a;'' -|-I3x = 45000. 
 
 -7 
 
 13 
 
 45000(25.404&c.=:x 
 
 53 
 
 1073 
 
 21460 
 
 113 
 
 3333 
 
 
 173 
 
 4273 
 
 23540 
 
 188 
 
 ■5288 
 
 21365 
 
 203 
 
 537568 
 
 
 218 
 
 5-l63«4 
 
 2175 
 
 2192 
 
 5464726448 
 
 2150272 
 
 2204 
 
 
 
 2216 
 
 
 24728 
 
 221612 
 
 
 21S58905792 
 286909420?, 
 
 "ind a 
 
 •oot of the equation 
 
 .rS^GOx- — 800x^360000 
 
 60 
 
 -800 
 
 60000(30.537&c.=z 
 
 90 
 
 1900 
 
 57000 
 
 120 
 
 5500 
 
 
 
 150 
 
 :,57525 
 
 3000 
 
 1505 
 
 565075 
 
 27876-23 
 
 1510 
 
 56552959 
 
 
 1515 
 
 56598427 
 
 212375 
 
 15153 
 
 5660903S79 
 
 160n5S877 
 
 15156 
 15159 
 
 
 
 
 42716123 
 
 151597 
 
 
 39626327153 
 30^979.^847. 
 
400 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 (312.) The method of proceeding, "when the first figure 
 is of a local value greater than ten, will be obvious. 
 
 We will add two examples in which the fii;st figure is in 
 the tenth's place. 
 
 10. Find a root of lOz^ _ 24x^ — 30x = — 6. 
 10 —24 —30 —6 (0.1768 &c.=x. 
 
 — 23 —323 —323 
 
 -21 
 
 — 35921 
 
 — 277 
 
 -203 
 
 — 37293 
 
 — 251447 
 
 -196 
 
 — 189 
 
 — 3740604 
 -3751872 
 
 
 — 25553 
 
 -1884 
 
 — 275336896 
 
 — 22443624 
 
 -1878 
 — 1872 
 
 
 
 
 — 3109376 
 
 — 18712 
 
 
 — 3002695168 
 
 
 — 106680832. 
 
 Find a root of the equation x^-\- 9a; = 6. 
 
 iO 
 
 9 
 
 6 (0.6378 &c.=a: 
 
 0.6 
 
 936 
 
 5616 
 
 12 
 
 1008 
 
 
 18 
 
 101349 
 
 384 
 
 183 
 
 101907 
 
 304047 
 
 186 
 
 10203979 
 
 
 189 
 
 10217307 
 
 79953 
 
 1897 
 1904 
 
 1021883644 
 
 71427853 
 
 
 
 1911 
 
 
 8525147 
 
 19118 
 
 
 8175069152 
 
 350077848. 
 (313.) By reviewing the foregoing examples, we discover 
 that the last terms under tlie third coefficient, or the trial 
 divisors^ remain unchanged in several of its kft-hand fig- 
 ures, thus in example 1, 740 is common to the left-hand of 
 the last two terms under the third coefficient. In example 
 2, the figures common are 8233. In example 3, the figures 
 common, are 815, and so for the succeeding examples. 
 
NUMERICAL SOLUTION OF HIGHER IQUATIONS. 
 
 401 
 
 the constant figures of the last term under the third co- 
 efficient, and also omit from the right of the last dividend, 
 the same number of figures save one, we may then divide the 
 remaining figures of the dividend by the remaining figures 
 of the last term under the third coefficient, by long division ; 
 and as many additional figures of the root may in this way 
 be found as there are figures in the divisor thus used. 
 
 12.. Find a root, to 8 decimals, of the equation 
 x3+x- = 500. 
 
 500(7. 61727975&c.=x. 
 
 1 
 
 8 
 15 
 22 
 226 
 232 
 238 
 2381 
 2382 
 2383 
 23837 
 23844 
 23851 
 238512 
 258514 
 
 iO 
 
 56 
 161 
 17456 
 18848 
 1887181 
 1889563 
 189123159 
 189290067 
 18929483724 
 18929.960752 
 
 108 
 104736 
 
 3264 
 
 1887181 
 
 1376819 
 1323862113 
 
 52956887 
 37858967448 
 
 150979.19552 
 132509 
 
 18470 
 17036 
 
 1434 
 1325 
 
 109 
 95 
 
 61 
 
402 
 
 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 EXPLANATION. 
 
 After tinding 4 decimal places in the root by the prece- 
 ding rule, we cut off 6 figures from the right of the last trial 
 divisor, thus leaving the constant figures 18929 ; and from the 
 right of the dividend we cut off 5 figures, leaving 150979 ; 
 we then divided 150979 by 18929, by the rule for abridged 
 division, (see Higher Arithmetic,) and thus obtained the 
 additional figures of the root. 
 
 13. Find a root, to 10 decimals, of the equation 
 r^ — 17a;2-|-54x = 350. 
 
 17 
 
 54 
 
 350)14.9540686096. 
 
 -7 
 
 — 16 
 
 — 160 
 
 3 
 
 14 
 
 
 13 
 
 82 
 
 510 
 
 17 
 
 166 
 
 328 
 
 21 
 
 18931 
 
 
 
 25 
 
 21343 
 
 182 
 
 259 
 
 2148175 
 
 70379 
 
 268 
 
 2162075 
 
 
 277 
 
 2163189 
 
 11621 
 
 2775 
 
 216430348 
 
 10740875 
 
 2780 
 
 2164320197236 
 
 
 2785 
 
 
 880125 
 
 27854 
 
 
 865275664 
 
 27858 
 
 
 
 27862 
 
 
 14849336 
 
 2786206 
 
 
 12985921183416 
 
 
 186341.4816584 
 
 
 
 173147 
 
 
 
 13194 
 
 
 
 12986 
 
 
 
 208 
 
 
 
 194 
 
 
 
 14 
 
 
 
 13 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 403 
 
 314.) Thus far we have sought only one of the roots of 
 v)ur equations. If we wish the three roots, we may divide 
 the given equation, when all the terms are transposed to one 
 side, by the unknown quantity ^niyius the value of the root 
 found by the above method, we shall thus depress the cubic 
 equation to a quadratic. (See Art. 255.) 
 
 14. Find the three roots of the equation 
 
 Here, we soon discover that one of the roots lies between 
 1 and 2 ; seeking this root by the above process, we find 
 
 -15 
 
 63 
 
 50(1.028039231 &c.= 
 
 — 14 
 
 49 
 
 49 
 
 — 13 
 
 36 
 
 — 
 
 — 12 
 
 357604 
 
 I 
 
 — 1198 
 
 355212 
 
 715208 
 
 1196 
 
 35425744 
 
 
 — 1194 
 
 35330352 
 
 284792 
 
 — 11932 
 
 353299945209. 
 
 283405952 
 
 — 11924 
 
 — 11916 
 
 35329 6370427 
 
 
 ^o*j^j»\JOt yj^jirl 
 
 1386048 
 
 — 1191597 
 
 • 
 
 1059899835627 
 
 — 1191594 
 
 
 
 326148.164373 
 317967 
 
 8181 
 7066 
 
 1115 
 1060 
 
 Now, dividing x^ — 15x= + 63x — 50 by x—1 .02803923 
 we find, for a quotient, the following : 
 
404 XIMEKICAL SOLUTION OF HIGHER EQUATIONS. 
 
 >r'^— 13.97 196077X+48.6362762. 
 Hence, we have this quadratic equation, 
 
 x3— 13.97196077a: = — 48.6362762. 
 
 This solved by the usual rule for quadratics, gives the 
 following values : 
 
 x = 6.576535; a: = 7.395426. 
 Therefore, the three roots of x^ — 15a;"--|-63a: = 505 are 
 1.028039 ; 6.576535 : 7.395426. 
 
 (315.) From the work of the last example, we see that 
 we need only seek one of the roots of a cubic equation by 
 the foregoing rule, as the other two may then be found by 
 the solution of a quadratic. When all the three roots are 
 real, it will frequently be as simple to find them by the 
 foregoing general method. But when two of the roots are 
 imaginary, we must proceed agreeably to the last Art. 
 
 15. Find the three roots of the equation t'* — 15x= — 21. 
 
 Applying the principle §f Sturm's Theorem, we find^ 
 
 X =x^—\5x-\-2\, 
 Zi = a:^ — 5, 
 X, = lOx — 21, 
 ^3 = 59. 
 
 For X = — 00, we find j h = ^ variations. 
 
 " x=H-oo, " -4- -I- -j--^=0 « 
 
 Therefore, this equation has three real roots. 
 
 ""or X = 
 
 — 5, we fir 
 
 id 1 p =3 va 
 
 riat 
 
 " a:=. 
 
 -4, " 
 
 + + - + =2 
 
 u 
 
 « x = 
 
 1, " 
 
 + +=2 
 
 (( 
 
 " x = 
 
 2, " 
 
 + =1 
 
 u 
 
 « x = 
 
 3, " 
 
 -f--f4-+=o 
 
 a 
 
NUMERICAL SOLITIOX OF HIGHER EQUATIONS, 40J 
 
 Hence, there is one negative root between — 4 and — 5 ; 
 one positive root between 1 and 2 j and one positive root 
 between 2 and 3. 
 
 For the positive root between 1 and 2, we have the fol- 
 lowing 
 
 
 OPERATION. 
 
 ±0 
 
 — 15 
 
 — 21(1.769149632 &c.=x 
 
 1 
 
 — 14 
 
 — 14 
 
 2 
 
 — 12 
 
 
 3 
 
 — 941 
 
 -7 
 
 37 
 
 — 633 
 
 — 6587 
 
 44 
 
 — 60204 
 
 
 51 
 
 — 57072 
 
 — 413 
 
 516 
 
 — 5659599 
 
 — 361224 
 
 522 
 
 — 5611917 
 
 
 
 528 
 
 — 561138629 
 
 — 51776 
 
 5289 
 
 — 561085557 
 
 — 50936391 
 
 5298 
 
 — 5610.6432764 
 
 
 5307 
 
 — 839609 
 
 53071 
 
 
 — 561138629 
 
 53072 
 
 
 
 53073 
 
 
 — 278470371 
 
 530734 
 
 
 — 224425731056 
 
 
 — 54044.639944 
 
 
 
 - 50495 
 
 
 
 — 3549 
 
 
 
 — 3366 
 
 
 
 — 183 
 
 
 
 -168 
 
 
 — 15 
 
 
 
 — 11 
 
406 NUMERICAL SOLUTION OF HIGHER EQUATIONS 
 
 For the negative root, we have the following 
 
 
 OPERATION. 
 
 lO 
 
 -15 
 
 -21(— 4.4416216£ 
 
 -4 
 
 1 
 
 — 4 
 
 -8 
 
 33 
 
 
 12 
 
 3796 
 
 — 17 
 
 124 
 
 4308 
 
 — 15184 
 
 128 
 
 436096 
 
 
 132 
 
 441408 
 
 -1816 
 
 1324 
 
 44154121 
 
 — 1744384 
 
 1328 
 1332 
 
 44167443 
 4417543716 
 
 
 — 71616 
 
 13321 
 
 4418343168 
 
 — 44154121 
 
 13322 
 13323 
 
 4418.36981764 
 
 
 — 27461879 
 
 133236 
 133242 
 
 
 — 26505262296 
 
 
 
 133248 
 
 
 — 956616704 
 
 1332482 
 
 
 — 883673963528 
 
 
 — 7294.2740472 
 
 
 
 — 4418 
 
 
 
 — 2876 
 
 
 
 — 2651 
 
 
 — 225 
 
 
 
 — 221 
 
 
 
 — 4 
 
 
 
 — 4 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 40' 
 
 For the positive root between 2 and 3, we have this 
 
 
 OPERATION. 
 
 ±0 
 
 —15 
 
 -21(2.67247201 
 
 2 
 4 
 6 
 
 -11 
 
 — 3 
 
 96 
 
 — 22 
 
 1 
 
 66 
 
 528 
 
 576 
 
 72 
 
 58309 
 
 
 
 78 
 
 63867 
 
 424 
 
 787 
 
 6402724 
 
 408163 
 
 794 
 
 6418752 
 
 
 801 
 
 642195856 
 
 15837 
 
 8012 
 
 642516528 
 
 12805448 
 
 8014 
 
 6425.7264889 
 
 
 8016 
 
 3041552 
 
 80164 
 
 
 2568783424 
 
 80168 
 
 
 
 80172 
 
 
 462768576 
 
 801727 
 
 
 449800854223 
 
 
 12967.721777 
 
 
 
 12851 
 
 116 
 
 Hence, the three roots of x^ — 15a; = — 21, are 
 x = — 4.441621651 ; 1.769149632; 2.672472018. 
 16. Find the three roots of the equation 
 
 lOOOOr^ — 4519x2-f- 665a: = 32. 
 Applying Sturm's Theorem to tiiis equation, we find 
 X = 10000x3 — 4519x2+665x— 32, 
 Xi = BOOOOx-'— 9038x-f665, 
 X2=942722x— 125135, 
 X3 = 5425404570000. 
 
408 NUMERICAL SOLUTION .OF HIGHER EQUATIONS. 
 
 When a: ^ 0, w'^ find ] h = ^ variations, 
 
 « X=l, " +4- + +::=.0 " 
 
 Hence, the equation has three positive roots, each less 
 than 1. 
 
 When X =0,1, we find 1 h = ^ variations, 
 
 " a:=0.2, " + + -f+ = " 
 
 Which shows that the first figure of each root is 0.1. 
 
 Again, when x = 0.11, we find 1 [-= 3 variations. 
 
 " a: = 0.12, " -i--\ \-=2 
 
 + + =2 
 
 + + = 1 
 
 a: = 0.19, " — + + -}- = l 
 From this, we discover that the first two figures of the 
 least root are 0.11 ; the first two figures of the next root are 
 0.13 ; the first two figures of the other root are 0.19. 
 For the first root we have the following : 
 
 " a; = 0.13, 
 " x = 0.14. 
 
 10000 -4519 
 
 665 
 
 32 (0.119503816&C. 
 
 — 3519 
 
 3131 
 
 3131 
 
 — 2519 
 
 612 
 
 
 -1519 
 
 4701 
 
 69 
 
 — 1419 
 
 3382 
 
 4701 
 
 -1319 
 
 23659 
 
 
 — 1219 
 
 14308 
 
 2199 
 
 — 1129 
 
 138360 
 
 212931 
 
 — 1039 
 
 133665 
 
 
 — 949 
 
 1336.369809 
 
 6969 
 
 — 944 
 
 
 6918 
 
 —939 
 
 
 
 — 934 
 
 
 61 
 
 —93397 
 
 
 400910942V 
 
 
 10908.90573 
 
 
 
 10691 
 
 
 
 217 
 
 
 
 134 
 
 
 
 .. 
 
 
 
 83 
 
 
 
 80 
 
 
 
 m^ 
 
 
 
 1. 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 409 
 
 For the second root we have the following 
 10000 
 
 
 OPERATION. 
 
 
 -4519 
 
 665 
 
 32 (0.137139216&C 
 
 — 3519 
 
 3131 
 
 3131 
 
 — 2519 
 
 612 
 
 
 — 1519 
 
 2463 
 
 69 
 
 -1219 
 
 — 294 
 
 7389 
 
 — 919 
 
 — 6783 
 
 
 — 619 
 
 — 10136 
 
 — 489 
 
 — 549 
 
 — 101768 
 
 — 47481 
 
 — 479 
 
 — 102175 
 — 10229671 
 
 
 — 409 
 
 — 1419 
 
 —408 
 
 — 10241833 
 
 - 101768 
 
 — 407 
 
 — 1024.547809 
 
 . 
 
 — 406 
 
 
 — 40132 
 
 — 4057 
 
 
 — 30689013 
 
 — 4054 
 
 
 
 — 4051 
 
 
 — 9442987 
 
 — 4030] 
 
 
 — 9220930281 
 
 
 — 2220.56719 
 
 
 
 — 2049 
 
 
 — 171 
 
 
 
 — 102 
 
 — 61 
 
 48 
 
410 NUMERICAL SOLUTION OF HIGHEB EQUATIONS. 
 
 For the third root we have the following 
 10000 
 
 -4519 
 
 665 
 
 32 (0.195256967 &c 
 
 — 3519 
 
 3131 
 
 3131 
 
 — 2519 
 
 612 
 
 
 — 1519 
 
 549 
 
 69 
 
 -619 
 
 3078 
 
 4941 
 
 281 
 
 36935 
 
 
 1181 
 
 43340 
 
 1959 
 
 1231 
 
 436066 
 
 184675 
 
 1281 
 
 438736 
 
 
 1331 
 
 43940475 
 
 11225 
 
 1333 
 
 44007375 
 
 872132 
 
 1335 
 1337 
 
 440.1540636 
 
 
 25036S 
 
 13375 
 
 
 219702375 
 
 13380 
 13383 
 
 
 
 
 30665625 
 
 133856 
 
 
 26409243816 
 
 4256.381184 
 3962 
 
 294 
 
 Hence, the three roots of the equation 
 
 10000x3 — 45 1 9x2+6650: = 32, 
 are 0.119503816; 0.137139216; 0.195256967. 
 
 17. Find the three roots of the equation x^-{-2x- — 3x=9 
 
 Applying to this equation the Theorem of Sturm, we find 
 
 X =x'-|-2x'— 3a:— 9, 
 Xi= 3x3-|-4x — 3, 
 X2 = 26x-j-75, 
 X3 = — 7047. 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 411 
 
 When X = — oo, we find 1 = 2 variations, 
 
 " x = +oo, " +4-+_^l " 
 " a:= 1, " — -f +— = 2 " 
 
 " X = 2, " 4- + -I- — = 1 " 
 
 Then this equation has but one real root, which lies 
 between 1 and 2, the other roots being imaginary. 
 We find the real root by the following : 
 
 1 
 
 2 
 
 3 
 
 9(1.939465 &c. 
 
 = X. 
 
 3 
 
 
 
 
 
 
 4 
 
 4 
 
 _ 
 
 
 5 
 
 931 
 
 9 
 
 
 59 
 
 1543 
 
 8379 
 
 
 68 
 
 156619 
 
 
 
 
 77 
 
 158947 
 
 621 
 
 
 773 
 
 15964891 
 
 469857 
 
 
 776 
 779 
 
 16035163 
 1603.828996 
 
 
 
 151143 
 
 
 7799 
 
 
 143684019 
 
 
 7808 
 7817 
 
 
 
 
 
 7458981 
 
 
 78174 
 
 
 6415315984 
 
 
 10436.65016 
 9623 
 
 813 
 801 
 
 Dividing r^+2x2 — 3a: — 9 by a:- 
 T^ + 3.939465X -f 4.640455 for the quotient. 
 Therefore, solving the quadratic 
 
 x2-f3.939465x = — 4.640455, 
 we find the following imaginary roots, 
 
 1.96973 + 0.87213 v/^, 
 .96973 — 0.87213 v^—1. 
 
 (-1.9 
 ^-1.9 
 
412 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 18. Find the three roots of the equation x^ — 5x^-\-8x=l. 
 By Sturm's Theorem we have already found, page 381, 
 
 X =x3 — 5a;2+8x— 1, 
 Xt = Zx''—10x-{-S, 
 Xo = 2x — 31, 
 X3 = — 2295. 
 
 When x = — oo , we find 1 := 2 variations, 
 
 + + +- = 1 " 
 — H =2 ' " 
 
 (( 
 
 X=: + CO, 
 
 u 
 
 u 
 
 x = 0, 
 
 C( 
 
 11 
 
 .r = l. 
 
 u 
 
 + + 
 
 = 1 
 
 Hence, this equation has one positive root which lies be- 
 tween and 1, and two imaginary roots. 
 Its real root is found by the following 
 
 1 — 
 
 
 OPERATION. 
 
 
 5 
 
 8 
 
 1 (0.1362934&C 
 
 49 
 
 751 
 
 751 
 
 48 
 
 703 
 
 — 
 
 47 
 
 68899 
 
 249 
 
 467 
 
 67507 
 
 206697 
 
 464 
 
 6723076 
 
 
 461 
 
 6695488 
 
 42303 
 
 4604 
 
 669.456964 
 
 40338456 
 
 4598 
 
 
 
 
 4592 
 
 
 1964544 
 
 45918 
 
 
 1338913928 
 
 6256.30072 
 
 231 
 201 
 
 30 
 27 
 
 By dividing x'' — 5x'+8x— 1 by x — 0.1362934, 
 
NUMERICAL SOLUTION Or IIIGHEU EQUATIONS, 413 
 
 we find the quadratic x^ — 4.8637066a: = — 7.3371089, 
 
 which gives the following imaginary roots : 
 2.43185 + 1. 19298 v^^, 
 2.43185 — 1.19298 v/—l. 
 
 x = - 
 
 19. Find one of the roots of x^ — •2x = 5. 
 
 Ans. a: =2.09455 148&C. 
 
 20. Find one of the roots of 2x^-}-3x = 90. 
 
 ^ Ans. a;=3.41639726&c. 
 
 21. Find one of the roots of x^-\-X'-\-x = 100. 
 
 Ans. a: = 4.26442997&c. 
 
 22. Find one of the roots of x^-|-x=: 500. 
 
 Ans. a; = 7.89500828&c. 
 
 23. Find one of the roots of a:^+10x'+5x = 2600. 
 
 Ans. 11.00679933&C. 
 
 SOLUTION OF EQUATIONS ABOVE THE THIRD DEGREE. 
 
 (316.) It is obvious that the above method which we have 
 employed for cubic equations, will apply equally well to 
 equations of a superior degree. By carefully studying the 
 preceding method, we shall be able to deduce, for equations 
 of the nth degree, this general 
 
 RULE. 
 
 1. Having found the first figure of the root, multiply it 
 into the first coefficient and add the product to the second 
 coefficient, which sum multiply by the same figure and add 
 the product to the third coefficient, and this sum must be 
 multiplied by the same figure and the product added to the 
 fourth coefficient; and so continue to multiply the last re- 
 sult by this same figure and to add the product to the next 
 
414 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 succeeding coefficient^ until the last coefficient is reached^ 
 which last sum must he multiplied by the same figure and 
 the product subtracted from the term constituting the right- 
 hand member of the equation ; the remainder we will call 
 
 the FIRST DIVIDEND. 
 
 Jigain, multiply the first coefficient by the same figure^ 
 and add the product to the number under the second coeffi- 
 cie7it, which sum must be multiplied by the same figure^ 
 and the product added to the term under the third coefficient ; 
 and thus we must continue to multiply and add, until we 
 have obtained the second term under the last coefficient, winch 
 result we shall call the first trial divisor. 
 
 Again, multiply the first coefficient by the same figure of 
 the root, and add the product to the last term under the 
 second coefficient, which result must be multiplied by the 
 same figure, and the product added to the last number under 
 the third coefficient; and thus we must continue to multiply 
 and add until we reach the coefficient next to the last, wheji 
 we must again begin with the first coefficient and multiply 
 and add as before, until we reach the n — 2th coefficient; 
 then, again, commencing with the first coefficient, we must 
 multiply and add until we reach the n — 3rf coefficient; ive 
 must continue this process, until we have thus obtained n 
 terms under the second coefficient, n — 1 terms under the 
 third coefficient, n — 2 terms under the fourth coefficient, 
 n — 3 terms under the fifth coefficient, and so of the rest. 
 
 II. Find the second figure of the root, by dividing the 
 FIRST dividend by the first trial divisor, proceed with 
 this second figure, precisely as was done with the first figure, 
 observing to keep the work so that units shall stand under 
 units, tens under tens, Sfc, S)C. 
 
 EXAMPLES. 
 
 1. Find one of the roots of the equation 
 3x^_|.x3-f 4a:-'-f 5x = 375. 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 415 
 
 1 
 10 
 19 
 28 
 37 
 373 
 376 
 379 
 382 
 3829 
 3838 
 3847 
 3856 
 38569 
 
 4 
 
 34 
 
 91 
 175 
 
 17873 
 
 17249 
 
 18628 
 
 1874287 
 
 1885801 
 
 1897342 
 
 189849907 
 
 OPERATION. 
 5 
 107 
 380 
 397873 
 416122 
 421744861 
 427402264 
 427.971813721 
 
 375(3.13364 &c.=i. 
 
 321 
 
 156035417 
 1283915441 163 
 
 2764.387288^7 
 2567 
 
 2. Find the four roots of the equation 
 X* — 80x3 -|_ l99Sa;2 _ 14937a; = 
 
 197 
 171 
 
 — 5000. 
 
 1 — 
 
 — 80 
 
 — 50 
 
 — 20 
 10 
 40 
 44 
 48 
 52 
 56 
 668 
 576 
 684 
 592 
 5923 
 5926 
 
 1998 
 498 
 -102 
 198 
 374 
 566 
 774 
 81944 
 86552 
 91224 
 9140169 
 9157947 
 9175734 
 917742041 
 917860692 
 
 FIRST OPERATION. 
 — 14937 
 3 
 
 - 5000(34.83228 &c. = x 
 90 
 
 — 1561 - 
 703 - 
 
 -5090 
 6244 
 
 i050968 
 ■2G783.SS507 
 210586234S 
 21076978320S8 
 " 210.9533553472 
 
 1154 
 
 10868416 
 
 671584 
 6235165521 
 
 480674479 
 4215395664176 
 
 591.349125824 
 
 168 
 t. 
 
416 NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 SECOND OPERATION. 
 
 1 — 
 
 —80 
 
 1998 
 
 — 14937 
 
 — 5000(32.06029&c 
 
 -60 
 
 498 
 
 3 
 
 90 
 
 —20 
 
 — 102 
 
 -3057 
 
 
 10 
 
 198 
 
 -2493 
 
 — 5090 
 
 40 
 
 282 
 
 -1753 
 
 — 4986 
 
 42 
 
 370 
 
 — 1725106984 
 
 
 44 
 
 462 
 
 — 169.7040736 
 
 — 104 
 
 46 
 
 4648836 
 
 
 — 10350641904 
 
 48 
 
 4806 
 
 4677708 
 
 
 
 
 — 49.358096 
 
 4812 
 
 
 
 — 34 
 
 — 15 
 
 — 15 
 
 THIRD OPERATION. 
 
 -80 
 
 1998 
 
 _ 14937 
 
 — 5000(1 2.75644&C 
 
 —70 
 
 1298 
 
 — 1967 
 
 — 19570 
 
 — 60 
 —CO 
 
 698 
 198 
 
 5023 
 5267 
 
 
 14570 
 
 —40 
 
 122 
 
 5367 
 
 10534 
 
 — 88 
 
 50 
 
 5339063 
 
 
 -36 
 
 -18 
 
 5296132 
 
 4036 
 
 — 34 
 
 — 3991 
 
 5291946125 
 
 37373441 
 
 -32 
 -313 
 
 — 6133 
 
 — 8226 
 
 5287.687500 
 
 
 2986559 
 
 — 306 
 
 — 837175 
 
 
 26459730626 
 
 — 299 
 
 — 292 
 
 — 861726 
 
 
 
 
 34058.59375 
 
 — 2915 
 
 
 
 31726 
 
 -2910 
 
 
 
 
 2332 
 2115 
 
 217 
 212 
 
NUMERICAL SOLUTION OF HIGHER El^UATlONS 
 
 AV 
 
 FOURTH OPEUAiTON. 
 
 80 
 
 1993 
 
 — 14937 
 
 — 5000 (0.3509SSCC 
 
 797 
 
 197409 
 
 — 14344773 
 
 — '13034319 
 
 794 
 
 195027 
 
 13759C92 
 
 
 791 
 
 192C54 
 
 — 13663561875 
 
 — 69656S1 
 
 788 
 
 19226025 
 
 — 135.G763S500 
 
 — 68317809375 
 
 7875 
 
 19186675 
 
 
 
 7870 
 
 
 — 133.9000625 
 
 122 
 
 -11 
 
 — 10 
 
 Hence, the four roots, true to 5 decimal places, are 
 
 34.83228; 32.06029; 12.75644; 0.35098. 
 
 3. Find the four roots of the equation 
 
 a;i_j_4x ' - 4a- — 1 la;+4 = 0. 
 
 By Sturm's Theorem ^ve have 
 
 X =a;'-|-4.r< — 4j:-— lLi-f-4:=0, 
 X, = 4x3 -j- i2x^ _Sx—U, 
 
 X, 
 
 ==20x 
 
 -4-25X- — 27, 
 
 
 X,- 
 
 = 227x-|-31, 
 
 
 X,-. 
 
 = 154' 
 
 798S 
 
 
 
 riicn x = - 
 
 — 5, we fin 
 
 d + - H 1-=.4 V 
 
 ariations 
 
 " x = - 
 
 -4, 
 
 a 
 
 -++-+-3 
 
 a 
 
 " x = - 
 
 _ o 
 
 a 
 
 h + -+ = 3 
 
 a 
 
 « x = - 
 
 -h 
 
 ii 
 
 + + -- + -2 
 
 " 
 
 " x = 
 
 0, 
 
 u 
 
 + + +--2 
 
 a 
 
 " x = 
 
 1, 
 
 u 
 
 ++ + = 1 
 
 u 
 
 " x = 
 
 2, 
 
 u 
 
 + 4-4- + +--0 
 
 u 
 
 Hence, there is one negative root between — 5 and - 4 ; 
 one negative root between —2 and — 1 ; one positive root 
 between and 1 ; and one positive root between 1 and 2. 
 
41S NUMERICAL SOLI TION OK HIGHER EQliATIONS. 
 
 These roots wiien found are 
 
 ar = — 4.2834, 
 x = — 1.6908, 
 x= 0.3373, 
 x= 1.6369. 
 
 4. Find one. of the roots of the equation 
 
 2x'-j-ox'-}- 6x-^+ 2x' — 3x = 300. 
 
 
 
 OPERATION. 
 
 
 .) 
 
 6 
 
 2 
 
 — 3 
 
 300(2.2233498cc. 
 
 9 
 
 24 
 
 50 
 
 97 
 
 194 
 
 13 
 
 50 
 
 150 
 
 397 
 
 — 
 
 17 
 
 84 
 
 318 
 
 465S432 
 
 106 
 
 21 
 
 120 
 
 344210 
 
 540131^0 
 
 9310804 
 
 95 
 
 1310S 
 
 37I4G4 
 
 54819U13C32 
 
 
 254 
 
 13024 
 
 399700 
 
 5503O3425C0 
 
 12SGI3G 
 
 25S 
 
 14148 
 
 402700816 
 
 5575300G592'i562 
 
 109038027204 
 
 202 
 
 140S0 
 
 405064464 
 
 558.759248434410 
 
 
 20 '• 
 
 1473408 
 
 409632960 
 
 
 18075572736 
 
 270 
 
 147^«24 
 
 409080108854 
 
 
 1072591397779680 
 
 2704 
 2708 
 
 1484248 
 1489060 
 
 409527502010 
 
 
 
 
 1949.05275820214 
 
 2712 
 
 149019filS 
 
 
 
 1676 
 
 2716 
 
 149131254 
 
 
 
 
 
 2720 
 
 
 
 
 273 
 
 27200 
 
 
 
 
 223 
 
 27212 
 
 
 
 
 — 
 
 5. Find one of ihe roots of the equation 
 2x5 — 7i3+10x = 9 
 
NUMERICAL SOLUTION OF HIGHER EQUATIONS. 
 
 419 
 
 
 
 
 OPERATION. 
 
 
 - 
 
 -7 
 
 
 
 10 
 
 9(1.630101025 &c. 
 
 2 _ 
 
 — 5 
 
 — 5 
 
 5 
 
 5 
 
 4 - 
 
 -1 
 
 -6 
 
 - 1 
 
 - 
 
 6 
 
 5 
 
 - 1 
 
 54992 
 
 4 
 
 8 
 
 13 
 
 10832 
 
 217760 
 
 329952 
 
 10 
 
 1972 
 
 27128 
 
 2326581362 
 
 
 
 J12 
 
 2716 
 
 48320 
 
 2479627610 
 
 70048 
 
 124 
 
 3532 
 
 49660454 
 
 248015150553963002 
 
 6979744' 'Se 
 
 136 
 
 4120 
 
 51015416 
 
 24806.7544722052010 
 
 
 148 
 
 4.16S1S 
 
 52384940 
 
 
 25055914 
 
 160 
 
 451654 
 
 52389553963002 
 
 248015150553963002 
 
 1606 
 1612 
 
 45650S 
 4613S0 
 
 ;iO'-fo 1 1 fitin>iQnr»>3 
 
 
 O/oy-J loSl 
 
 
 2543.989446036998 
 
 1618 
 
 4613063002 
 
 
 2481 
 
 1624 
 
 4614126006 
 
 
 
 1630 
 
 
 
 
 62 
 
 163002 
 
 
 
 50 
 
 163004 
 
 
 
 12 
 
 
 
 
 
 12 
 
 6. Find one of the roots of the equation 
 
 x' + 2x* + 3x3 -^ ix' + 5x = 54321. 
 
 2 
 10 
 18 
 26 
 34 
 42 
 424 
 428 
 432 
 436 
 440 
 4401 
 4402 
 
 3 
 
 83 
 227 
 435 
 707 
 72396 
 74108 
 75836 
 77580 
 7762101 
 7766803 
 
 OPERATION. 
 
 4 
 668 
 2 84 
 5964 
 6253584 
 6550016 
 6853360 
 6861122401 
 6868889204 
 
 5 
 
 5349 
 25221 
 277224336 
 303424400 
 3041105122401 
 304.7974011605 
 
 54321 (8.4144 &c. 
 
 42792 
 
 11529 
 1108897344 
 
 44002656 
 3041105122401 
 
 1359.160477599 
 1219 
 
 140 
 122 
 
 18. 
 
420 NUMERICAl- SOLUTION OK HIGHER EC^L ATIONS. 
 
 7. Find a root of the equation 
 
 26a.* + 28la.'3 - 576a:2 ^ 29Sx :^ 25. 
 
 Ans. a: =0.77933994 &c. 
 
 8. Find a root of the equation 
 
 x^ — 5x'3 -f-5a:^ 1. 
 
 Ans. X = 0.20905692 &c. 
 
 9. Find a root of the equation 
 
 x^-}.2x^' + 3x* + 4x' + 5a:2+6.r= 654321. 
 
 Ans. a: = 8.95697957 &c. 
 
 10. Find a root of the equation 
 
 2x'^ — 6x« — 5x^ + 20r» -j- 2x^ — 18a,- + 4x = 4. 
 Ans. a; = 2.62599736 &c. 
 
r 
 
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 RCIfi. ^C30 
 
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