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Lea disgrammes suivanta iiiuatrent la m^thoda. 1 2 3 1 2 3 4 5 6 / y # A,< • • .• ,V •- ^:^iV*«*f *-.v •vV « * « « • f ^:4^ f \ « « ^ P---!>7 i: *^»^*.iiii#*^ 'A ^ • t •*^, •'V-, I \ / i^f t > \vv / Ni ' i^- -, • » '* # r r "" 1 m- I i V \ / ' ' / / Hi ' 1 m- ♦■ L0VELL*8 SERIES OP SCHOOL BOQKS. ELEMENTS Of AL GE BR A; DKSIONID FOH TBI VtM OF CAMDIAK (fRAHAR AND COUOK SCHOOU BY JOHN HERBERT SANOSTEB, H.A.,. XATHSXATIOAI. XASTXB AHD LXOTUBKB IIT OHKMISTRT AlTD HATUBAXi FHI1:A>80PHT in THB normal BOHOOL fob UFPBB CANADA. PRINTED AND PUBLISHED B7 JOHN LO.VELL, AND SOLD BY B. MILLEB. ADAH MILLER, 62 KINO STRiBBT BAST. 1864, ' Yntered, M«ordiiig to tlM Act of the Proriccial Parliunent, in tbji ff^sf oi» ^loiisapd iifht haodred «nd liitj-foar, bj John LotbIiL, ia the Office of the Registrar of the Provinoe of Oanada. Erkata. Fago 58 last line fbr X "f 3x - 4 readx' + 3«-4. •• 106 Quest. 18 "- " =0. 1 " 490 «« 49c» 7"x+a " x+1 13«+2«» " ~2x* 36"(4r» 1 " * oc y 21" 8 24" 61 « • 123 II II 158 II II 165 II a no II II 187 II II 361 II II 181 II II 386 II It It H .- a (4«*+3) xoc Vy pikU with " paid with 100 pieces in itlnth " fourth A PREFACE. The following Treatise is respectfully sabmitted by the author to the teachers of Canada, in the confident belief that it will materially lighten the labor of the instructor, and, at the same time, facilitate the pupil's progress and hia thorough comprehension of the principles of the science of algebra. It is the earnest hope of the author that it may meet with the same flattering reception, and very general introduction into the schools of the country, that his fellow-teachers have so kindly accorded to his previous productions. The order of succession of the different chapters depends of course mainly on their importance and difficulty, and that here adopted is the one that appears preferable to the author} but, as every chapter is nearly independent of the others, the teacher can easily modify the arrangement to suit himself. The aim of the work is to embitce all that can be pro- fitably discussed in the time usually allotted to a common and grammar school course ; and, indeed, this volume will be found to contain at least as much of the subject, as is required to be read for the ordinftry degree of B. A. in the British and Canadian Universities. Ghaptcnrs on con- tinued fractions, logarithmic series, probabilities, and IT nUVAOB. the general theory of equations were prepared, bat, in accordanoe with the advice of some of the leading educators of the province, they were omitted as unsuited to the design of the work, and to the requirements of common or gram- mar schools. The author has approached the subject with the con- viction, founded on many years' experience as a teacher of mathematics, that the science of algebra tries, beyond all others, the powers and patience of the learner. The pupil is commonly introduced to it while his mind is yet in an undeveloped state ; its language is new to him, and he is unprepared by previous training to comprehend its abstractions. The difficulties which thus beset his path are, of course, for the most part, only to be overcome by his own perseverance, aided by the knowledge and ingen- uity of his instructor, yet it appears to the author that very much also depends upon the 8tyle and thoroughness and adaptation of the text-book employed. Accordingly in the preparation of this yolume no pains have been spared in rendering the statement of principles, and the demonstra- tion of theorems as clear and concise as possible, or in fnlly illustrating each rule by numerous examples carefully worked out and explained, or in selecting and arranging the examples of an exercise so as to begin with the simple, and gradually pass on to the more difficult. The author hopes that while he has insisted upon great thoroughness by numerous and appropriate problems, he has, at the same time, rendered the pupil's advancement easy and certain by the many explanations and illustra- tions introduced. The great majority of the problems and exercises are new, — ^being now published for the first time, but there are FRBVAOB. alio a number already familiar to the teacher. In 8eleet> ing these the author has, he bdieves, in eyery ease rigidly adhered to the rule, adopted by Todhunter, Colenso, and others, of not inserting a problem unless it had already appeared in at least two British authors — in which case it is to be regarded as common property. Becognizing the fact that very many of the pupils of our common and grammar schools study with the view of com- pleting their education at some one of our excellent Cana- dian universities, the author has, at the end of the book, introduced a collection of problems and theorems, embracing among others all or nearly all of the pass and honor work in algebra which has been given on the examination papers of the university of Toronto during the last eight or ten years. These will serve to shew the pupil the style of questions he is expected to answer at our universities, and will, at the same time, in a measure prepare him for his examinations. As no teacher would think of introducing his pupils to arithmetic without, to some extent at least, first drilling them in notation and numeration, so no inteUigent teacher will neglect to drill his pupils in algehraic notation and numeration before introducing them to the ordinary rules. The teacher is respectfully referred to exercises ii, iii, and iv, and is recommended to extend and continue these until his pupil is thoroughly and practically acquainted with the definitions. ' Well knowing the great inconvenience to both teacher and pupils of inaccuracies and mistakes in a work on algebra, the author has subjected this treatise to a searching revision ; and he believes that the few corrections marked on the back of the title page are the only errors in the VI FRITAOB. kMer-prsM of the eiereiMt and uiswen of the work The teacher is reipeetftiUy reoommended to canae hia pnpili to make the six or eight trifling alterations there indicated in the body of the work with pen and ink. A key, containing fall solutions to all the more difficult problems, is in press and will be issued almost immediately Toronto, January, 1864. CONTENTS. Paoi DeflnHions 10 Axioms 16 Bxercises on Definitions, kc 1 Y Addition 22 Subtraction 26 Use of Brackets 2f IfDltiplioation 38 Division St Division and Multiplication by detached coeAcients 42 Synthetic Division 44 Theorems in Division and Multiplication 46 Factoring 61 Greatest Common Measure 67 Least Common Multiple 64 Fractions 66 Reduction of Fractions 68 Addition and Subtraction of Fractions 72 Multiplication of Fractions 78 Division of Fractions, 76 Reduction oC Complex Fractions 76 Viscellaneous Theorems in Fractions 78 Simple Equations. . . 81 Problems in Simple Equations 88 l^multaneous Equations of two unknown quantities 100 Elimination by Addition or Subtraction lo4 SUminatioQ by SabstUution • 108 viii OONTBNTS. Pau Elimination by Oompariion 104 Simuitaneoai Equation! of more than two unknown! 106 Problems producing Simultaaeou! Equation! 109 Involution • 114 Inrolution of Monomial! 116 Inrolution of Binomial! 116 Involution of Polynomial! 117 Evolution of Monomial! 119 Evolution of Polynomial! 120 Extraction of root! in general 129 Theory of Indice! 130 Surds 137 Reduction of Surds. 189 Addition and Subtraction of Surd! 140 Multiplication and Division of Surd! 141 Ralioaalization of Surds 141 Theorems in Surds 144 Square root of Binomial Surd! 147 Imaginary Quantitie! 148 Quadratic Equations 160 Theory of Quadratic Equation! 161 Equation! which may be !olved like Quadratic! 166 Simultaneou! Equation! of the Second degree 171 Problems producing Quadratic Equations 1 78 Ratio 186 Proportion 188 Variation , 192 Arithmetical Progression 198 Qeometrical Progression 204 Harmonical Progression 209 Permutations 213 Oombinations 217 Binomial Theorem. 221 Notation and Properties of Number! 236 Inequalities 240 Vanishing Fractions 243 Indeterminate Equations 244 Miscellaneou! Exercises , 262 Answers to Exercises 270 ALOEBRA SECTION I. DEFINITIONS AND AXIOMS. 1. Algebra is Arithmetio generalized; or, in oUier wordB, it is a kind of Arithmetic in which the nimi^ert (xr qyantUiet under consideration are yeprutintei iy htiert^ and the operatuyM to be performed on these indicated hjf tifffis, 2. The symbols employed in Algebra are of five kinds Tis. : — 1st. Symbols of Quantity, 2nd. Symbols of Qp^rafion. 3rd. Symhola of Relation, 4th. Symbols of Aggregation, 6th. Symbols of Deduction, STMBOLS OF QUANTITY. 3* The symbols of quantity are the Arabic numorals and the letters of the alphabet. B ^ 10 DJfiJflMXTiONS. [BmoT. I. 4. Algttbraic quantities are of two kinds, viz. : — let. Known or determined quantities^ or those which may be assumed to be of any value whatever. 2nd. Unknown or widetenMned guantitiesy or those whose value oan be determined only by actually performing the operations involved in the solution of the problem, &c. 6. The first letters of the alphabet, a^bj c^ d^ &o., are used to represent known quantities, and the last letters of the alphabet, as, ^, «, to, v, &c., are employed to represent unknown quantities. 6. The symbol is called zero, and indicates the ab- sence of quantity, or it represents a quantity infinitely tmaUj i.e. less than ^ny assignable quantity. 7. The symbol oc is called infinity j and denotes a qawa- \M'^ infinitely great, i.e. greater than any assignable quantity. NoTX.--The qrmbol « is also employed to indicate that one quantity Taries ai another. [See the Motion on Variation.] SYMBOLS OF OPERATION. 8. The symbols of operation are + , - , '^j x ^^j***.*". h h h *o., V, J^, V, &c. 8. The sign + is called plus or the sign of addition, and indicates that the quantities between which it is written are to be added together. Thus, 7 + 0, read *l plus 9, means that 7 and 9 are to be added together. a •)• 6, read a plus 6, denotes that a and ( are to be added together^ AbTC. i-12.] PEFINITIONS. 11 10. The sign - is called mintu or the sign of tubtrac' tiarty and indicates the subtraction of the quantity following it from the quantity preceding it. , Thus, 11-6, read 11 minw 6, means that 6 is to be taken from 11. a-b, read a minus 6, implies that the quantity a has to be decreased by the quantity b. 11. The multiplication of one algebraic quantity by another may be indicated — 1st. By writing the sign x between them. 2nd. By writing a dot . between them. 3rd. By writing them in juxtaposition. Thus, axb and a . 6 and ab each indicate the multiplication of the quantity a by the quantity b, and are read a multiplied into b, or simply a into b. The last is the method commonly em- ployed to indicate multiplication in algebra. Arithmetical multiplication is expressed only by the sij^n X, the other methods being obviously inapplicable to numbers. Note.— Quantities connected by the fdgn + or x may be read in any order. Thus 6 + 8 is the same in valae as 8 + 6, for each is eqaal to 9 ; so 6 X 6 is the same in value as 6 x 6, for each is equal to 80. 12. There are three modes of representing the division of one quantity by another, namely, by'writing between them the common arithmetical- sign of division -f- or by writing between them either the sign : or the sign - Thus, o-r 6, and a: 6, and -J each represent the division of the quantity a by the quantity b. The last method, i.e. writing the quantities in a fractional form is that usually made use of in algebra. Nora.— Quantities oonneoted by the sign — or -r- must be read Just as they are written. Thus 8— 8 is very different in value from 8—8; so 12-f4 is quite distinct from 4— 12. 12 DlVINinONS. [SXCT. I. 18. The symbol « written between two quantities indi- cates that the less is to be subtracted from the greater. Thns, 7-3 or 3-7, read the difference between 3 and 7, denotes that 3 is to be taken from 7. So a-& or fr-a indi- cates that a is to be taken from 6 or 6 from a, according as a is less or greater than b. NOTX.— The symbol ~ is employed only when it is uot known whioh of the two quantities ia the greater. 14. An exponent is a small figure or letter placed to the right of a quantity to show how often it is taken as a/actor. Thus, a^ - a4uiy the 3 indicating that a is to be taken three times as factor. m' = mmmmmmmf the 7 showing that m is to be taken seven times as factor. (a +6)* = (0+6) (a + d) (a + 6), &c., to n terms, the n denot- ing that the quantity (a + 6) is to be taken as factor as manj times as there are units in n. NoTB.— When the exponent is unity, it is not commonly expressed. 16. The extraction of a root is indicated either by writ- ing It with a fractional index or by placing it under the TQ lical »ign *j. Thus, V7 or 7^ denotes the iquare root of 7. J^a or a^ denotes the cube root of a. ^a or a" denotes the n'* root of a, Ac. 16. The number 3, or 4, or 5, &c., placed in the radical sign or as denominator in the fractional exponent, is called the index of the root. The index 2 is never used in con- nection with the radical sign ; thus, ^Ja is the same as ^a. 17. When a fractional exponent is employed the nume- rator denotes the power and the denominator the root to be taken. Axn. 18-24.] DEFINITIONS. 13 Thus, a^ denotes the 4^ power of the 1^ root of a or the 7^ root of the 4*>> power of a. X ■ indicates the n*>^ root of the m<* power of x, or the m*** power of the n*>> root of X. 8TMB0LS OP RELATION. A^ > —< — ' 18. The symbols of relation are : , = , : : , > , and < . 19. The symbol : denotes ratio. Thus, a :b denotes the ratio of a to b. 20. The symbol = is the sign of equality. Thus, 7-i- 4 = 6 + 6 denotes that the sum of 1 and 4 is equal to the sum of 6 and 6. a = 6 denotes that a is equal in value to 6. 21. The symbol :: is also a sign of equality, butisus^jd only to denote the equality of ratios. Thus 9 : 27 :: 5 : 15 denotes that the ratio of 9 to 27 is equal to that of 5 to 16. a\h:: c: d denotes that the ratio of a to 6 is equal to that of c to greater tharij and the symbol < leu tharij are signs of inequality. Thus 7 > 6 denotes that 7 is greater than 6. a > 6 denotes that a is greater than h, 6 < 7 denotes that 5 is less than 7. a < 6 denotes that a is less than 6. NoTB.— The opening of the angle is alwayB toward! the greater qnmtity. 8TMB0LS OF AOGBIOATION. 28. The symbols of aggregation are — , | > ( )i { j , and []• 24. The symbol — is called a vtncuZum, and Indioates that the quantities oyer which it is placed are to be regarded as constituting but one quantity. 14 DEFINinONS. [SaoT. I. Thus, a + 6 - e X (f means that the quantity formed by the ■nbtraotion of c from the sum of a and b is to be multiplied by d, ym+x+y denotes that the square root of the sum of m, Xf and y is to be taken. 26. The symbol | is called a bar, and indicates that the quantities in the column directly preceding it are to be considered as forming but one quantity. + a Thus, + h -c denotes that the quantity formed by the subtrac- tion of c from the sum of a and 6 is to be squared. 26. T\lq parmtheseB ( ), braces \ j, and brackets [ ], denote that the quantities contaii^ed within them are to be regarded as constituting one quantity. Thus (a + b)x denotes that the sum of a and b is to be multi- plied by X. {a - (6 ■{- c) }> indicates that the sum of b and c is to be taken from a and the remainder cubed. [a- [m - (b+c)x]]y denotes that (b + c)x is to be taken from m and the remainder subtracted from a, and that this final remainder is to be multiplied by y. SYMBOLS OP DEDUCTION. 27. The symbols of deduction are .'. and *.* 28. The symbol .*. is equivalent to there/ore, wJtence, ihencCy consequently J from which we infer , &c. Thus, a-b and c = b .'.a = c. 20. The symbol '.' signifies since or becaiise. Thus, o = c •.* a = 6 and c = b. >j,»^mMmim)msi- -A»**i AbM.96-«I.] DIUriMiTlONS. 16 • 30. The parts of an algebraic ezpressioii separated from each other by the sign of addition or subtraction^ expressed or nnderstood, are called terms. Thus, a is an algebraic ezpreBsion of one term and is called a monomial. a 4- 6 is an algebraic expression of two terms, and is called a hinomial. a+b-c is an algebraic expression of turee terms, and is called a /rmomioZ. 2a 4- 36 -40 4- a; -y is an algebraic expression of fiye terms, and is called a muUimmial or polynomial. 31. The parts of an algebraic expression connected by the sign of multiplication, expressed or understood, are called /aqtors. Thus, the factors of the expression ab are a and 6. The factors of the expression a*bc* are a, a, 6, c, c, and c. The factors of the expression (x-y)*(a>my)' ure (x-y), (x- y), (a -my), (o -my), and (a -my). I 32. The terms of an algebraic expression which are preceded by the sign + are called additive or positive terms ; those preceded by the sign - are called suibtractive or negative terms. Thus, in the expression 7a-3<;-4(2 + 5m + 7x4-8y-mx-a6, the terms 7a, 6m, 7x, and 8y are additire or positiye, and the terms 3c, 4<2, mx, and ab are subtractire or negatiye. |70TB.— When no sign ii expressed before a quantity it is understood to be addittre. Thus, in the above expression, la is written for + 7a. 33. A coefficient is a number or letter written to the lef^ of a quantity to show how often it is to be taken u CKMiSfta* Thas, 7a indicates that the sum of seven a's is to be taken in an additiTe sense. -fo denotes that the sum of five -x*s is to be taken in an additlTt tense. Here 7 is called the coefiBcient of a, 5 the coefficient of «, fte. ( i Sfp w wi I 11 -I f I ' ■ ii m mo ' *' * I ■ n l « i ft<(lpi.a.««i». . ' - J I 16 AXI<»I6. f Sw. I. y 84. Like algebraic quwiHHe* are those that oonsibt of the same letters affected hy the same exponents. Thus, - 3a, - 2a, 4a, - 5a are like quantities. a*hCf ^a*bCt - 3a'6c are like quantities. 6 (a«-6+c»), 7(a*-6 + c«) and tV (a*-6 + c») are like quan- tities. But a'frc, and ab^c are unlike quantities, because the same letter is not affected by the exponent 2. So also a'b^c*, a>6'c*, and a*b^c^ are unlike quantities. 36. Homogeneous .terms are those in which the sum of the exponents of the literal factors in each are equal. Thus 2a*y and 7a*y> are homogeneous, and the sum of the exponents of the literal factors in each being 6, they are called homogeneous terms of ^ve dimenrions. 3ax^y^f4M*x*y^f da^y, 7a£y', and y''^ are homogeneous, the sum of the exponents of the literal factors in each term being 7, and they are called homogeneous terms of stven dinumioru. V 86. The reciprocal of a quantity is unity divided by that quantity. Thus, the reciprocal of 3 is i, of a is i-, of ;^ is f, of | is ^, &c. AXIOMS. 87. An axiom is a theorem which cannot he reduced to a simpler theorem. The following are the principal axioms made use/eiTin algebra: — .1 ' ■» I. I%£ whole i$ equal to the turn of aUUi parti. II. If equal quantities or the same quantity be added tb equal quantities^ the sums will be equal. ' III. If equal quantities or the same quantity be subtracted from equal quantitieSf the remainders wiU be tfftffll... ly. If equals be multi^ied by equals or by the MHii«';ll#]^»it>- duets will be equal. ,i,m4M* i-fcJUu 8i-«7.] EXBROISBS. IT y. JftquaU he divided by et/uals or by the Mme^ the quotientt wiU be equal. YI. If the same qtumHty be both added to and subtracted from another^ the latter unll not be altered in value. YII. If equals or the «ame he added to or subtracted from unequal quantitieSf the sums or remainders will be unequal. YIII. If unequals he mnltiplied or divided by equals or by the samet the products or the quotients toill be ur*equal. IX. Equimultiples of the sam£ quantities or of equal quantities are equal to one another. X. Equal powers or equal roots of the same or of equal quan- tities are equal to one another. XL Things which are equal to the same thing are equal to one another. ExiROISB I. 1. Whatisals^bra? (1) 2. Classify algebraic symbols. (2) 3. What are the symbols of quantity ? (3) 4. What are the symbols of operation 7 (8) 6. Write down the symbols of relation. (18) 6. Express the symbols of aggregation. (23) 7. What are the symbols of deduction ? (2*7) 8. What letters are employed to denote known quantities 7 Unknown quantities ? (5) 9. What is the meaning of the symbol 7 Of the sym- bol «? (6 and 7) 10. What is an exponent ? (14) 11. What is a coefficient ? (33) 12. What are the terms of an algebraic expression 7 (30) 13. What are the factors of an algebraic expression 7 (31) 14. What is a monomial 7 A binomial 7 A multinomial 7 (30) 16. What are like quantities 7 (34) 16. What are homogeneous terms 7 (35) 17. What are additire terms 7 (32) 18. What are subtractive terms 7 (32) wt matmitttmtiitt-^vtmntmi fttftxu^. 18 IZBROISBS ON [Saop.L 19. What ate positire and negatire terms ? (33) 20. When no sign ii expressed before a term how is it re- garded? (32) 21. How many ways have we of indicating the extraction of a root 7 (16) 22. What is the index of the root ? (16) 23. What does the denominator of a fractional index denote ? What the numerator 7 (17) 24. How are quantities connected by the sign + or X to be read 7 How those connected by the sign - or -f- 7 (11 & 12, Notes) 26. What are axioms 7 (37) 26. Oive the principal axioms employed in algebra. (37) BXIBCIBK II. Read the following expressions and explain what each indi- cates : — If ^^ 1. a, 6fl, 9c», 4o', »% Ka+6), 6a:(y + « - c), - 3 m Ax 2. 3a + 4-7c, («-y-«)», abcj^, Vab (m + xy*) xz ; c* + 2fl6-«* 3. (m + X) - (« + y), a-,a«-6* = (a+6) ('^•'^)^J^^:p^- 4. 7 + o>o-3, a*6and6>c.*.cy', a'm'y (a + 6)', a^x* (a -m)> into their simple factors. •54 BXBROISBB \>N (8B0T.1. 22. Express the dirisioa of the sum of mx* and y' by the square of the sum of a and 6. 23. What is the coefficient and what the exponents of a and x in the expression ax T 88. To find the numerioal value of an algebraic expres- sion, when the value of each letter entering into it is given : — B,viM.~^Sub$tiiut9 for coefc Uttir it» numerical voJim, and per- form upon the retuUing numbere the <^eratione indicated by the eigne connecting them. Thus, in the following exercise, wherever a occurs in an expression, we write its assumed value, 1 ; for 6 we write 2 ; for c we write 3 ; for d we write 4 ; and for m we write : then we multiply, divide, add or subtract these quantities as directed by the connecting signs. For example, taking a = 1, 6 = 2,c = 3i and m = 7, we thus find the value of the expression :— • , _. 6c + o Vi»(3a - 4c + 26») - m = V7(3x 1-4x3 + 2x2* - 2x3 + 1 = V'?(8- 12 + 16) - 6 + 1 7 ' ^' 7 88. We are said to show that one algebraic quantity is numerically equal to another, When by eubetHuHng thevalueefor the individual letteri in each toe thow that the numerical value ofthefiret expreeeUm w the same €u that of the other. For example, if a = 4, 6 = 3, d = 7, and/ = tfibdf+ «* - d = 2d - (a + 26) + 1 Here we at once throw out the quantity a'di//*, because /being = p, the whole quantity into which it enters as a factor must - 0, and, therefore, as an addend, it disappears ; then substituting tiieir values for the others, 4 X 3 - 7 = 2 X 7 - (4 + 2 X 3) + 1 12-7= 14- (4 + 6) + 1 12-7 = 16-10 5 = 6 ABTik 88,89.] DBFINinONS, BTC. 21 Ezisoiu lY. If a- 1, 6 = 2, e = 3, d^A^ and m=: 0, find the ralue of :— 1. a»-l. 2. c»-3c. 3. ab+cd. 4. o«6*-(c-o). 5. V6+7+X 6. a'««*d» 7. 6 ( o«c« ). 8. (6»d«-m«)». 9. (o + 6)(<«-m)». 10. 4io-(d-c)j^ 11. (pU*d*)K 12. (d«-6c)«(c»-6«i)«. a+1 6+1 c+1 13. Show that ^— j- = o, -^—i =*» 'xTT "" ^f **• 14. Show that 14a- (3b + c) < d«-6 (6+c). 16. Show that (o«6 - c«rf + a6c)m = a*b^d*m. 16. Show that Va6«c»- 4 (6+ dfc > j (6 + c) (d*+e*) j*' a6Se'-6i 17. Show that ^:i^:p7:|:^ = 6 (6 + c)+a6«c»m. , o«c« + 2a6cd»»-(d-c)» ,. . ■ ^. 18. Show that ,, .. -t-=1 — = }(fc-((i+c +6+a)| V2(d« + c«) + 6(c+d) * Find the numeral value of the following ezpreuions : — 19. (2-6) (30 + 46 - c) + |a6 + (3d- 2c)}- 4a (2c - 36) ~\abc*-(3c-\-a)]-{-{dbd-(c-\-d)a}b. 20. (c«-a«)(6«-m«)+m{6cd(a-6«)d} + 3a{a + c(d-3a)}. 21. |(o-6)+(c+d)}« + {(c+w)-(6-o)p-{(«i+d) + (26-c)}«. 22. V(o+c)d + ^c*XaVh) + 1 2 (d + 6c)« + ( 7(1 -6«c ) J * - (6cd + a)«. 1(am)i+Z^^r (6d +4c) , o«6«c«-7 d + \ d» (o+c)j* ^^; ^ i a6'c+ (cdw)^ i (6-a) + i« | jd- (i+m)"}" ^abcd-d*. 24. i { a6 (a + 6) I -i { 6c (c + o) I + i I (ca-6) (a«6 + 3) j + i|(d+c)(I+36-2c + d)«| OR g(<* + fr- g)* + 11 j (3o+2c)(2a- 6 + jd)} , j(3c + 6)-Vdj(d+"c+"6«-TO) "^ I (a4-3d) '«-(c »+66)-(c+d) jt (2a6 + cd- 6d )(d+c) a6»»+Vdc*-a 7(d + fl6«) # 22 ADDITION. [8iOT.IL SECTION II. ADDITION, SUBTRACTION, USB OF BRAOKETS, MULTIPLIOATION, AND DIVISION. ADDITION. 40. When the quantities are similar and have the same sign:— RuLi. — *idd the coefficients, annex the literal part, and prefix the proper $ign. (1) (2) (3) 7a -2cd 6(x+y) 3a ^Zcd 2(x + y) 6a - cd 5(x+y) 11a -Red 8(x+y) 8a . cd (x+y) 2a -Bed ll(x+y) (4) -8(cd-a') -4(cd-a«) -3(cd-a«) - (cd-a*) -7(cd-aa) -2(cd-o«) 2a-3nt+ y - 6va + b 3a-6m+6y- S^tT+b 8a-1m+3y - 5^a + b 6a-3m + 2y- ^a + b 3a-2m+ y a-lm 31a ~20o£ 33 (x+y) -.26 (cd-a') 22a-27m+13y-16^a + 6 EzssoisK y. Find the sum of :— 1. 3a, 2a, 9a, 11a, a and I7a. i. - 4a6«, - 7a6*, - Ha6«, - a6«, and - 3a6«. 3. 3(a + 6-c«), 6(a+6-c«), 2(a+6-c«), (a + 6-c«), and 7(a + 6-c«). , , , , 4. 4a (X - y«)*, »« (« - y*r , 3o (x -y«)^, and 11a (x- y«)». 6. 3a-4y+7, 6a-3y+3, 6a-3y+3, 7a-y+2, and6a-2y+8^ 6. 3 (x+ y) + 7a> a6c, 6 (x+ y) + 5a- 3a6c, 2 (x+y) + 11a -7a6c, (x+y) + 2a-o6c, 2 (x + y) + a^6a6c, and 3(x + y) + 2a -3a6c. 7. (a+6)x-^c+d)y-(d+/)«, 6(a+6)x-6(c+d)y-7(d+/)«, 2(a+6)x-3(c+d)y-4(d+/)«, 4(a + 6)x-6(f + d)y- 6(i+/)«, and 3(a+6)x-4(c + d)y-6((i+/)«. 8. a»6»x?+a»6«xS-a«6lx»-a»6Sx«, 3a«6'xl+7a»6«x*-6a«63x» -ea3&lx*, 7a*63xi + 3a>6'xf-6as&ix3-2a36§x% and Vi'ft^xl + a» b*aA -2a«6lx» - 8a»6fx«. ABM. », 41] ADDITION. 28 (1) (2) (3) 4a 6a- 3e 5a6 + 6cy- 3 -7a 2a 4- 4c -8a6-8cy+ll -3a -3a+9c -7a6+4cy- 6 -2a 6a- 5c llab~8cy+ 7 5a 4a+3c 3a6-4cy+ 6 6a -7a-12c -7a6+ cy- 1 Sa 7a-4c -3a6-4cy+14 41. When the quantities are similar, but all have not the same sign : — RuLi. — Jrrange the quantitiea $o that timilar termt thaU bi in the tafM vertietd column. Jdd uparately the poeUive and negative coefficiente ; to the difference of theee ttoo eume prefix the eign of the greater and affix the common literal part. (4) 6(a+x) -3a'xy + 7Va+6 9(a+x)-6a»xy-8(a+6)* - 7(a+x) + 6a"ary - 6(a+6)^ 3(a+x) -3o'xy - 5(a+6)' ll(a+a:)-6a'*y + 3(a+6)" - 13(a+x) + 6a*xy - 8Va+6 8(o+«)-6a»xy-17(a+6)i Ezflahatioh.— In (1) the sum of the positive ooefBcienta 6, 6, 4 s 16, ram of the negative ooef. 2, 8, 7 = 12; then 16 -• 12 = 8, which is positive, beoAuae 16, the greater, is the anm of the positive coelDoient. In (2), left hand column, the sum of pos. ooef 4, 6, 2, 6 =b 17, and of neg. eoef 7, 8 = 10 ; then 10 - 17 =r 7, which is pos. because 17 is pos. In right hand column ram of pos. coef. 8, 9, and 4 =: 16, and of neg. coef. 12, 6, and 8 =b 20; then 20 - 16 == 4, which is neg., because 20, the sum of th« neg., is the greater. EXXBOIBB VI. ' '■' Find the sum of: — 1. a4-6anda-&; 2a+6-c anda-6+4c; 4a-36+cand 76- 8c. 2. 2a6 + 3ay - cd, 6a6 - 2ay + 6cd, ^ - 6ay + 2cd and -3a&-2ay4-7cd. 3. 5a»x!-3(a+6)-7«§y+7, a^x\ - 7(a+6)-8xiy-ll, and - 7a'«l + 3(a+ 6) + 3x§y - 16. 4. a-k-h-^c-d^ a-6-c+(2, a-h-vc-d^ -a-6K+rf, -a + 6-c+d anda-6+c-(/. . 6. 3xy+7a6-3, 6xy+3a6+7,4xy-7a6 + ll,and-7xy+lla6+2. 6. 3+70-66+C, 7a+3-46*2c, 76-3a-7-)-3c, and 6c-264' 6-30, - ^i^-iv* 24 ADPITiaN. '^^* 7. ab~xy+cd-m+Cf 6c-3xy+Am'-cd-3abf 6cd-6m + 5c + )6a6-3xy, 6m+6c-3c(2+2xy-3afr ftnd llxy - 3m ~2e-v3ab~ led. 8. 6m^+3a:y-7, 7a:y+3-8iii^ +ya, ll-yz+lxy-llm^ and -ll«^ + 3«y + 4. 9. 6minl-9a}di + 10mixi, ealrfi - 6inixt - mini, 2a}di - 3flii«i-3i}|}n} and -mini-mixi +a§c{i. 10. V2+'V3+V4-V«+c*» llV2-9V3+7i^«-6V4+Vc. -3^3 + 7V2+V4-.7o^+8Vc, lla^-V2+3^3 + 7V4, and 9c^-4a*"+llV4. 11. 3ajy-7ay+2cx-xi+3^y, 2a:y+llV* - 7ay, 13yi-llca;+ 2oy, 12^y - 7xi + 3cx - ay, llxy+ 3ay + 6cx a nd 4xy — /« - 3^ y. <>^ 12. (ox + fry - cz)i -V to+w -(a?-y), 7Vi ft + n + 3(a; - y)- J^/axiby-^Zt 1(x-y)+8i^ax+by-cz-ll(m+n)if 6^/m+n+11 (ax^by '-cz)k - (x- y),- 12(0 3? + 6y -c?)i - 3 (x - y) + 4 (m +n)i and 7Vm + n - 9)yax+6y-c«+ ll(x-y). 42. When the quantities are unlike : — RuLi. — Connect them together by their proper ngiu. > . - 0) 3a -4c 7d -5i» Sum = 3o-4c+ 7d - 6j» (2) l55'.+3c-6Va+& 2m-4o26 + 3a6a -6xy + 3oJ6i Suni=5o+3c-6Va+fr + 2OT-4o26 + 3o6''-6xy+3ai6i 43. WheMlte quantities are partially similar : — RuLi. — Md the similar quantities by jtrt. 38, 39, and to the partial sum, thus formed, affix the unlike .quantities by their proper signs. . ^ ▲sm.4a-M.] SUBTBAOnON. 26 • (3) 2a-4c-f-& 7a- 3c -I- m -9a+6c+3a6 14 + 1am, 3ax^ + 7ay - lOi-i + Sd'p -iax'y + 8x1+ 7-6y» + 3xy, 2y'» - 6x*y2 + 2y, - 3y« -xV+ 4y», and 6 - 6y». 8. 6(xy+x«-y«)i + 3(o+y)c-7o'y, 8(xy+x«-y«)i- 7(o+y)c + 3Tft, 8^xz+xy - y« - 4am, 7(a + y)c - 1 l^xz-yz+vy^ 5am - 3m ~3(a4-y)c-(x«-y« + xy)i andx*y-m'. SUBTRACTION. 44. Thiobbm.— TAe mbtractionofanypositiife quantity it tqmo' aktU to the addition of the tt,me quantity taken negatively; and the subtraction of any negative quantity i» equivalent to the addiiion of the came quantity taken positively. Dbmomstbation I. a=a+b-b (Ax. vi) ; subtract +6 from Mch« Then (Ax.m) a-(+b)sa-bma+(j'b) << IL a^a+b-b (Ax. vi) ; subtract- 6 from eatb. Then (Ax. ni) a- (-i) s a +6 aa-f (+ 1) 26 SUBTRACTION. tdaoT. tt. 46. To subtract one algebraic quantity from another. — B(7LB.— C%anj:e all the $igtu of the stdttrahend or imagine them to be chantfed, and then proceed a$ in addition. NOTK.— Once the sigiM of the snbtrahend are changed, the question is no longer one in subtraction, but is converted into an equivalent problem in addition. Prom 7a-13xy + 27 Take 5a-llxy + 19 EqoiTalMt Remainder 2a- 2xy+ 8 '(1) From dab+ 3xy~23 Take 6o6- *lxy+11 Rem. 4a6 + 10a:y-40 t To Ia-I3xy+21 I Add-5a+llxy-19 Sum 2a- 2xy+ 8 (2) Prom 3x^- 1xy'+3z^-4 Take dxhf+ ^xy'-'dz^+m Rein. - 6x^ - llajy' + 8«' - 4 - m (3) 2(«-y) + «'*(a-6) From Take -*l(x-y)-a^+l1 Rem. »(x-y)+z3(a-6)+o%-l7 EZBBOISK VIII. 1. Prom 4ay«- Yxy» + Sa**- Ixy + 13m - 1 1 Take 3aV«+4«y'-6aa«-llxy- 7m -11 2. Prom 3a-7c + 4a;y^-7Va-6'' Take-llo+ 7c - m^+eVa-t" 3. From (a +b)^3s'-y + ^am^ -cd Take lam' - 3cd + 4(a+ 6) (^x^-y)i 4. Prom9(xy+y'-«»)H3Vx^^+'7aixi-ll-ym+ 17«V«+* Take 5(xy-a3 4.y2)i + i7a.(5+a)J + 3OTJ_7aM+3(«^y»)i 6. From3+V2-5x+^4-7y+8i-6V«-6 TakeV2-13 + 4i-6V8-5x + 16y+3(a-6)J 6. Prom5a-66-7c+4d-Ile + 7ni-16x+y-72 Take Ad-lzi-Sa-eb + m-Bc+Ox-lly + abcd Jam.4B»4A.] BllAOKITSi 27 USE OF BRACKETS.. 46. Much difficulty is commonly experienced by a beginner in the management of brackets. His attention is therefore particularly directed to the following rules^ remarks and exercise) y RuLK 1,—^If any number of quantities f enclosed withifl brackets^ be preceded by the sign -I-, the brackets may be struck out as of no value. This arises from the fact that when a quantity is added the signs of its terms are not changed. / RuLB 2. — If any number of quantities^ inclosed toithin bracket Sf be preceded by the sign -, the brackets may be removed if all the included sigm be first changed^ i.e. + into - and - into +. The necessity of thus changing the signs is manifest from the following illustration : — a-(b + c) means that we are to subtract the whole qnantity b+c from a. If we subtract b alone the remainder a-b ia too great by c, for we were to subtract the sum of b and c. Hence to obtain the correct remainder we must take c from a-b, but this gives a-b-c. Therefore o-(6 + c)=o-6-c. Again (t-(&-c) means that b is to be decreased hj c, and the remainder taken from a. If now we take b from a, the remainder a -6 is too small hj c, because we have subtracted a quantity too great by c. Hence to make the remainder a-b what it ought to be we must add c, but this gives us a~b+c. There- fore a-(b-c) = a-b + c. Bkmark 1.— The learner must oarefUUy note that in every case in which he meets with [ or { or ( he must look for the counter part ) or | or ] and that the above rules apply only to the signs of the quantities, simple or compound, included within the complete or outer bracket. Bkmark 2.— In removing the brackets from a quantity it is to be care- fblly remembered that the first sign within the bracket, when +, is always understood, and that the rules above given apply to it as well as to the other signs. 28 USB Of BttACkBTS. [SlOT. II. Bx. 1. Simplifya4-(6-c + d) OPIRATIOK. a+(b-c+d) = a+b-c+d Bx. 2. Simplify 3o-(4c-d+3a-m) OPRBATIOK. 3a-(4c-d + 3a-ni) = 3a-4c+rf-3a + «i= -4c + rf+7ii Ex. 3. Simplify 3m - { a + (c - m) } OPERATION. 3»i-|a+(c-»») | = 3m-«-(c-m) - 3ni-a-c + 7/i = 4m-a-- + u> + « in brackets. 1. Taking the terms ttoo together. 2. Taking the terms three together. 3. Taking the terms /our together. 4. Taking the terms six together. 6. Three together, using an inner bracket after the model, 6. Three together, using an inner bracket after the model, 7. Four together, using an inner bracket after the model, f*±(«i«i*)} 8. Four together, using an inner bracket after the model, 9. Four together, using an inner bracket after the model, {•i(*±*)±*} 10. Six together, using an inner bracket after the model, |*±*i*± (•±*±*)} 11. Six together, using an inner bracket after the model, , f(±*±*±*±*)i*±*} 12. Six together, using two inner brackets after the model, {* + (*±*)i*±(»±*)} NoTK.— The asterisk is used merely to denote the position to be ooonpied by the given letters with reference to the brackets, the sign i, read plus or mhwu, implies here that the student is to 'letermine which one of these 0f)pu Is to be employed, AST. 48.] USB OF BRACKETS. 31 48. A number or a letter written directly before or after a bracket, inclosing one or more quantities, implies that each of the included terms is to be multiplied by that number or letter. So the line that separates the numerator and denominator of an algebraic fraction acts as a vinculum in uniting the terms of the numerator into one quantity, and hence when the several terms of the numerator are written separately the denominator must be placed under each. Ex. 1. Remove the bracket from 6 (a - am + by^-c). OPIBATION. 6(a-am + 6y'-c) = 6o-6aj» + 66y'-6c Ex. 2. Remove the bracket from 4{a-6-(cx + dy-6>)o|i» OPIBATION. 4\a-b-(cx+dy-b^)a\ms4m{a''h''(cx+dy-b*)a] s4am'-^m-Am(ex+dy-b')a = 4ani - 46m - 4am (ex + dy - 6*) = 4am - 4bm - 4acmj; - 4admy + 4a6>«}i Ex. 3. Remove the vinculum from 3a-»m~y-m''+a:)y 26*Vc 3g-wi--(c'- OT'+a;)y _ 3a m c'y-m ^ + xy 26Vc' " 26Vc ~ 26Vc " 26Vc 3a m chf mhf xy ~ '2& Vc ~ 26Vc " 26 Vc ^ Ib^c " 2b^^/e NoTB.— In the first step of this operation, when the bracket inclosing the last three terms is struck out, the inoladed signs are not changed, because the Tinoulom written under these .terms still binds them into one, but when in the next step this vinculum is removed, the minu$ sign preceding it has the eflbct of changing the signs of the terms as exhibited in the operation. EXKBCIBI XL* Remove the brackets and vincula from the following expres- sions :— 1. 3(a - 6) ; 4x(a + 6" - x=») ; 3p*x (1 - 6 - c') 2. ni(a-6«+«i/>)+x''«(l-3a-6)-m»x"(3-6-»i'«a?) « See Arts. 52, 58, and 67. 82 USB OF RRAOKITS. [8aoT.II. 8. 8{l-(iB-y)aJ+4fl + (a-.6 + y)«}-e«jo-(-8-m)fJ 5. a-&> g-fy-(C'-rf-w) ^ 6. m + xyz 6a - (m — 3p) 7. a{(m-y)x-c(o + 6)j + oy ^^j^ ^ 6. 36 f - (a - c)d + (wi - «)/ j ■ g^, 49. Two or more terms of an algebraic expression that have a common factor are often written in an abWyiated form by the aid of brackets, placing the factor common to the several terms directly before or after the bracket, and the remaining part of each term with its proper sign within. Ex. 1.— >GolI«ct the coeflScients of x^z in* the following ex- ' pression into one quantity : Scu^z - 3x^yz + bd^m^xh/z + Zahc^x*yz -sAfz. OPIRATION. 5ax^t -3x^yz+5dl*m^x'yz + 3ab<^x^z~xhfz = (^5a-3x + 5a^m* + 3ab<^-l)xh/z 60. Any factor of an algebraic term may be regarded as the coefficient of the remaining factor. This is at once evident from the meaning of the expression coefficient = 4^n " together with," and efficiens " making " or " operat- ing," i. e., the part which cooperates with the remainder to make the complete term. Thus, in the term Sabxy^ 3 is the coef. of abxy ; 3a is the oo^f. of bxy ; 3ab is the coef. of xy ; 3a6a: is the coef. of y ; 3aby is the l^oef. of X ; abxy is the coef. of 3 ; 3xy is the coef. of ab. ^c, ^c\ ',0^^ ▲MS.4»^.] MULTIPLICATION. 88 51. When termt involTing brackets are to be added or Bubtraoted it is commonly best first to strike out tbe brackets by Art. 46, and then after performing the addition or subtraction re-bracket the terms, if necessary. Ex. 1. Add 2a(x-y + 3), 6(?«-r*-«j:), and 2(a + ay-4»i) OPIRATION. 2o(x-y + 3) =2ax-2oy + 6o 5(i»-(^-«c) = -6aa: +6/n-6c''' 2(a + oy - 4ot)= 2ay + 2fl-8/n Sum = - Soof + 8a - 3wi - 6c* = - 3(oi: + m) + 8a - Be* Ex. 2. From i>(x-y) + g(y-«) take a(a;-«)-6(y+«) OPERATION. i?(x-y) + g(y-«)=px-py + 9y-9« fl(x - «) - 6 (y + «) = ox - a« - 6y - 6« ?r^ Diff. -j9x-j>y + 5y-9«-ax+as+|||y + 6« = /»x-ax-j?y +gy + 6y- g« +0* -f 6z = (p-o)x-(p-9-6)y-(9-a-6)« EXEROISB XII.* Find the value of: — 1. 3(ani-x+y) + Bo(x + 3y) + 2(a-y)w + 4x(a+l). 2. (a-x + y)m + 3(»i + a)x + 4(a-y) + 3(o+x)y. 3. T(a+6-c)-5(6 + x-6c)-3(TO-a-c). 4. (a-^-m) X - 3 (am + c) xy + 2 (a-cm) y" added to (x + y^) a + (c + a)xy-(6+/)y'''. 5. 3(x + y + «)am + 2c (x + «) + (y-s)rtc subtracted from 3(0-6 + c)y - (2m - c)x - 3»i(ax + ay - a«). 6. 2a(p + xy)c - 3(m - 2xy + y^)c - 3rt(y + c) subtracted from 1 1(0 + 6)?tty - 3xy(tt - 6 I- c). MULTIPLICATION. 52. Thborbm. — Quantities having like signs, give, when multiplied together, a product which is positive ; and quantities having unlike iigttSf give, when multiplied together, a product which is negative. Or, as it is sometimes expressed for the sake of brevity,— In Multiplication, like signs give plds, and unlike signs, minus. * See Arte. 62 and 68. -t*'"'^. z***^. 84 HULTIPLIOATION. [SaoT. U. DmoKiTBATioii I. •!• a X 4-6 means that 4-0 ii to be taken in an additive lenae, i. e., is to M added as often as there are units in b. Bat +. a added once gives + a; +a added two times gives + 2a] -fa added three times gives -t- 3a, and so on. Hence + a added b times gives + a6, that is, + a x 4- 6 = + a6. II. -a X 4-6 means that - a is to be talcen in an additive sense as often as there are units in b, but -a added once gives -a ; -a added two times gives - 2a ; -a added three times gives -3a, and 80 on. Hence - a added b times g||^ - ob that is - a x 4- 6 = -a6. Otherwise, -a4-a = ; multiply each of these equals by 4- ft. Then -ax4-&4-al''30; subtract 4- ab Arom each of these equals. Then -ax4-& = -a&, which was to be proved. III. 4- a X - 6 is equivalent to - 6 x 4- a since quantities connected by the sign of multiplication can be read in any order whatever. But -dx4-a = -a&by last case. Therefore also 4-ax-is.aft. IV. -a4-a=:0 ; multiply each of these equals by -6. Then -ax-&-a6 = 0; add 4- od- to each of these equals. Then -ax'-frs4-a&, which was to be proved. 68. Thiobbm n. — Different powers of the same quanii'i; are multiplied together by adding their exponents, DniojiBTBATiON.— a* X a' = aaaaxaaa = aaaaaaasa^ = a**9^ and the same is true in all other cases, hence generally a" x a^sa"* ". Case I. 54. When multiplicand and multiplier are both simple alge- braic quantities, Bulk. — Multiply together the numerical coefficients and write the Utters in juxtaposition after this product. Thus 3aft X 5cy = 3 X 5 X t^cy = I5abcy ; - 2a& x 3c = - 6abc ; 2xy X - 1 Im = - 22mxy ; - Axy x - *!am = 28amxy, Gase II. 56. When the multiplier is a simple quantity and the multi- plicand is a polynomial, RvLK.'—Multiply each term of the multiplicand by the multiplier ^ and anmect the several partial products by their proper signs. Ann. SB-M.] MULTIPLIOATION. 85 Ex. 1. Haltiplioftnd, 4a«-2ay+8xV Multiplier, 2axy Product, So^x'y - ia*xy^ + 6tu*y* Ex. 2. Hultiplioand, 4am'-3acx-4xy+7 Multiplier, -3ay> Product, - 12a>}nV + Qd'cxy' + 12aary' - 2 lay* 4 Gabi III. 60. When both multiplier and multiplicand are polynomials, B.vht.'— Multiply each term of the multiplicand by each term of the multiplier f and add the teveral partial produete together, Ex. 3. a*-ab-b^ a " b a'^aV>-ai>^ a*~2a*b Tb* Ex. 4. 3ax>-3a*a;4-2a'r' 5a '~2x 16oV- 16oa«+ 10o«r» - 6aa;»+ 6a'«»-4o'«» 21a'ar»- 4o''x» - 6aa:» - 15o»x + lOa'x* Ex. 6. 2a6*-o%»+a36» Sab -2a&'-3a% 6a»6»-3o36» + 3a<6* -4a'6* + 2o36*-2o*6» -.6a8&»+3o*6»~3fl»6* 6a=*68 - 9o»65 - 4a»6* + 3a*6* + 3o*69 - 2o«6« - 3o»6* Ex. 6. a^-2ab + b* a*+2at+6» 2o»6-4o%» + 2a6» o^- 2a6» + 6* a* ^2oV +6* 86 MULTIPLIOAnOV. [SBor. n. Ex. 1, «"-(a-6)x + a6 af 3_ (d _ ifyx» + abx - mx* + (ma - mb)x - oftw x*-(a-6+ni)x' + (ma-mfr + a6)x-a6M Ex. 8. x^-ax^-bx + c X -nk X* - ox' - 6x* + ex * - wix ' + amx' 4- 67HX- cm x*-(o+TO)x'-(6-affi)x''+ (c+6m)x-c//i EXBRCISB XIII. 1. Multiply 0=*- 2tty + y» by o' - 2ay h 2y' ; and a^- 3a'b + 3a6»-6» byo"+2a6 + &«. 2. Multiply 2a*fli' + 12amxy + 9x'V by om -xy ; and Sa'x - 3ax* by 3o»x>-x»-l. 3. Multiply a* - a'm + ahn!^ - ma^ + m* by a + wi ; and 2o' - 2axy + 2^* by d^-ax+ y". 4. Multiply x^-Sx- 7 by x-4 and a^' + a^+o^ by o^'-l. 6. Multiply a' + 2a^b + 3ab^ + 46» by a=* - 2ab - 3b\ •6. Multiply ab-ac + bchj ab + ac-bc. 7. Multiply a* - 20^6 - So'ft^ - 20^3 +6* by a» + 2a6 +6". 8. Multiply 3x2- 2a6x-2rt='6'' ^y x+2a6; and x2+2x-3 by x«-x+l. 9. Multiply X* + 2x3 + Sx" + 2x + 1 by x* - 2x3 + 3x2 - 2x + 1 10. Multiply 3y^ + 2xV+ 3x^ by 2y'* - 3xy v Sx^ ; and a*+6- by a- + 6« . ' 11. Multiply 2a + 3, 3o + 4, Sa^- 2, and «~ 3 together. 12. Multiply ax-vby by ax+cy; and a" -6" ; c' by a"**' -6»-'. 13. Multiply o"-c'+j'' by aj-m^ +x« . 14. Multiply o?^ax-\-x^ by a^ - a^x + ox* - x^ . J5. Multiply 2a- b, 36 + c, 2c- vn, and 3»i-x together. ▲M».67-4».J DIVISION. 87 DIVISION. 67. Division is the process of resolving a given quantity into two factors when one of the latter is given. As in Arithmetic, the given quanlity to be resolved or divided is called the dividend^ the given factor is called the divisor^ and the factor to be obtained, the quotient. Since the divisor x qnotiont = dividend^ the sign of the quo- tient must be such that the sign of its product hy the divisor shall bo the sign of tlio dividend. + a6 Thus,— j- = +a -ab . -ab — -- =+rt •.•-6x + o = -ah: ~~r- --a '.• -ax + b - -ah. ■ -0 ' + + ax 4-6 = + ao; — v- -a •.• -rtx-6 = + fl6; Hencef the ride of signs for division is the same as for multiplica- tion ; that M, like signs in divisor and dividend give plus in the quotient^ unlike signs in divisor and dividend give minus in the quotient. 68. Since a* x o' = o* + ' = a'', it follows that o' -f o* = o^, that is, o^ 7 o* = a'" * = a"* ; or generally, since a^x a" = a"+", it follows that a**** V a''=a' or o' m+ M a'' = a" Hence, one power of any quantity is divided by another power of the same quantity, by subtracting the exponent of the divisor from the exponent of the dividend. Thus, o66« V fl'6' = a*63 j x'a" %- ars' = x*; ab^c^m* -f fcm' = ahc^'m, &c. Case I. 69. When both dividend and divisor are simple quanti-^ ties or monomials, RuLK. — Divide separately the coefficient of the dividend by the coef. of the divisor, and the literal part of the dividend by the literal part of the divisor} write the partial quotients thus obtained injuxta- position, and prefx the proper sign. m DtVISIOlf* (SkM. n. Thug, 14a'6«cH ^ ?o»6c*, 14 * 7 = 2, and a'ft'c' -f o'ftc* = o*6c«, and the quotient is - 2a*bc*f because the signs of divisor and diyidend are uplike. Similarly - 21a%« -f 3a«6 ^ -Ix] - IBxy'z* i- - 2xa? - 9fZf kc. l^oiTB.— If both eoef. and literal part of the diviBor are not oontained as fiMton in the dividend, we can only indicate the division by writing the two quantities in the form of a fraction. For example, lab^cx^-r-Umy can only be expressed thus, limy But when we have thus expressed the quotient we can cancel any factors that are common to both numerator and denominator. Thu8,24a«a?y«-j-16aa?s«=^i^ — ^ = r t-S~~ T^ ExiBOiSB Xiy. Find the quotients of : 1. 15a6c' T 5ac ; 42ax'y' a laxy* ; 24a^xy t %axy ; - 20*'y'«"^ •5- 20a:y'«'. 2. -14a6'cfli* ^ Ta6m»; -14a6«' -s- 146x; -27ma:'y i- -3*='; - 12«''y 4^ - 4r'y. 3. 12o6'c 4- 20aa:y ; -I7a6x» -^ Horn* ; -21a6x3y 4^ -SSftr^z* j db*e/^~Uacfx\ Gasi II. 60. When the divisor is a simple quantity but the diyidend a compound quantity, i. e., a polynomial, Rdlk. — Divide each term, of the polynomial by the dtvisor, as directed in Case I, and connect the several partial quotients thus obtained by their proper signs. ExAMPLi.— Divide Aa^b^c - 3a6c* + Uab^ex - 8a6y» by - iab. 4a%»c-3a5c* + 12a6»cx- Softy" +4a»6»c ,-3aJc* Here r-. = — r-T-» and — r-jTi ***^ -4ab -4a6 -4ab + 12a6'ca; - 8o6y» , 3<^ , ., -— T— , and — - .- = - abc, and + ■— , and - 36'ca:, and + 2y* - 4ao ' - 4«o ' 4 ' 3c" - abe + -— - Bb^cx + 2y'. 4 AitM.00,61.1 Dinsioir. 89 EXIBOISK XY. Find the quotients of :— 1. l2axy^-'2*labc^+12axh/-8acm^Aacx, 2. 2\xf-lla+lA3Phf'-A9f^35axy. 3. -64a Inttor. in fixam^ 6, fio^vtt', ink iiiebViyeiiifende irtkeft ^m the Act ^hat the terms of both diviior and dividend are not arranged aeeording to ile, to* if We had atrioig^ ttei diViit6t(dt)»fii(1^9ti -f at) weaboi^iiave obtained 14-a.lbr the ^notlent. The itud4nt then mmt be eaif the principal letter, or Utter of rtferenee, as it ii called; and that not ^nlj at starting, but throughout the whole process he must take care to rrang« the pArtial dividend* iedbrdiilg to th^ mtat jjilltt tfthat aUbi^ted i th« dlvlBor. ExfROisi XYL Find the quotient! of :•— 1. «»-2»f + »" dlTiatd by x-y; Md aH3a^+3ii*»+fc» di- vided by a + ft. 2. m* + 4m'x + 6>«V + 4mx' + x* divided by wi* -v 2aix + x*. • 3. 9x«-46x» +96x*+lB0x divided by x*-4x-5. 42 DIVISION. [8icT. n. 4. a^ + 6aV> + b*+ 5ab^ dirided by a 4- 6 ; and - l+x^y' dirided by-l+xy. ^ 5. *• + 10« - 33 dirided by 3 + x" - 2*. 6. a" + 2d*«i» - 2o*iii* - 2a'' m + «i" - 2ai»' + 2a*m' divided by fl* + «* - ahn, - om*. 7. 1 divided by 1 + a ; a divided by 1 - a ; 1 - m divided by m + l] and 1^2x4- Sx'rl+x-x'. 8. 6a* - 10o»m - 22y> + 36y* divided by 4a; + 3y. 12. 2a** + 20-6' - 4a"c" - 3a"6 - 36'+» + 66c" divided by rf»+6'-2c». Nora.— If the teacher it dedrouB of giving his pupils a greater nnmher of questions in division he can And material Ibr snoh in Bzeroise xm, in which the product may he regarded as the dividend, and either the mniti* pHer or multiplicand as the divisor. Similarly, the questions in Ezerdse ^VI. may be made to ftamish additional material for practice in mult^- oiMlon. PIYISION BT DITAOHED OOlf FI0ISNT8. 62. It is ;K)metimes conyenient in diyision, as also in mnltiplioation, to employ only the ooeffidents. The mode of proceeding is shown in the following rule and illnstra- tion : — RuLi.-- iXmug arratigtd the divitotand dividend a* in ordinary diviiioHf omi^ the lettered and eet d^um the eoefficitntSf each preceded by iti proper eign^ and place zero for every term of either dvnew or dividend that may chance to be abeent. Proceed vdth theu coefficients ae in ordinary dtvinon^ and the retuU voiU be the eoefficienti of the quotient with their proper a^pm ; the literal part to attach to^each of these is easUy determined by inspection. Avr.62.] DIVISION. 48 Ex. 1. Divide »** - 144 by 3x - 6; OPIBATIOM. 3-6)9+ 0+ 0+ 0-144 (3 + 6 + 12 + 24 9-18 18+0 18-36 36+ 36-72 72-144 72-144 Hence the quotient = 3di" + Gr' + 12a; + 24. ExPLAVATioir.— We piece three dphera in the dividend to oeonpy the plMM of the abient terms x^, x*, and x. Weluioertaiii the litend parts to attach, by observing that x* ■r-x = x», which wfr ^l*i^ efter the irst coeffldent, and the others of course follow in regn|p||cier. Bx. 2. Divide «« + 4a;»- 8«* - 26«» + 36*2 + 2 Ix - 28 by x^ 6a? + 4. OPUATION. ^ 1 + 5 + 4) 1 + 4-8-25 + 35 + 21-28 ( 1-1-7-K14-Y 1 + 5 + 4 -1- -1- 12- 5- -25 . 7- 7- -21 + 35 -36-28 14 + 63 + 21 14+70 + 66 - 7-35-28 - 7-35-28 Henoe quotient = a;* - a:* - 7x* + 14a: - 7. The stedent is reoommraded to apply this method to the ejcainptof in ^eraiaeXyi^ 44 SYNTHlttIO DIVISION. [i^ioT. tl. gTMTHETIO DIVISION. 68. The following is a still shorter method of division, and is peculiarly applicable when the first coefficient of the divisor is unity. It is frequently called " Homer's Method;" after the name of its inventor.* RuLB.— v^cr properly arranging divUor and dividend^ if the fir$t coefficient of the divieor be not unity, divide both dividend and di/oitor hy the fir 9t coefficient of the UUtir. Then eet down the first term of the dividend for first ttfm of the quotient. ' Arrange the divieor in a vertkai column to the left of the divi- dend, and chaise the eign of every term in it except the first. Multiply all the terms of the divisor^ so changed^ by the first term of the quotient, and arrange the products diagonally under the Me^tBkdfoUowing vertical columns of the dividend. JtM the terms in the second colutiin and the sum will be the eeeond term of the ^tient. Multiply the changed term* ef tlu divieor by the ueond term of the quotient, and arrange the products ^MdUr (ke third und following verticiU colualm of the diifidend, Cotitiwue this process until the remaining vertical columns added give zero for sum, oruntU, in other cases, the division is carried ae fir ai dieited. Non.— It if unud in synlhetic divitioii to pedbrm tHe work by d«t«oli«d ooettdentB, lememlMriiig to place Os for the abwni ferau in both divisor and dividend. Bx. I. Divide o* - 3a*x^ + Sa"** - x« by a" - 3«»»+ 3aa^-a^. 1 + 3 -3 + 1 OPIBATION. 1+0-3+0+3+0-1 3 + 9 + 8^ + al -3-9-9-3 +1+3+3+1 Quot. = 1 + 3 + 3 + 1 + + + = a^ + 3aH + Zaa? + at* * Synthetic divliion demancte the) attention of the student not oidy on •accosnt of ite brevity and elegance, but alio ibr its great value in many of fli» bigber AOptalmamU of reaeireh, rach afe in obtalntaig faeliln vmfen.- tory to the integration ef finite differences, in oenstmctinjl A reioarrin/i; series, in th» treatment of reciprocal equations, *o. '- Art. 68.] STNTH97IO Bpr^ION. 45 ExPLAHATioir.— Using only tke ooeffldenti we write a fbr eaob abient term, 1. e.i for the temu inyolrlng a*x, as«>,^uid ax. The flnt coef. of the diTlaor being unity, the tut atep of the rale ii not reqoiied. We set down the ditisor vertically on the right of the dividend, and change aU iti aignf except ihe flrat. We place the first term of the divid'* ad for first term of quotient. We multiply the changed terms of the divisor by the tnktemfk of the quotient, and arrange the products, 8, - 8, and 1, diagonally as represented, so that the first is under the second term o< the dividend, and so that each is horizontally opposite that term of the divisor from which it was obtained. We add the second column, and get + 8 for the second term of the quotient. We multiply the changed terms of divisor by this + 8, and anaiME* the products + 0> — ^1 and + 8, diagonally, as represented. We add the fhird column, and thus get + 8 for the third term of the quotient, and so on. Lastly we attach the proper literal part to «adi term. Ex. 2. Divide 6a* - «» + 2a» 4- 13a + 4 by 2a^ - 3a •!• 4. OPIBATIOK. ?t3 + 4) 6-1+24-13 + 4 1 + u - 2 Quot. =:3 + 4+l+0+0 = 3a' + 4a + I. • £zpi:Zero divided by zero givee any quantity le^a^- everfmr quotient. Dbmonstbation.-- Since the divisor x quotient = dividend, and the diTidend and divisor are both sero, it follows that ^e qoo^ tient may be any quantity whatever, or in other words, Jh j| a a, because x a s o. . m» >i'Mr, * An algebraic theorem is an algebraic property required to be dettOi^ Btrated. ^ ii l l ii ii ryiifwjjyi ii i l lllV i ;!', 0tm ni t mmm ji t ii.i»..i>!»tei».. sioT. in. AltTt. 94-71.1 THEOREMS. 47 ily mas- w«* zero snd, and dirisor Hence n«ntlt7, • t an m- f gires gnitnde creases. = 0, the ^Oa oc. ityinfi' ^ word* id, it it rerthe When • must d, and eqnop 09. Thiobim v.— 7%e ztro power of any quantity i$ tgual to unity. DsMONSTBATiON. — Since one power of a quantity is divided by another power of the same quantity by subtracting the exponent of the dirisor from that of the dividend, it follows that a^ra^a^'^ = aO; but any quantity divided by itself equals unity, hence a -^ a - 1. Since then ai- a = aP and also = 1, it is evident that a^ si. Cor. Similarly it may be shown that a And a~^ are equiva- lent expressions :*— for — = — = 0" a a ^=«-». NoTS.— It follows from the foiegoing thoorenu that a being any finite quuititj whatever, 0, — • and — - are equivalent symbols, each representing no quantity, or the absence of quantity, or a quantity less than any assignable quantity. •^ and oc are equivalent symbols, each representing a quantity greater than any assignable quantity. Hence also, zero and infinity are the reeip* rocals of eaeh other. -_ ■ ., ' - ao, and — and 1 are equivalent symbols, eaeh representing unity. rr- is a symbol of indetermination, i. e. is employed to des^puite a quantity which admits of an hiflnite number of values, or, as we shall see hereafter, a quantity whose value depends upon its origin. 70. Thiorim YI. — The equare of the turn qf ahiy two quan- tities i$ equal to the sum of .the squares of the two qtumtities to- gether urith twice their product. Dbmomstbation.— Let a and b be the two quantities ; then a + ft s their imn, and (a + by = the square of their sum. Now (a + 6)«= (0+6) (0 + 6) = a» + 2a6 + 6». / fl. Thiobim VIL— 7%e sqtuare of the difference of any two quan- titUe U equal to the sum of the squares of the two quantities ^■i^ t^tuUk edby twice their product. ■ tj^ — ^Let a and b be the two quantities; then Mlc VIII.— >7%< product of the »um of any two qwM' tUuM 6y M< diff'erence of. the tame two quantitit* it equal to the dijfereme qf the tquaret of the two quantitiet. DiMovaTRATiov.— Let a wid 6 be the two quantities, a being the greater; then (a-i-6) = the earn, ^d (a-fr) = the difference of the quantitiefi and , (a 4- 6) (a - 6) = a> - 6> = diflT. of their Bquares. 73. Thsobim IX.— 'I%e product of two binomiali having the tame quantity for ftrtt term but their ucond termt unlike^ it eqwU to the tquare of the ftrtt term together with the product of the two tteomf terme and alto the product of the ftrtt term by the turn of the two eeeond termt. DmoMSTBATiON. — ^Let (x ■)- «) and (« - 6) be the two binomials, then by Mtpftl multiplication (x + a) (x-b)^ a^+(a~b)x~iib. Siiimf1jiF(x-a) and («~6) are the two binomials, their ptodnol ^rBl be «* -I- (- a - fr)x + a& = ai>- (a + 6)« + a6. 74-. TanqQMM X.— 2%« difference' of the n^powert qftffto fnon* titUt UeKhoofft divieible by the difference of the eimple powere of the tome two quantitiet whether the exponent n be an odd number or an «f sfinpnftsr. Obhomtbation. We are tQ show that the two quantities beinjg a and x, and the difVsrende of theirn*^ powers being a"-«", then a* -«" is diriiiblt hja - x whether n be an odd number or an eren numbv. «"-«■ _a»-i + as-x" = a»-» + x(a»-»-x»-») «« X. a-x a-^x Now it is erident that when a"- ^ -x*- ^ it divisible bj a-x then a"-x" must also be dirisible by a * x. But when n = 2, n> 1 » 1, and it is manifi»st that a-x is dirisible by a -x, therefore c? - x' is diTisible by x. Again if n s 3, n - 1 = 2, and sinc^ a* - x' is divisible by a - x, then also a'^x' is dirisible by a - x, and hence alib' a*^M* ii dlTisibie bya^-x, and hence also a'-«' and so on. Thmfcre a«* f X" is exaetly dirisible by a .- x, wbether n bf m fi4i^f A% even number j: c. "I'^l'ii mi'itiiiii if not dir. by « - X. But when 7t = 3, n - 1 - 2, and since a' + x' is not dir. by a- «i therefore a' + x* is not div. by a- x. Bat when n s 4, 9.1 = 3, and since a' + x' is not dir. by o-x, therefoce a* + x* is not diy. by a - x. And therefore an-i-x' is not dir. by a-x, and therefoo af -f «* is n^t dir. by a - x, and so on. . Therefore whether n be eren or odd| a" + 9" is QQt dir. by ai-'f* 76. Thiobim Xll.'~Tii*.diff§r4netoftlUn^potoer»efaMgimQ ftutntitiea w not divisible by the sum of the qtumiUiu whtn m Um DiMONSTRATION. = a'*-i-«*-*x + a+x a+x Now a' -«" is div. l^ a + X only When a" " * - x" " ■ if 4iT. by a + *. Takinff » = 3, n-2 = Ij and a* > -x"-' r a - x, wbieh to eiH* dently not div. by a + x, and therefore a> - x' is not diT;iiiy o + x., ^ But 'Allien M - 5, n - a = 3, and since a* -x> is not dir. by a-l-«r, therefore also a" - x' is not div. by a + x. Bat when n = 7, n - 2 = 5, and since a" <- x'^ is not d|T. by a^ Xt therefore also a^ - x^ is not diy. by a + x, and so on. Thep^^^wlten n is an odd number, a"- x"is not div. by a-f x. 77. lfii»UM XIII. — 7%< sum of the n*i> powers of any two qua^l^^ f$ fiii>t divisible by the w^ of the ^wmtUifis whm^'n ia an fl" + X" DMCOmTRATION. ; = 0" a + x x(o"-»-x"-0 a + X « '. lW*» rSsUVt^f/^- 50 TR10RBM8. IBwtrt. HI. Now in order Uiftt o* + x» ih«U be dir. bj a + «, o* ->-«'♦- » miuit be diT. hja + x. When fi s an eren number, ii - 1 must s an odd number ; nnd we hftTe ihown (Theor. xu.) that the dIArence of the odd powers of twoqoantitiei ii not dir. hj the fum of the quantities. Therefore when n is en eren number, a* ~ ^ - x" - ^ is not dir. hj a +«, and therefore a* + x* is not dir. bj a 4- « when n is an eren number. 78. TnoBiM XIT.— TUc difertncB of the n^ pow«r$ of any two fuanUtiet U txaetly dintibU fry the nm of tht fuaniitUi when niean wen number, o* - «* x(a* •* + x» -*) DiMOMiTBATIOV. rO»-»- a+x a-fx KowwhenoV» + x»-i is dir. bya-t-x, then also a* - x" is dir. by a +x. But when n = 2, n- 1 = 1, and a -i- x is evidently dir. by a -f x, thtrefore o^ - x* is diT. by a + x. And by first step of next theorem a* + x* is dir. by a •<• x, and therefiMre also a« -x* is dir. by a + x, and so on. Therefore a** -(• x*^ is divisible by a + x, when n is an eren number. Novn.— Hm MTenri itep« of thii end of the fbllowing demonatntion mntaalfy depend upon one saolher. Thvi, the lat step of the Ibllowing depends Is divisible by a + x. ■ ■*■ . . ■ j«*'.»^, .^•. Abts. 7S-M.] THBORBMS. $1 And sino« (3nd ftop of Theorem xiv.) a< - j;Mi diriaible bjr a+x, therefore alio a' -fx' ii divisible bj a-f x ; and lo on. Therefore a" + x" is dir. bj a+x when n = an odd namber. 80. The following is a recapitulation of the latter tf these theorems: — a" - »" is div. by a - as when n is odd. o« - x" is div. by a - as when n is even. «» + x« is div. hy a + X when n is odd. «« - a?» is div. by a + aj when n is even. All other nth powers are indivisible by either a + x or a — X. Illubtrativi Bxamplib. Thiorim VI. (2x + 3y»)» = (2x)» + 2(2x) (3f) + ( V)» = 4x* + 12xv» + 9y*. (2ox + 6y«)» = (2f(3x^)«= 16aV-S4a6xV»«V. Oonversely m' - 2)»y + y* = (m -. yXm - y) ; 4»y - 4«c«y + «^^ a (2xy-ae)(2xy'-ac). Thbobim YIII. (m » xy) (m + xy) s m' - (xy )' = m" - xV (3a + 7y) (3o- 7y) = (3«)»-(ty)» = 9o«-49y». <4a^-3a>d) (4(A:y-i-3a'6) a (4) (x+a)(x.a). Jul* -iiifrjis = (m« + o»6«) (o« -««6») = («• + a*6«)(»«4-a«fr«) (»♦ -«*6«) a (m« + a«6«) (m* +o*6*) (m«+aV) (m*-.<^ « (Ml* +a»6«) (m*+a<6*)(i»«+a«6")(m + o6) (m-oft). s'>-<'7«*(*BBlS«(t»;«~- S-WWBW"1*«KI»>WP- s« TfiBOBKMS. ISMtn. Ut. Taiompf IX. (X - 7) (at + 9) - x" + (9 - V)x - 63 - ar» + 2* - 63. (x -3) (* - 7) = x' - (3 + 7)x + 21 = x»- 10« + 21.' * C|oiiT«n»ly. Find the factors of r* + 14x + 33.' HeM sinee 14 is tbs s.nm and 33 the product of the two last terms^ we iwek to find by inspection what numbers added will make 14 and multi- plied together will make 38. Evidently 11 and 3. therefore x'+ 14x +33 = (x + 11) (X + 3) ««+x-42 = (x+7)(x-6)'.- 7+(-6)=land7x-6 = -42. «"-»«+20r?(x-8)(«-4)v-6+(-4) = -9 and-6x-4 = + 20. «"^«-l»6=:(x-13) (x+12) •.• - 13+ J2 =-1 and-13x 12 =- A6. Thiobims X., XIV., and XV. — By actual division, 5«'+a'x+ox'''+x* ; ——. — = a' -a'x + ax''-x'. ar-9 , a + x a* + X* 3 tf*+«*x+a'x'+ax' + x4 : - = a*-o'x+a*x'^-ax'+x*, ' a + x 61, In order to be enabled to write these and similar quotients without actually dividing, observe the following points : — I. The number of terms in the quotient always =: the expo« nent of a in the dividend t exponent of a in the divisqr. ;^ II. The coef. of each term of the quotient is unity. f III. The exponent of a decrefkses and that of x increases i% the several terms of the quotient, by unity, or more generally b^"^ the exponent of the corresponding term of the divisor. lY. When the connecting pign of the divisor is inmMi, all tb9 signs of the quotient are +, but when the connecting fiif n oj| the divisor iapliu, the sign9 of the quotient Arc + And - alt!»||^ nately. Y. The sum of tjlje ej^ponents of eitcb term = tbe 4ifl<»eiliiil between ^e exponent of a in the dividend and that of a in th« divisor. W^ilt-W-^ csissaEs TSKOT.m. AAm. 81, 83.] fmotmh. 5» 1/ "e inek to md multi- 12. ■4 = + 20. a =-1*6. ) export uses ia II the Find hj iaspoelloii fhe yalue of :— ^1. (a-ay)»; (8«+2«y»; (3»y-t)«; (2a«»-8«)»; (2a+8a»>»)». 2. (tt-ftpH^+f*); (2a+3y)<2a-3y); (Sofc-^yXapffSaft) ; (2ni^.3ai!y'>(9il^4- 3«y'). 8. (8a - 2«y) (2«y + 3a) ; (2a - 7X7+ 2a) ; (x + 3) (3 - «) j '(2 + 6<^)»; (3a-4a»y»)». " 4. (x-6) (ar+ 11) ; (3a- 2) (3o + 5) ; (» - 4) (x - 9) J (« + 3) («-7); (x-2)(«-l). 5. (o' - «») ^ (o- x) ; (a« -ar») + (a+ x) ; (m» + a») i (mf a) j (c* + a;*) + (c + aO. -^ 6. (a^ '+«» »y» ») * (a -hJ^) ; (a^m* - r») + (om-r) ; (a«+iii*t«> •f (o - «m) ; (a* - y *«•) * (a *« ^«). 7. («•+ ftt + 20) + (X + 6) ; (^4 7x -fi^ * (x - 1) ; (^4 «« - 4) + (3x+ 4) ; (Co*x*+a9x - o*)>(2/«• Bx. 2 (2x-3y - 2« )( 2x +.3y ^ 2«)= {(2x - 2«)-3y }[(2*«*aO<«|| = (2x - 2«)»- (3y)2 = 4x2- 8x« + 4«^ - 9y». Bx. 3. (a- 2b + 3c) (0+ 2b - 3c) ={ a - (25-3c)|f a+ (2fr-3e)} = a»-(2ft-3c)*=' fl^-(45*- I26c+9c*)= a*-4»»+l26i;-9i^; Bx, 4i (>#+3e» + 3c ^S3?«"^^*>(*'**-^)K«-6 + c)(a+6-c);(a+6+c)(a-6-c 2; (3a- 2c+ 4) (4 - 3a + 2c) ; ( 2a - x + 3 w" ) (2a + x - 3m*) 5 (JA-3y +2xy) (3y - 2a + 2xy), ■'Iw- 54 lACTOSINe. [Bmat. HI. 8. (Is- 8e -f 2« - 3y) <3y - 2x-3c + 2d)\ (a + 2e + 4m -¥ d4) (••i-M-ie-'im), ' 4. (8a-iM-.2+«y)(2-m> + 3a-«yX^ (l+?a»- 3i^^ iBiil^l%vtlw ibUowing expressions, i. e. perfonid tie dp«r»- tieni lai H c iiiUji d uid ledaoe the result tP its simplest fbnn :-«* ' 9. (ds-Sft) (2a4-36) - (2a - 46)> - 4(3 - a) (1>^3) -4(2a-i)>. 6. (4a-> 3«y) (3xy - 4a) + 3 (2a + xy)' - 7 (^ + «y) («y^ 8a>^ 4(2a-8i:y)S, -> ^ 1 (1 -») (1+*) (l+x") (1 +«*) (1 +«■) (1 +«»«). . . .8 i»YiRi. 8. (a - »y) (a + ary) (a' -I- a^) (a^ + x«y<) . . . . to n termSi* 89. AlUkoYif h ^ liATe Men (Theor. zi and zn) tl^at the snsi 4{f tlM eiwn poiv;er» of aaj two quantities is not diy^jide either % tiiesum or the difR»renee of the quantities, it sometiipes happ^ nipt we can resolve the sum of two even powers into its omiB^pit- nwA^fitetors. This occurs whenever the exponent n contaiiii^ an odd fthctor, as for example when it is 6, or 10, or 12, ht 14, '^ , .p|. L-^ltMolve a> - x'y> into its olementarj factors. : . yi^ " Thseri X. a* -*«y» « o* - (xy)* = (a-«y) (o^ +a«y + 45y).r^ Bz. 2.— Itesplve a> - m<* into its elementary factors. \(9,*'-'m* )a. divisible Yxfa^mf and therefore its factors («-») (o* +a*iii+ a%iH«»' +»*). Bz.3.^What are the factors of «' + y»*? «»+y»*=«»+(y»)»=(»+y)(x6-«»y^x*y*-«'»' +«'^"-* y*°+y^^*). Observe liei9 the Mqponentt ot x. in the Becond flu>tor deoreMe.hy Ittpi' ni1)tiinttOB eatthet of x tai the flnt fitotor, while the exponents of y in the aeeond ftetor inereese hy the addition of thtt of y in the flnt Iketojr. Jbt 4.^-^hat are the fiictors of a» « - «» 6ci « T By Theor. VIII. a^^-{mey^^ { a« + (mc)* I !«' -(««)'j «d. a* - (mc)' B { a* + (mc)* j { a* - {fncy\ ; and so on. Ther^fore^ , c»«-»»«c*» = (a»+»«c») (a*+m*c*)(a»+»i»c»)(a+mc)(a--iiic). « AieerleiB by inqieotion what power of 2 expreawi the eiqpoiitttt eeditiMn of the product of the flnt two of thece Awfbn, thctf ^ iii4 hence ^n ftoton. ...* .v« If.* • ' AmT. 88.] ifA(ys(mme. 55 Bx. S.-^W]iftt Are the fkcton of 32a;* -I- 243y« t aa«« + «48»» « (2*)* + (8y)» « (2» + 3y) {(2«)* - (2«)»(3y) + (2x)»(ay)"- (2»)(8Sf)* + iW] » (to + 3y) (!««♦ - 24»»y + 86«y- Bx. •.— ResoWe d^ * -t- m^ * Into ite two elemeattty flMtort. a*' 4-»*' s> oio+jpVOs (a»)» +(x«)» = (o^+as?^ («•-.«•««+•♦»♦-«%•+«•), and reiolTing (a* +«*) and a* - «f bilo tl^ir iheton, we iad.ihat ,..-■ .' . : ' - :■ "■ i .1 •■■'■- .: - -. .■: - : .-. - - it .GJ-: * Bx. S,.-!^9eiol^ :< <«^ ^-|r««^%«ighielementai7faetor8. «» + «» ^i^init^a) (m'-iite.'l-^. • , ... -via '"' ■■ Thereft)W| f(i» +«•» = («»• -«t»*» +«»•)(»»- iii»*»^+«^|)^i- And ^B^iirly i" - «" = (lii ^ • +«*^*^]»*^ Therefiwe^Mk^ ^ r «' !. = the abore eight fiMtmn. . BxnoiBi ZIX. ||9|Qi^^ into elementliry fiictori :— , ,. ■' ■'■>'■■■ ' ■ ' ' ~? ' f . f *?*•; 2, o» + c» ; 3. o* + x* ; 4. a* ^ |* J" e.ou.ftu. 7. o*-i»***j 8. 32(i^+«^; ^iec«; 10. 243«»-32c*; 11. i^-^-x^-^ l^ff •+»«<»; :%i?*S 14. x" + ei»o . y 16. oM - c*« ; 16. |l»*fH»»«^ir li?*|i*c»»«; 18l «|U»+^^ 19. »^* + iiii*; 20.<^)**^>. ■^'i'i *S-«iCi, # »*«!* ' '•'iikMi'A-'^''' u FAOTOBINO. [BWixr^ni. ;^f!^^*^^''S^^ ^^p^^P* ( ■r >>-^ i pi 1. S|mplif5r«-«rH(-*>*i-{-(-|-a-(-f-(-»-a) t,mmtiarf ^» «.») (m + «) - a(a- JU?)* - (3d - 2«) (9(K-. 3a) - 3. Add tofetber V3. •)- aV6 •*- 3V6 - ya; ; 2V3 - 3V5 - 4a«*- V'> aV* - 8V* +iAb« - Va ; »nd 4V6 - 3a?x - 3Vx. ^S. Diride a* - x'^hf a + s to S temi. illi^v, ,^Ct. W^ii^ 4W^fii«to]« ilf x> H I4x ^ 61 ? •^> E#olte «»i'*s»" iato iti lix •lemehUigr foctota. ^•. mride «%^«ili^-^4iAnfti:y •!> 4p«iii*>:» into iH ftstOM. ~%> ^t If • jj^ j» « a, e = 4| (2 =: 1, and m - 0, find the vahib of afc»4Vcl(6c + d) - b - (tf + 6 4^) "fe. r r.. JB^HP^I, wid •tt^i?- i<* - 35* j&y d» + 2a% + 3<«» + 4i*. S' ii.#^^ tT&tliAtieAUj x«. ii^ + »^»- ca(> + «»^ 4 OCX - W^. |/18. JlewlTe a«* - »•* into iti elementary factors* . :,§, j/U. jNa^ 1^ iup^ction the value of (a' + c^)(a^c){aT^^ ^« -«> *c^ + «) {»l» - tt«c + «*«* - tfV + a«c* - i»«c» + «M y 15. If a s i| and a + 5 -f j(> s y^l3; If#ia4-ft+^-' "^^ ^^'^ ''^''^ -■.■■,.'. lo ,-a i^df > 86. TnoBiui l.-'-iiy aqwmiU^ fMOture mothtr quant^ Uiinll aho meantrt any mtUtipU oflM qiiua/dity. Dnxoimnukiioir.—We ue to show tiiat if m meanire otlMHi it wlU also measure Ithanir multiple of a. » ' ^; <^\.: : .1 Let «» be ewtidned n times in a. Thai « •■mmti.. And ttf^fm. 'Inmv M CTideBtirfMiniM fmii, therefore itidso measuei its 9q[iu^t«, , 87. TnoipM'tr;^!^ one qtuaiUUit laeawre ttoo other qUaiAUiM thiS^ wUl J^ niMture the tmn or diferettee of my m^ilHpUe qf titilm0lliamoi«r-^'We are to riiow that if m measure a and also b, it wli ll i l i gp ip t Mrtia inre na±pb, ^l^ipaaw^ <* and » by hypothesis, it also (Theor. I) measnne iM i|«Bi>i» be contained t times in na and s times in pb; tbfiilfHI "^ * 3ssm. Therefore fnadpb « «mi 9mss{t±$}m. /Slag^im b(< t •) timee in na ipkvad is therefore a measnreiioii /'ti ./ :^.Wt«»MHS'^»».VA!-V.,,';;n; .,flrj*," »." 68 0RBA¥EI9¥ COMMOy MEASURE. [Stcr.IV. 88. The G. C. M. of two or more qiuntitieB can often be found by inspection or by the following : — Ruu.— JImo^m ench o/tht quantities into itt component factor$ : tkeHTke product of thou factors common to all the given quantities wiU he their O, C, M. Ex. 1. What is the G. C. M. of 49a%>c« and 68a«6»c" ? 49a'6'c« = 7a%V x 1c and 68a*»V = WH^ x 9^, whence it is erident that the O. 0. M. required is 7a'6V. Bx. a. Tht G. . M. of m\a^ - m^y and (ahn + a«»)» ; that is, of m*(a^-m>) (a* -m') and {am(a+m)f, that is, of w^ia 4- mX^ - m)(a + la) (a - m) and tNi" (a + m}^-m) (o+m)} that is, of m*(a-\-my {a > »)=< and m\a + my (a+ m> d^ is m*(a^\-my, Bx. 8.— The G. 0. M. of 15(a;'- 2a«-8a') and 36(aH* + a-a^-f of) ; that is, of 5(«+a) x 3(» - 3a) and Bix+a) x 7 (a^- oar -i- c^ is 5(«+a). BxiBOUi XXI. * Find by foctoring the G. 0. M. of * 1. ISaMa and 24tfibl*m\ a. ai«*ia», 18«flia» and ISa^m*. _3. 8«Ac*y -h lYwRxy - 3a^^ and Sxy •(- 3axy - 14«(^x^. /-4. a»4-a»-m«'-a««anda5» + 4»+4+ox + a«. , f^i,^ 5. 8<^ (<^-«^ and iw"** (o-«)». « "f J **.e. 3siV -»»*)(«+"•)» iwC^^-w')" *nd 4ja»(a»-m») (a-ii^^. ^. «»-4«-ai, «*-ia»+35 and «»+5«-84. 8. (aat - «)* and fl^x* - 3« + a). 9. afV3*-4, «*-a«+ 1 and x*-l. \ r^i^pntmmivmvmximr. lililjlliit'imiiiMli Abt8.88,89.] ORBATBST COMMON MEASURE. 59 88. To find the G. 0. M. of two polynomial:-- Rule. I. Striki out M< greaiut monomial f actor {\f thortbt any) vtktt^ U common to all the tcrm» of both polynomialif and nterve U. II. Rtject from each of the polynomiaU anv rema4ning monomuU factor that may be common to aU U$ temu. III. Arrange thereeuUingpolynomiah aefor divieion, ue.^ according to the powers of the sAme Utter of refitrence^ and make that one the divisor tohose first term is of lower , or of not hig^ . dimentionif as to the Utter ofreference^ than the first term of Mother. IV. Multiply (if necesiary) the dividend by the Uaet monomial that unU render its first term exactly divis^U by the first termofthedivuor, y. Divide the dividend by the divisor and continue the divition until the highest exponent of the letter of reference in the remainder is less than the exponent of the letter of reference in the first term of the divisor j observing that if the coef. of the first term of any partial rem. shotdd happen not to be divisibU by the coef, of the first term of the divisor, in order to avoid fractions, the rem. u to be multipUed by eudt a number as will render the coef. of Us first term exactly dioisiMe by. the coef. of the first term of the divisor, VI. RiQectfrom the remainder its^eatest monomial factor, aMif its first tkrt^ is negative, change all Us signs: eoneidef^Ui* result as constituting a new divisor and the former divisor a neio dividend : proceed as before, and continue theopera' tion uatU there is no remainder, 111, Mul^y the last divisor by the reserved monomial^,i^ a$iii . dful the product wUl be the G. C, M. of the given polynomiaU, \ |lvut.— T)ie G. C. M. of ^wo quantltief lit evidently tiiei^Aet ^ ^^oksMiamionto})oVh . . . ,, ,. ^ . . . rto both (M we may do for we ^e of eopvenieiiee) wi 'wi0l^ |pl niM4 iUi llMtor m eniering Into we G. C. U., mi4 tlier0fi>N Ire'fll^t tt. 60 ORBATB0!r COMMON BfBAStllB. [Sior. IV. d)oir dr n.— Slnoe tlw O. C. M. of two qtiuitltfM is Um prodvet of all tbt tetora whiob an ommnm Io bolh qmmtttlM, tt !• erldoit ttpat a flMlor whiek beloBgi obIj to OM of the two euuM»t Item a pait of ttMir Q. 0. X., and thonfcre we may, fl>r tlie aake of abbreviatisg the work, i^Jeet at dlfeeted inn. IT.— HaTtaglijrII itrnekovteTerjr monoinial tliat is a ftolor of either of the qaaatttles, tt is erldent that if we anMipljthe divldead by any MonoMial in order to make its flrst exaetly divisible by the L.^(term of the divisor, this monomial not being a ftetor of eaetaof tlie torms of tlie divisor (Mioagh It is of tlie flrst term) eannot be a ftetor eommon to botii dividend and divisor, rnd therefore eannot Uram put of tiieir O. C. X. m, y, VII.^Let the given polynonials whose O. C. M. is required be iii>iM and n*/b, where m*, n and /are monomials. After striking out and reserving the oommon ihetor m*, and r^Jeoting from the remainders na and fb, the flwtors n and/ whleh are not oommon to both; then the rednoed polynomlahi whose O.C.M. is sovght are a and b. Pappose these being properly arranged, the leading letter of Ms of lower or not higher dimensions than tiial of a. Then divide and suppose a-i-b gives a qnotlent p with rem. 0; — also b-i-t gives quotient q and rem. d; also o -S- d gives quotient r and no rem. ThendlitheO.C.X. ofaaadd. We shall flist show that d ii a oommon measure of a and (. Beoause d aa e as ur es e, siaoe it goes into it without a remainder, therefore (Theor. I) it measures qp a multiple of 0. 1 JBeeaase d measures d and also qe, thei^fore (Theor. II) it meamres / their mim, whleh is 5. y" Beoaose d measures ft it also measures j»fr, a muMple of fr. BeoalNe d measures pft and also it measures their sum wUoh is a. ihiitvitin d measures both ft and a, and is a common measure of them. Neirt we diall show that d being a eommon measure is the gttattit oMnrnwi measure of a and ft. For tf d be not the'G. 0. M. of a and ft let there M a greater as df. Than beoause df measures ft it measures |>ft, a maH^e of ft. Beeanse # measures • and also jift, it measuius (Theor. II) their diflbr* enee, whidi is o. Beeanse df measures it also measures qe, a muH^e of 0. Beeanse d' measures ft and also go it measurestheir diftrenoe, whiehis d< TheieAne df measuresd,tiiatls,agreatorquantityaieasttrss alessiwhieh is absurd. Thenfore df l§ not » coliliiion auosilre of a and fti and In like aatter ttni9arlNiilK>wnfhalnoquanti(ygrMerthan disaoommon measnt of (lattlK Therefore dis the G.OiX. of a and ft« AST. 89.] ORBATEST COMMON BOBASURB. 61 v.— W« vuj mult^^ly Miy naudnder Iqr an j nvaiber in Mder to nak* itiflntMtl. •XMttjdiTlribtobjtlMflntMel.oi the dirlMr, beoavM the O. C. M. of a Mtd 6 ii the MUM ai the 0. 0. M. of any diTlwr 6 end rem. 0. If now m KoH^ly this rem. o bjr uj monomial ae/» thedirlmr 6 haviag no monomial ftetor, eon have no Ihetor In eoaunon with /, nor thatefuo anyineommonwlthyktaitwhatitmayhaTelneommonwItho. That if, theO. O.K. of bukd/B will be the lame ai the O Xoibaado, and theieibre the Mune aa the G. C. M. of o and b. VI.— We ndoot the monomial ftetor of the remainder befbre makinf It a dlrltor, beeaaae ttw fbimer diriaor, which has now become a dirldendi contains no monomial fhctor, aad.therefbre can contain no ftctw in com> mem with the monomial rejected from what now becomes the dfriaor, and therefciotho e. 0. M. of the dlTldend (lart divisor) and the nnredneed difisor (t e. hMt rem.) is the same as the G. 0. M. of the dividend and diTisor rednoed as directed. We can change aU the signs of the divisor becaoae this ii equivalent mersly to dividing it by - 1. Bz. 1. What is the G. 0. M. of «>-10«-l-21 uida;'>-2«-35T OPIBATIO*. a:" - 10« + 21 ) x« - 2x - 30 ( 1 «■-!•« -I- 31 8a; - 66 = 8(a; - 7) «-7)a:»-10x + 21(«-3 -Ss + 21 -3X-I-21 /. G. C. M. = X - 7. EzpiAKATiov.— There is no monomial flurtor common to both, nor is there any monomial flutor common to alt the terms of either. Therefbre we at once proceed to divide, x being taken as letter of reference ; the first terms of the given qnanttties are of the same dimensions, and conseqaejutly it makes no dUbrraoe which is taken as divisor. I After the flrst step of the division we obtain a remainder 8x - 86, and before using this for divisor we strike out its menoariol iaetw 8. Hits gives us X - 7 Ibr 2nd divisor. We make the last ditlsor the new divi- dend, and finding that we now obtain no rem., weeMfriiide thfit the G.C.M.is«-7. ** ?^' ^ ! ^-*^ ^^ ^^ * ^ ^'* HJ^ '* f '■''T^- '^ ^•-^•N '''*"' 02 0BIATI8T COMMON MIASVRE. [8iot.IT. Iz. a«-J'iBd tiM O. 0. M. of a«« 4- SoRs- Qa^s" Md 6a«x OPIBATIOM. ta« + 80^ - 9«A? ~ ax a(aa> •!• 8m - to') aa^ + 8ax-9x*)6a^- lV 18x 6a* 4- Sa'x-aYax* -l«a%r + 41«s*-3c* -a6aftt-S9a«"+llT«» 80««> - laox" a 40x>(aa- 8«) 2a - 3« ) aa' + Sox - Ox* ( a + 8a» ao'-Sac 6ax * 9x^ 6ax^93^ O. 0. M. of the reduced polynomialB s aa - 3x and reaerred common fJMtor » a. Therefore 0. 0. M. of giren quantities - a(2a - 3x). Bacn.AirAT»m.—Herewe strikeout and reaerve the monomial fttetor a, whieh !■ eommon to both qoantttiM, and itrike out end reJeot the monomial llMtor « of the leeond quintity end remaining monomial llMtor a of the flnt We Mleet the dLfiaor as shown in the margin, bedaune as, iti Itrst tmm, is of loww dimenrions than a', the llfit term of the other. Our first rem. is 90tue* " 190»3 fiNMn whieh we reJeet its greatest monomial faotor 40»t ^ aad this gives us Sa- 8a; fbr a newcUvifor, the last diTisor heeomlngthe new dividend. Aut.W.] greatest COM1IO|| MEASURE. 68 Bx. 8.->Find th* O. 0. M. of to« - «^- 8«V •(• Sxy* - y« ab4 9x* - toV - a«v + Say - »*. oratATiON. 6** - *"v - 8»y + 8«y»- y« ) •«« - 8x*y - a«y 4- 8xy" . y« ( 3 2 - s««i + faV - 8«/ + ir* = - y(8«* - 8x^ 4- Sxy* - y*) 3x" - 5j:«y + Say - y* ) 6a:* - «V - SxV + Saty* - y« ( 2« -f 8y eas* - 10««i + ««y - 2a5y» 9s^.^ »a:y+Bay-y« 9«^-rsxV-i-9«y'-8y« e«y-4ay+ay* 8a» - 3afy +.y* ) S«» - ftaf«y + Say- y» ( » - y Sat* - 2aj»y -f ay - ?a:^ + 3aty* - y' - S«*y + aay - y" Th«nfor« O. 0. M. « Sac' - Oary + y* ExTLi HATioii.— Here, after leeing tiMt the tenu are profi^y arraaged and that there IB no monbipilal ftetor to njeot, we mnltlply the dtrldend by S in order to make Its int term ezestly dlTMMf hy the inlterm «f the divifor. Belbre making the rem. a div. we oast out ite monomial faetor y and ehange all its ligni, or, what anionnti to the same thing, we eait ont the monomial fhetor -y. JMm making the next rem. a new dlviaor we eait out iti monomial fketprSy*. Ezmotsi XXII. Find the Q. G. M. of— 1. 2^ . 6x - 14 and ae* > X - 6. 2. af* - SJc* + 21«* - 20x + 4 and 2x'^ - 12a:« + 21a' - 10 ^. a^-^ax- 7a -I-. *fx and a" - 3a f Sac - 6 - 6(i«y - 26y'-f 2a6y* and 12a^6 + Sfry' - I6ahy. ^. oF 4- 9a* + 2Ta - 88 and oF -f I2a - 28. 10. 8a%« - 24 120* - 48a> -t* 22a + 12 and a* + 4a« - 8a« -160^+110^+ 12a -8. 12. 2<^ - 2aH - 1606* -f 126* and 3a*tf - 9a^6<; - 24aF6*e + 54a6*c - a46«c. 00. To find tbe G. 0. M. of three qoantitieB : — Find the G. 0. M. of two of them, and then of this G. 0. M. •nd the third quantity. To find the G. 0. M. of fi»ur qiumtitieB :^-Find the G. 0. M. of any two of them, and tiien the G. 0. M. of the other two, and lastly the G.O.M. of the two greatest common measures thus found. LBAST COMMON MULTIPLE. 91. The Xei|st Common Multiple (1. o. m.) of two or more algehiaio quantities is the quantity of lowest dimfn- sions, as to the letter or letters of reference, which exa^y contUBf e9K)h of the given quantities. Noiy.— Of oonne there 1b the same olijeotion to the lue of Mie woid '* letut " here m to the word "greate$t *' in regard to eommon mettam. it would be more oorreot to use the term lowest common mnltiplt. 02. To find the 1. c. m. of two or more algebraic quantities: — RfTLi. — Divide their product by their O.C.M,'- OTf Divide one of the given quantitiee bythiir^ik C, JT., and multiply the quotient and remaining quantity or quantiHei together for their l.c.^^. m Am. 9MB.) lAAST OOHMON IIULTIPUI. 66 Fmnmt om Bou.— Ltt it U Nqvlrtd to tad th« 1. «. a. of wof two QMBtltitt a aad ^ ud tot m bo tiM 0. 0. K. of thuo VMBtMtf. Lit a as JM ud * at ffw, aad M bflBf tho 0. 0. M. of • nd ft, it Mlowi oreowMthatjioadf hMToaooouMmiMtor. Thtajif atoMtqwuitlty that eontoliis botiip and q, mA w^ ss tht iMit qvaatitj that ooBtolai p, q, ud m, OBd thoraloio a tho 1. 0. m. of a and b. TboB 1. o. a. s pqm ymxqn ax b a 6 or ^ xba. a X Ex. 1. Find the 1. c. m. of ISoRx^ and 16a;tV'' OPmATIOV. O. 0. M. of ISo^xV And 15a«Vx a SoxXy. 18-x, x«- 1 ud «*•(■ 1 ; that if, of «(«*'- > >. a;"- ly and x*4- 1 ; that ii x(x - 1)(« -f 1), («- ])(x*+x •!• l)Mui («+l)(«"-x + l) = x(x . 1) («> -1. X + 1) (x + 1) (x»-x-Vl) = x(x»- 1) (x»+ 1) = x(x* - 1) = x» -X. No«B<-~or «OHw tlM saiM iiMtor it onl7 to b« taken oNee iA tiM 1. f . n. altlHMf hit n^ oeeor in eaeh of the fiven quantftiee. Ex. a^-'^tho 1. c. m. of 4(x» - xf»), aO(x« + «^ - »|" - ai*)! \%{ff +y), ia(x» + xy)» and 8 (x» -x*y) ; th»f ii^ of 4x(x«-. j^) ; 20 { («•+ xhf) -(x»»+ y»)| ; I2y«(« + y) \ lax^x + y)* and 8x'(x - y) ; -»^-'**n''T'»^»« 66 IRACTIONS. tSSCT. V. Ihftt if, of 4«(« + y)(af - f) ; 20 1«»(« + y) - y»<« + y)) j iaf» (x+y); 12«*(x-l-yV,and8x*(«-y); tlitt if, of *t(« + y) (X - y) ; aO(»+y) («» - y») ; 12y»(» +y) ; 12a:>(x +y)V imd 8x«(x - y) ; tiMt if, of 4«(« + y)(« - y ) 5 20(x + y)» (* - y) ; 12y»(« + y) ; ia«', (x» - lOx + 21), and x!- tx. i* («^- x»), and (a^+x - ox - a). ""T^. iF - 9i?+ 26a- 24, and a» - «i^ + 19a - 12. MK 8(i^-»»), 4(a-6)», 6(a«-6*), ^ia-by, and (oa-60*. -> SECTION V. PRAOTrONS. 94. Algebraip firaotions are in all essential respects simi- lar to arithmeliotl finodons, and the rules for opera^g iipoln them are the same as those for common arithmetic, and are dedacecl in the same manner. 7 •5. Since the value of a firaotion is the quotiMit, whidi is obtained by dividing the numerator bytlie denominaior, we infor the fbllowing principles, upon which the princ^Mil .rules are foan4e4 ;—^ '/•■■yw.,-vvit'iX3^^^'V*-*^vo-f...^Vji.> >'^ fHr | B HW ili Btfn-; 4ll ll » r t ' -l 'I W I| lJMMMII» l l il!» Abts. MtW] FRACTIONS. OT I. T%at mult^lying the nmuratoTf or dividing the dmombuUw of a fraction by any quantUy^ muUipliee the fraetUm 6y that fuan' toy. II. I^at dividing the numerator, or mtUfiplying the denomina- tor, of any fraction by a (^uaMtity, divides the fraction by that quantity, III. 2%a^ multiplyii^f or dividing both nutneraior and denomina^ tor of 9 fraction by the same quantity does not change its value, 86. These principles are, however, susceptible of general proof, as follows : — a cim a I. Let -T be ray fk«otion rad m ray integer, then -~ = -=■ x m. For * a am 6 6 in eacli of tbe frtotions -r rad -^ the unit ii divided into 6 eqwl pftrte, and m times as mray of these parts are Indioated hy the latter flraill- = a a wn = — X TO, conversrty i- = --r- 6 6 TO. O 6m X TO, conversely -r— ■ ^ -r -i- n, um o ni. Since both multiplying rad dividing ray quantity by the numbfidees not chrage its value, if we both multiply rad divide ■?■ by m, its vttiift will remain unaltered. Bnt (I) ^ x m ^. ^, rad (U) 2SL «f If *^ ^^ =B T*' ^- *•> althongh the parts In' the former flraetton are -^ th of each of those in the latter, m times more of them are takaa. 68 BXDVOTIOJX OF FRACTIONS. [Sbot. y. Or, -— - =a — r — (Art.i)it>»(2e both numerator and denominator by their 0,C, M. Nom.-rTlie itiideiit ihovld alwayi endeavour to ftctor the numemtor end denomiaetor lo m to find by impeotion the O. C< M. when it eia be M> found. Otherwiie he moit And the 0. C. M. of the two terau hy Artm, J!i}f^f*\^\ >-J,'V lainatorft. Tatar and Abt0.O7,96.] RBDUOnON OF FRACTIONS, a^nuey amx x ay 69 Bz. 1. Ex. 2. Ex. 3. 0^(1 +8«) l+8ar 2o" - Sa^oi + oV a'(2-3iiifyO 2-3«+y«* (a-«)(a-x) <^-2ax + x^ o*+a'x + ax* + «» a-x. * a»L6a-27 (o + 3)(a-9) a-9 *• o?« + 8a + 16 * (a + 3)(a-f6) ~ a + 6' a-x Ex. 6. x' - xy -l- flix - my x'(x - y) + m (x - y) X* + xy + mx + «y - X (x + y) + « (X + y) (X - y) (x + m) X - y (X + y) (X + «i) " X + yV x' "■ 8x "t* 3 ^*' ®- g .g ft. .V Hew (Art. 89) the G. G, M. of the numerator and denominator is x + 3, and dividing both terms (x »~8x + 3) 4- (X't-3) x»-3xfl by X + 3 we get (^+3,. +, + 3)^.(^+3) = x« + i ' Exnoiu XXIV. Redaee the following fhustions to their lowest terms :— v^-ab 2am + m'x - m' c -f oc 1. 4. t. U. 14. 17. ^ 2. ox + ay mx -t- 6x + X 21xV-35xV 14xV 12. 3a»»i + i»' 6. 3. 6. n4-an ax^ 06 + 6c*"/- ' o^r'in+axy+xyt' »^o — J» 8. 9. o» + 6» 10. a*-2a&-l-fr' aa-6»* *"• a^-6« 13. 7x«-21x+36 . x»-llx + 28 .>^ llx« - 38x + 66* x»+2xV + 3xy ax*-.8«'^-6x*y* y 16. K 18. 4x» 4- 12x -f 9 x»- 4x"r21* ^^^x'-ftx-u'^ a«-2a% + 2o6«-6» a* + aV + ** -.>s Sfcs iSSli •!*k •ev 70 RBDUOnON OF FRACTIONS. tSlOT. V. 19. 21. 23. a*-«* 22. o^ - ahi - am? + m'* «^ + (6 + c)« + 6c' (a + m) (a + m + x) (a + m-- x) 2 . , . . . 7^ am 4- 26p 4- 2<9i 4- 6m 2«" 4- X* - 8* 4- 5 Yx* - 12* 4- 6 24. - ..«o 4'« «o* 89. To rodttoe a mixed quantity to a fraotional form : — Bulb.— JfolfQily the entire part of the quantity by the denomina- tor o/thefraetionf and to the product connect the numerator of the firaetional part by Ue proper eign. Beneath the whole expression thusformedf write the denominator, X + y a^m - abm 4- (x 4- y) ahn - abm + x +y i-' i Bx.l. 0-6 4- Ex. 2. a" - 2ay - ^^ 4aY - Say" - 3x 4- 2am am am aim 3x - 2am 4aV - 8«j' - (3« - 2om) 4ya 4y2 EXBBOISK XXV. Beduee the following mixed quantities to their equivalent firaotions:— 3-2a , 2 3a>-30 1. 2aiK-y4- . 2. o'' + o4-l4- -, 3. 3fl-y- 4. 8«4-v- ax 2a4-xy «-y * 5. Sox "y'^ + m - o- 1 " ' X4-3 3a»'4-xy' a4-x «y«-«'m-2m'« -^ ^ ^^-. (o-6)' ] jj, ^ 8. 1- a^'-m^ 1^ 9. r- o* - 2ax 4- x^ V^t: 100. To reduce a fraction to a mixed quantity : — RuLi.— •Dioitie the ntumrofor by the denominator^ and place the rewutindtrt if aniy^ over tht denominator for th% fractional part. Connect thefraetUm thus obtained to the entire part of ^ fuot^i by the sign plus. .}i W « >-w yaii " ;»m ABTS.W-101.] FRACtlONS. 71 Ex. 1. r = a- 46-3+^. 3a 3a 9x^~ax - 2« - ox 2x + ax Ex. 2. -r — — ;- = 2« + ^ J i '- = 2« - 3x+ 1 3X + 1 • Sx + l BionoiBl XXYI. Reduce the following fractions to mixed quantities :— 20fli'-20m + l. - «' + «* «'+2ary + y«+x»-y* . • 3. 1. 5m 2. u.' 6;ii''-5p» + 3 /, l-o- flft + o"* x + y / m-i-ab + 5am 6; ^ jw-p a6-^ m + 6. 101. To reduce fraetions to a common denominator: — Rdli. — Find the I. c. m. ofaU the denominaiori ; then taking each fraction in n 1-a 1 0? Bx. 1. T-T-r + r— +: (1-ay 1+a ofl + •: 5 + l + o • l-o' l-a?» l-rf» l-o» l-2a+«^ + l+a+a^ 2-a + 2a* l-a» Sz. 2. (l-ai?)* l + 2x« + a:* l-x« l+x» 1-x* 1-x* 1-aJ* l-2x*+x* l + 2« + «*-l + 2«-«* 4x !•«* Bz. 3. a» 0? ^ 0(1 -g)* a»(l-a) 1-0 " (1-a)^"^ (l-*)*" (!-»)• " (l-«/ 0(1 - o)'- <^(1 -a) + o" a(l - 2a -f «^-(aP-a^ a» (l-a)» (1-^ a-2a^+aF-io*+a"+a? c-Srf'+Stt* (!-«)• (l.a)» \l^af * Abts.102,108.] MULTIPLIOATIOK Ol* FBAOTIOKS. 78 xy(«-y)=' « y a;y(x-y)' xy(x~yy 2xy(g-y)' x'ix-y)'^ xy(x-y)" ~ xy^x-y)* X* + 2g V •>• y* - y' (g" ~ 2a;y-Hy')-2ay(a* - 2gy + y'')-g''(x''-2 JiH-y*) «y(« - y)* «* + 2*'y" + y* - asV + ixy" - y* - 2«"y + 4a; V - 2a:y» - «* + 2x'V--a;'V 4*y 4a:y xy(x-yy~ xyix-yy (x-yf EXBRCISK XXVIII. Find the value of :-~ 2a 3 c a; 2 (a -6) a-6 a + ^' T "*■ 26" m* ^* y**" !^T«'+3)* ^* ^Tft " o^* x' y «y 2x 5x 7 9 («+y) «+y (x+yy 0-6 6-c a-c m ,p 6, — r— + — T — • — . 7. ; — ■ - . 00 be ac m+p m-^p 8 10 11. __3 4(1 -5a) __7^ x(lQ-x) 2x+B 2-3* n^ ' l+la" 4a»-l 2a-l* ®* x»-4 ''"2-a: *"x + 2 * "^ 1 ^1 ^ A+y «-y\ ^-C;^ «l+l> jp + x m + x (j)-x)(x-m) (x-to)(iii-j>) (»»-i>)(l»-:*) a- 6 6-c 2a6-2ac ^^* + 6 "*■ 6 + c " 6(a + e)+c(« + *)-Kc-*)* 1 1* 3 3 13. z -— —• + 14. 1-x 1+x l-2x l+2x HI fA m fl(a-6)(a^<:) 6(6-o)(6-c) c(c-o)(c-6)* 108k To multiply fractions together : — tLvm.-^'Multiply all the numeratort together for a new nUihera^ tor^ and all the denominator* together for a new denominator. F 74 MULTIPLICATION OF FRACTIONS. [8«ot. V. Nora 1.— If ray of th« givon quantitiM are mixed fraetions, thoy must bo rodttood to tho fl«otloiial form before mnltlplying. NoTB 2.— Before multiplying the student must, by attention to the prin- oiplos given in (Arts. 70, 80,) strilce out all the ftiotors common to a nume* rator and a denominator. Pboov ov Bulb.— Let it'be required to multiply r- by ^• a a Let — = a; and -r^Vt then ^- X -r — ^V- Also a = bx and c =: dp, a o a ac llonoe ac = bdxy, and dividing each of these by bd vro got -r^ = xy. But -^ X ^ = xy. Therefore " X ^ = ^-l^roductqfnunwratcTs b '^ d bd product qfdenomifMtor$ a (l-a)o a -a' Ex. 1. X . - . VI - L I x + y b (x + y)0 ox + by x" - 1"*" x^ + bx x'^-bx + b^ Ex. 2. ^ -.rr- X X -3 x« + 6» x-'b X* x\x* - 6") X a(a; + 6) x (x" -bx + fr") _ Ji^jx - 6) (X + 6).x x(x + b)x (g' ~bx + 6") (X + 6) (x» -6x +b'*) X (X - 6) X x^* * x»(x + 6) „ a-J, a^*-l _ (a- 1)^-1) a*-2a»+2a-l Exercise XXIX. Find tho value of :— 2x 3x J- T- >< TT-' 5 2a 2/A 2. — X — X — . 3. xy my x xy * 2(a+b) x(o-6) 36 + 3a ' a x + 1 x-1 a^-x" d^-b* 4. 3a X — — X 7.- 6. p X x . 2a a+b a + b o + x x(a~x) 0. X a^'-x" 4ax* 7, — X . 3ax a + x ABt. 104.) DIVISION OF FBAOtlONS. 76 imerators. ar'-13x + ». 3 x«-9x + 20 a b e d m_ 10. 11. o»-4 o»-l 0-2 o'-l ~2a~ 2 + o' x*-a* x' + 6x + ex + 6c X -77 x*+6x-ax-o6 x'-'+cx + rfx + cd* x»+'x-12 x2 + 2x-36 ^ 1 Iv v^ 12. a ,„ — ^rr X ir-r; 77- 13. (1 -a + o«) x (1 +- + -). <. x»-13x + 40 x^-7x-44 '^ ^ a o» ^ 5a a + 2m 4a»- 16m» 14. ^ X a -2m 20o» + SOam + 80m' 104. To divide one algebraic fraction by another : — Rdls. — Invert the divisor and proceed at in multiplication. Note 1.— If either of the given quantities bo a mixed fraction it must be reduced to a fractional form before applying the rule. Note 2.— Alter having inverted the ternu of the divisor, be carefiil to cancel as far as possible before multiplying. a c Fbooi* ov Bule for Divibiow.— Let it bo required to divide >- by — . Put — -i — b d C Cl o x; multiplying each of these by — we get -r- = * X t a a cos cuL £= — . Again multiplying each Of these by (2 we get -^ = ex, therefore X = -r--. But X ■ be . a c^. „ a c ■ ad ad -.- -r- -z-i therefore — -f- -r = -r— = -r X — b d b d be b c = dividend x divisor with terms inverted. Ex. 1. o»-62 o2-2a6 + 6' ft'-fti* a* -6* o»+6» • o*-6* o» + 6' a"- 206+6" ( o>6) (0 + 6) X (fl' + 6») (0-6) (0 + 6) = (oHfc^)(a-6)(a-6) ^^ (a+6)^ a? + y^ a^-ay + y^ a^ + y' (fl-yy aX. 2, .. T — :: r:~ == -t s X - o*-y" ' (a-y)' o'-y* a*-oy + y - («•^y)(<*'"<»y+y')(<*-y)('»-y) " (a-y)(a + y)(a»-ay + y*) = <»-y« 76 COMPLEX FRACTIONS. (8«0T. V. ! ExiROisi XXX. Find the value of :— 1 X a + x X a + b d* + 2ah + b'* ■A, ~~ + • 2. —^— Til— 1. 3, ~ T „ — — : ,„. X y a ^ o' a-b a'- 2a6 + 6^ /a^~x* y'-A\ a^+x'* x-3 x»-16x + 66 *• V2 + y ^ a-x j ^ "~3a~* ** x^i * X2-17X + 72* /_a _6_\ / a b \ ' \o + 6 a-bj ' \a- b " a + b J' / a*-x* 1 \ /q*+ax+x* o^-ox + x^X \o''-2ax + x* a + x J ' \ a-x 1 /' 3a'''- 3 x^-1 / y x\/x x\ 2(a + 6) 2a* + 2a6 \ x + y y/ \ y x+y/ X /a*+b'' d^-b'^\ /a + b a-b \ 106. To reduce complex algebraic fractions to simple fractions : — Hvhis.—'Redwe both numerator and denominator to simple frac- tionSf if they be not simple already i then having thw reduced the fraction whole expression to the form of— — p — t in-vltiply the extremes together for a numerator ^ and the m£ans together for a denominator. Ex. 1. 3-x ~3~ _ 4(3 - x) 12 -4x Bx. 2. 1-i^ . ^y_a " y-4a 3(y-4a) 3y - 12a 4 1 _1_ 1-a •1+a I + 1-0 l+o 2-a 1-a l-o (l + a)(2-a) l-»ii-o=* AuT. 106.] COMPLEX FRACTIONS. 7T a 1 + «-l Ex. 3. a-\ a-\ 1 + a 1 + a 2_ o-l tt' 3 _ r '" tt'-O 1- a a-1 a a^-2a+l d^-a EZBBOISB XXXI. Simplify the following complex fractions 3a + f 6 a; • •!• 3 l+2a 1 + 2x a ii + Kx-3)- l-2a U^Kf^a) ^ l-2a l + 2a 6 1+a l-o 1 - 2a 1 + 2a l + 2a l-2a 1-a 1 + a a- ■\-b^ - a *■ a a' + 6" 10, bdf+ be + cf d^-b' b 4- c U 'V d adf- -^ac 11. l-^xhf^ xy 9. xy 1 - 1- 1 xy (l-2my'+{2m+iy (l-4m')-(l-2OT)' V ^ (l+2TO)^-(l-4m«) (l-2»i)2-(2«i+l)'' ' r- 78 THIOBIMB IN FBAOTIONS. [8B0T. y. 106. TfiiORiii.— //* any twofraetiom art equal to one anothtlrf we may confine, in any manner tohateverf by addition and ni6/rac- tion, the numerator and denominator of the one, provided we at the eame time eimilarly combine the numercUor and denominator of the other f and the retultit^^ fractione wUl be eqwU. That is, if ^ = ^, then a+ b c + d a b 7^ r"rf-<'^>5 / (I); > a+ b a- b c- d ^ ^ h d (") ; ^r = r 0") i c a + b 6 d e -f d a-6 a e c^d (VI); c + d c a a-b a a±b a + 6 ^r " 715 ^*^ ' > 7+1 "^ ma mc ma f n6 mc f nd c-d ma±nb (xi), fto., Ac. me.± nd (XIV) ; nb nd m4i± nb mc jr nd pa ± qb pc ± qd . (XIII) ; (XV,), kc. Or, 3%< above propoeitions hold with any multiplee whatever of the tvfo numerators f and cdto any miUtiplee whatever of the two denominatore. a^ •v/ Also, M» = jji (xvi). That is, the above theorem is true of any similar combinations of the same powers of the numerator and denominator. DBM0M8TBATI0N. a a (i). Since r- = T-.*. r- + l=;T-+lor d "b d a c . Since T- = v a a c .*. 7 1 = — - 1 or a+b c + d _ a-b d c-d d l»)i;Since r = T--^^r = l^j ^■f7 b d * b d b d Ibat is, c b or 1 X - a 1 X a AaT.106.j THBORBMS IN FRACTIONS. • a e tt b e b a b r-x — = -rx— or — --r. 70 (iv). Since r =" -J -'- r ^ - i - "• - j ^ ' b a e a c c a (v). Since (I) a + b c + d . and (ill) — a-^-b b a e^d d a-¥b e + d —— X — or d c • a a-'b e - d b d a«6 (vi). Since (ii) — r— = —J- and (in) — = — .•. -7-— b c -d d a~b c-rf X — s — T— X — or = , a d c a c (yii). Since (11) d a~b c " d C'd d c-d and also (i) a-\-b b d a + b c + d inverting by (111) -r a + b b b e+d b c + d a-b or a-b c^d' a+b c+d (viii). Since (v) = (iz). Since (yi) .'. (ill) -----. = — — ;. c-d a c ^ a .-. (Ill) a - 6 ~ c-d* a (x). Since (viii) and (ix) o a±b V^^ c±d' (xi). Since (x) a+b a , , - — and also -r c+d c c-d -7-; .♦. alternately by a-b .'. (Ax. xi) a+b 0-6 c+d c-d or XI = VII taken alternately. m c m ma mc (xii). Since r- = T'"«r"X~~T'<"~or-7- = -r. ' bdbndnnbnd B8 r / ma TOC (xiii, &c.) Since ;;^ = -j /. all the above changes may be nb made on these fractions. a c a a (XVI). Since ^ - j .'. ^- >^^ o" a' H /.n c c a' Similarly rj = ;« and t;; = ;i^. And all the above chan o a a f,n f,n may be made on the equal fractions rz ~ %„* r- '^vi S^j- 80 THEOREMS IN FRACTIONS. tSKCT. V. 107. Thborbm.— /^/A«r« be any num6er ofeqtud fractiom^ then toe may combine in any manner whatever by addition or subtraction the numerators, or any multiples of the numerators, provided we similarly combine the denominators^ or the same multiples of the denominators, and the resfdting fractions toill be equal to any one of the given fractions and to one another. ace That i», if j- = J = J, a a±c ±e ma±nc ±pe Then b b.±d ±f~ mb±nd±pf' Dbhomstbation. a c f - X. .'. a = bx, c - dx, and e =fx, .'.a±c±e = bx±dx ±fx - (b ±d±f')x. Dividing each of these equals hj (b -^ d if) vra get o ±c ±e ^ a a a ±c ±e F X = b±d±f'^''^''- b " b - b±d±f' Again, since a=:bx,c- dx, and e =fx, .*. ma = m&x, nc = ndx, and pe = pfx. And ma±nc ±pe = mbx ± ndx + pfx = (mb ±nd ± pf) a ma ±nc ±pe b " * = ma ±-nc ±pe a mb ±nd±pf ~ b mb±nd ±pf' a a^ -n It follows.that if- = - = -, then j-„ = -r^ = r d" / m and therefore n a' 6» €l^ ic^ i «" a** wio** ± nc^ ± pe^ Ts^ — is-.-?;;ii and therefore also r= = —{zr-. — si ^. ABTS. 107-112.] SIMLLE EQUATIONS. 81 SECTION VI. SIMPLE EQUATIONS. ._ 108. An equation consists of two algebraic expressions connected by the sign of equality. Thus, 3a + a; = 6 — m« ; a;3 — a;* + 8 = i VaS"^^; ow; — 6 =s are equatious. NoTK.— The part that precedes the sign of equality is called the first member or l^ hand aide of the equation; the part that follows the sign of equality is called tJie second member, or right hand side of the equation. 109. An identical equuztionj or an identity as it is termed, is an equation such that any values whatever may be substituted fpr the letters it involves without destroying the equality of the two members. Thus, a« — a?* = (a — a;) (a + x) 4 (a + 6) (a -f 6) = 4a« + 8a6 + 46« i(a ■{■ X) -\- i{a — x) = a 1 are identities, because no matterwhat numerical value may be assigned to a and x I or to a and o, the members J are equal to one another. ^ 110. All other equations are called equations of condi- tion, and the equality existing between the members holds only when particular values are assigned to some particular letter or letters involved. 111. The letter or letters for which such particular values must be found, are called the unknown quantities^ and are generally represented by the last letters of the alphabet, oc, y, «, to, &c. 112. An equation is said to be satisfied by any value which may be substituted for the unknown quantity with- I out destroying the equality of the two members of the equation, 82 SIMPLE EQUATIONS. [Skot. VI. — 20a; — 12 = 0. 2, 6, and — 7, are the roots of the equation x'i — 89a; := — 70. / 116. An equation which involves only one unknown quantity is said to be of as many dimensions as is indicated by the e3q)onent of the highest power of the unknown quantity that occurs in it. ThiiB ix — ^ 11 '\ nre eqvM^ouBqf one dimension or simple ' (equations, or equations of the first 2a{x — TO) + a; =: 6* — »»J degree. to« — a; + 80=a0^are equations of tteo dimensions, or qutidratic ex* + 2aa; = & } equations, or equations of the secmd degree. 4a;3 — 112a!2 j. 1093. _ 27 = o) *™ equations of three dimensions, ~ V or cubic equations, or equations of x9 — 15a;8 + 74a; — 120 = o) the third degree. X* — 74a;>-l- 12% = ") are equations of four dimensions, or {.biquadratic equations, or equations 354 _4a;3 4- 6a;B - 4a; -5=0j of the/oM^ degree, ^116. It will be shown hereafter that an equation involv- ing only one unknown, has as many roots as it has dimen- sions, and only as many. Thus, a simple equation has only one root, a quadratie equation has only two toots, a ottbio equation bao only three roots, &c. /„ 117. The solution of simple equations involves the following principles : — . . I. Any term may be carried from, one side o/the equation to the other, or transposid, as it is termed, by changing its sign. Thus, if 4* — a = 7 + m, then 4a; = 7 + m ■{- a, this being equivalent to adding 4- a to each side of the equation (Ax. n). So if 2a; — a =: 46 + a;> tlien 2a; — a; = 4^ -I- a, this being equivalent to f^<)ding -f ^ a^iuation can be reduced to the form ax = b. If it be possible let ax = b, have two dissimilar roots p and y. Then a^^=:b and also ay = bftind by subtraction a^ — ay = 6 — & e= 0, that is, a(fi — y) = 0, which is absurd, bccaupo, by supposition, (i — y does not = 0, nor does a = 0. Tliorefore oa; = 6 cannot have two roots. , 119. From Art. 117 we get the following rule for solving 4 simple e(]|'aation involving only one unknown (][uantity. ■V 84 SIMPLE EQUATIONS. [Shot. VI. I. Clear the equation offractionSf and, if necessary, of brackets also. IT. Transpose all the terms involving the unknown quantity to the left hand member of the equation, and the remaining terms to the right hand member. III. Collect, by addition and subtraction, as far as possible, the several terms of each member into one term, i. e., reduce each member to its simplest form. IV. Divide each member by the coefficient of the unknoton quantity, Ex. I. Given 8a: + 7 = 2a; + 43, to find the value of ar. SOLUTION. 8x + 1 = 2a; + 43 .1) 8a; - 2a; = 43 - 7 • (n) = (i) transposed 6x = 36 (in) = (ii) collected. X = 6 (IV) = (III) T 6. a; a; X Ex. 2 . Given — + — = 7- + 14 to find the value of x. 2 3 4 SOLUTION. X X X = - + 14 4 6a; + 4a; = 3x + 168 6a; + 4x - 3a; ^ 168 7x = 168 a; = 24 Ex. 3. Given 3a; + of*. (I) (n) (III) (IV) (V) = (i) X 12, the l.c.m. of 2, 3, 4. = (11) transposed. = (ill) collected. = (IV) -f 7. 2x + 6 llx - 37 = 5 + — — to find the value 6 2 SOLUTION llx - 37 2a; + 6 3a; + — - — = 5 + 5 2 30a; + 4a; + 12 = 50 +'55.r - 185 30x + 4a; •* 55a; = 50 - 185 - 12 -21x = -147 a;=7 0) (n) (III) (IV) (V) = (i) xlO (l.c.m. of 2 and 5). = (11) transposed. = (III) collected. = (IV) T - 21. ART. 119.) SIMPLE EQUATIONS. 85 Ex. 4. Giren x + 21 -9x 5a: + 2 2x + 5 - = 5iV ;; to find the value of x. ' 6 fiOLUTIOM 29 + 4X 12 27-9* 6x+2 2x + 5 29 + 4* ^+__ -_ = 6^ - — ^-_ 12X+81 - 27x - 10a: - 4 = 61-8a: - 20-29-4a: 12x-27x-10x + 8x+4x = 61-20-29+4-81 -13x = -66 x= 5 (I). (II)* (III) (IV) (V) = (i)xl2. = (ii) transposed. = (m) collected. = (IV) T - 13. * Note.— The student must remember that the separating line of a frac- tion acts as a vinculum to the numerator, and that in clearing of fractions a minus sign before the fraction has the effect of changing all the signs of the numerator. Ex. 5. Given 6x + 1 2x - 1 2a; - 4 15 1x - 16* SOLnTION. id the value 6a; + 1 2x - 1 2x - 4 7x- •16 30x - ■ 60 7x- 16 30a:- 60 15 5 6x + 1 - 6a; + 3 = '* "^ 7a: . 16 f 28a; - 64 = 30a; - 60 28a; -30a; = -60 + 64 - 2a; = 4 X = -2 (1) (II)* (III) (IV) (V) (VI) (VII) = (i) X 15. = (ii) collected. = (III) X (7x - 16) = (iv) transposed. = (v) collected. = (VI) + - 2. * Note.— When one of the denominators is a binomibi or trinumial, it is commonly best to first multiply each member by the 1. c. m. of the other denominators, and reduce the resulting equation as much ae possible before multiplying by this compound denominator. ThJs is especially the case when one of the remaining denominators contains tV -i others, as in this example. Ex. 6. Given i (x - J) = ^(x - J) - i(x - ^) to fin t the value of X, M SIMPLE EQUAttONS. [SW3T. VI. SOLUTION. 3(«-^) = 4(x-:J)-6(x-^) 3x-^ = 4x-n-6x + 2n 3x -4x + 6x = -n+ 2n + ^ 5x = n + 1* 25x = 5n + 3n 25x s 8n X = -sftjU (0 (") (III) (IV) (V) (VI) (VII) (VIII) = (I) X 12. = (ii) cleared of brackets. - (in) transposed. = (iv) collected. = (v)x6. = (vi) collected. = (VII) T 25 a b Ex. 7. Given -r- + — bx ex + -— + -—= ff, to find the valu.. of x. dx fx ' SOBUTION. ah c ^ _ bx ex dx fx * UiiJj + fe^rf/+ bcY+ bed^ = bcdfgx acdf+ bHf-^ bcY+ bcd^ X = bcdfg (1)1 (r) = (i) X bcdfx (111) = (II) T 6c(//fi: (a + b)x Ex. 8. Given ^^ :- + X a--6 (a + 6)x SOLUTION X+ 1 X + 1 a + 6 to find the value of x . a-b a*^b'^ a + b (a + 6)*x -f X = (i-b) (x + 1) a*x + 2abx +b^x + x -• ax-bx + a~b d^x + 2a6x + 6'x + x-ax + 6x=a-fc (a* + 2aft + b^ + 1 - a + 6)x = a - 6i (v) a-6 (VI) (0 (n) (III) (IV) X = a* + 2a6 + 6" + 1 - a J 6 = (i)x (a*-6») = (ii) expanded. = (ill) transposed. = (iv) factored. = (v) T Goef. of X. EXBROISB XXXII. Solve the following equations ;— X x 1.. +-=»--. Arn^. 1ft. 1 SIMPLE EQUATIONS. 17 2. 2a; - — = X + 4. X 3. 2a; - - + - = 3x- 11 + X + 9. 4. 2x - 7 + 3x - 1 X + 8 - 2x. 5. 2 - - X — 5 = 3 - X -7 levalubof X. 2x 3x + 1 6. 4x - -- = -— — + X + 6. 3i 2 1. 2x - 16i = a^ + Jx. 8. X + 3 X + 4 - 16 = - X + 1 le value of X. §. 4x - 1x 2X + 19 5 = 15 - 7x + ll 3JX-7 10. - + 3^ = 21 ^2- 11. 12. 8x-l7 14X + 17 11 4x + 4 13 = 3x - 31 -X 3 X = 2 + 14 -3x 13. 3x - 4x-5 + §x = 17 + 2 -ex 8x + 1 14. 3^x - 5 2f X - 9 12 X 15. X + 10. 21 - 7 5 X "■I 3( 5 1^ 5(x- 1) 97 -7x = 1 - 8 7ix - X + 2 9 - Six 8 6 = 2x - 2^ = X - iV(3^- 11) -9« 'J* f* *** »j» *** lA/ ti(/ «!/ vv vv 6 5x 12" + 4. 20 -X 18. 2x + 10 = |x IJr. - 36 + 20x 26 - t^ = 3U - 5X + 20 9x -la* S(^ Simple iiQUAtio^d. tS«o*. XL 20. 33^tf - 3x - 12 + Ix 9 + 6x Sx-ia 11* -17 9 Ox + 20 4x - 12 6x-4 + — r 10 16 8 36 21. 22. Ix + ^(x + 3) - Kx - 4) = i(x + 6) + 31^. 7(x + 2) 23. 6x ^^ — - = 5 + 2(2x4-1) l'r-3x 24. 5x - 2(Bx2 - 9) 3 + 2x = 9- 6x + 9 3 + 4x* 25. 2(x + 2) 3 Yx - 13 6x + 7 3(1 + 2x) 9 26. ax + b = c. 27. Sax -b^ = bc- ^ax. 28. 46x - 3x = J(o -b^+ 3ox). 3a - ax b d •' bx + 4a a' - 36x 29. 2a2x - 30. 31. 32. 33. 34. 35. 3a - X = X - 2x + a 4a - 3x (a- b)i 2a ax -6 ex a Sa' - 66x - bx = ab^ + ax 2a b^x ax - be s -^ . b- a 11a -3x 6a -5x a + b 2x a + b (a + xy 0-6 a-b d^- b'^' - abx = Jx' abc i(a+6) bx d^b' a 36. 37. 3 + 1-72X - 2-21X = •203x (a+by = 3cx - b'x 2a + 6 (_a + by 38. •3x + x(6 - a) = 3a - •23x. 39. K* - i) + 4 { 1 - (a; + §) } - I {a; - (1 + ix){ = X + f X, 8ax -b 56 1c — . = 4 -, 6^ _ - 9 40. 41. (a=» - x) (6'' + x) - 3a6 (1 - x) = (x - o) (c - x), AftTB. 190, 121.] SIMPLE EQUATIONS. 89 PROBLEMS PRODUOIMO SIMPLB BQUATIOMS INVOLTINO ONLY ONI UMKNOWlf qUAHTITY. ^ 120. A Problem is a written statement of the relations existing between certain quantities whose values are given, and another quantity or other quantities whose values are to be found. The solution of problems consists of two distinct parts : I. Th e JJlgebraic State ment^ _or hnf^fLj the statement. This consistTin the translation of the problem into algehraic language^ i. 0., in expressing the conditions of the problem, the relations between the giren and th« unknown quantities, by means* of signs and symbols, bo as to indicate the operations described ia the problem. II. The solution of thejresjolting equation. 121. It is with the former of these parts, i. e., '' the statement," that the student experiences the chief difficulty, the naturb of problems being such that they admit of no general rule for their statement. The student must, there- fore, be left very much to his own ingenuity, and he can expect to acquire facility in the operation only by long continued practice. He will, however, be very much assisted in his efforts by attention to the following general in»tructions for making : — THE STATEMENT OF PROBLEMS. I. Read over the problem carefully^ until its conditions are clearly apprehended^ and it is distinctly understood what is given and what is required. II. Represent the unknown quantity by x, and set down in alge- braic language the relations existing between it and the given quantities, as described in the problem, or in other words, indicate upon x, by means of signs, the same opera- tion that would be necessary tc verify its value in the equation if that value were jlready determined. G 90 SIMPI4B EQUATIONS. [8«ct: VI. . i NoTK.— Before oommenoing the exercise the begianer is partioularljr directed to ntttdy canity the solntion of the prolimiiMrjr problema, iu order to observe the modes of proceeding to mako the statement. Ex. 1. What number is that from the double of which if 10 be subtracted the remainder is 44 ? SOLUTION. Here we have given that a certain number is such that when its double is diminished by 10 the remainder is 44. Let X = the number. Then 2x = its double, and 2x - 10 = its double diminished by 10 . Then, by the problem, 2x - 10 = 44, which is the required stAtement. 2x = 64, by transposition, x s 27, by division. Therefore 27 is the number required. Verification. (27 x 2) - 10 = 44 64 - 10 := 44 44 = 44 Ex. 2. Find a number such that one-half, one-third, and one- fourth of it added together Fbali exceed the Mumber itself by 4|. SOLFTION. Here we hare given that | + ) + i of a certain number > the number itself by 4i, or what amounts to the same thing, that i + ) + i of a certain number = the number itself + 4}. XX X Let X s the number ; then r- = i of it ; — = ^ of it; and — = 2 3 4 iofit. XXX And — + — + — = £ + 4}, which is the statement required. 234 6a: + 4a: + 3x = 12a: + 54 (n) = (i) x 12. 6a: + 4a; 4- 3x - 12a: = 54 (in) = (11) transposed. a: = 64 (iv) = (ni) collected. Therefore 64 is the required number. 54 64 54 Verificatim. -r- + -r- + -r = 54 + 41 ii •> 4 27 + 18 + 13i =581 681 = 58 [ Amr. 121.] SIMPLE EQUATIONS. 9& which if 10 ch that when Ex. 3. Diride the nnmber 112 into two such parts that if 21 be added to the less the' sum shall be less than one-third of the greater by the third part of unity. SOLUTION. Here 112 is to be divided Into two parts such that the less + 21 shall be equal to (^ of the greater) - ^. Let X = the greater part ; then since 112 is the sum of the two parts, 112 - X = the less. (112 - x) + 21 is 21 added to the less, and »- - i is ^ of unity less than ^ of greater. X , Then (112 -x) + 21 = — - J, which is the statement. 336 - 3a; + 63 = a: - 1 (ii) = (i) x 3 -3x-x = -l-63- 336 (iii) = (ii) transposed. - 4x = - 400 (iv) = (ill) collected. X = 100 = greater. 112 - a: = 112 ~ 100 = 12 = less. Verification. (112 - 100) + 21 = ^^^ - ^ 112 - 100 + 21 = 4** - i 133 - 100 = 33^ - i 33 = 33 Ex. 4. What sum of money is that from which if $46*20 be ' subtracted, one-half the remainder shall exceed one-third of the remainder by $50. SOLUTION. Here the sum of money is such that J (Sum - $46*20) is > by $50 than i (Sum - $46-20). Let X = the sum of money. Then x - $46-20 is $46*20 subtracted from the sum. X - $46-20 rt is half the rem., and X - $46-20 . Then X - $46*20 X - $46*20 is one-third of rem. (0. 2 '»"'- 3 3a; - $138-60 - $300 = 2x - $92-40 (ii) = (i) x 6. 3x - 2x = - $92-40 + 138*60 + $300. X = $346*20 := sum required. NoTB.— The student should verify the result in every case, as is done in the three preceding problems. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 1.1 12.2 m lit 14.0 1^ 1 L^lfuiJ^ ^ 4« ► Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716)872-4503 ^ '^ \ ^ o^ 92 SIMPLE EQUATIONS. [SlOT. VI. Ex. 5. A certain number consists of two digits, such that the right hand digit exceeds the left hand digit by 2 i and if the sum of the digits be increased by ^ of the numberi the digits will be inverted. Required the. number. SOLUTION. Let X = the left hand digit. Then x + 2 = the right baud digit. lOx + (X + 2) = the number.* X + X + 2 = the sum of the digits. 2x -f 2 + f (lOx + X + 2) = the sum of the digits increased by f of the number. , 10(x + 2) + X = number with its digits inverted. Then 2x + 2 + f (lOx + x + 2) = 10(x + 2) + x. 14x + 14 + 9(llx + 2) = 70(x + 2) + 1x. 14x -1- 14 + 99x + 18 = 70x + 140 + 1x. 99x 4- 14c - 70x - 7x = 140 > 14 - 18. sex = 108. X s 3 = left hand digit. X + 2 = 6 = right hand digit. Therefore the number is 36. Ex. 6. A can do a piece of work in 10 days, which Ji and B can together finish Jn 6 days. In what time can B working alone do the work? SOLUTION. Let X = number of days B would require to do the work. Since A does whole work in 10 days, in 1 day he would do ^ of it. Since B does whole work in x days, in 1 day he would do i^ of it. *NoTB.— If we take any number, m 6642, and ropremnt its digits raspao- lively by the letters d, c, b, and a, then d + o + 6 -f a will express, not the number, but merely the sum of its dij^ts. In order to express the number we must take into account the local as well as the absolute values of theldigits, i.e., we must remember that the first digit being so many units, the second is so many tens, the third so many hundreds, fcc. Hence (l-)^c + & + a = 6 + 6 + 4 + 2 = 17 = sum of digits. And lOOOd + 100c + lOd + a = 6000 + GOO + 40 + 2 =: 6642 = the number . And of course 1000a + 1006 + lOo + <2 = 2000 + 400 + 60 + 6 = 2i&6 = n^unber with its digits inverted. Abt. 121.] SIMPLE EQUATIONS. 98 Since A and B do the work in 6 days, in 1 day they would do ^ofit. Then J^z work for 1 day 4-B'8 work for 1 day = work of both A and B for 1 day. That is, -,»iT + - = * 3a: + 30 = 5x 3x - 5ar = - 30 - 2x =: - 30 0). (ii) = (i) X 30a: to clear of fractions. (Ill) = (ii) transposed, (iv) = (III) collected. a: = 15 = days B would require. Ex. 7. A person being asked how many ducks and geese he had, replied that if he had 8 more of each he would haye 7 geese for 8 ducks, but- that if he had 8 less of each he would only have 6 geese for 7 ducks. How many had he of each? SOLUTION. Let X - the number of ducks he had. Then x + 8 = number of ducks increased by 8. a; + 8 — g- = number of tiroes he had 7 geese. X +8 — ^ X 7 = number of geese he had when increased by 8. o X 4- 8 Hence number of geese = 8 less than — g- x 7 = 5(x + 8)- 8. Also X - 8 = number of ducks diminished by 8. ^x + 8) - 16 = number of geese diminished by 8; and by x-8 i(x + 8)-16 the question, — -— = . 6(x - 8) = 7 f i (X + 8) - 16 }. 6X-48 = *i^(x + 8)-112. 49X+392 6X + 64 = ^ . 48x + 512 = 49x + 392. X = 120 = number of ducks. i (120 + 8) - 8 = (i of 128) - 8 = (7 X 16) - 8 = 112 - 8 = 104 = number of geese. 94 SIMPLB EQUATIONS. [SaoT. VI. Ex. 8. A merchant has tea worth 4s. 3d. and 6s. 9d. per lb. How many lbs. of each must there bo in a chest of 126 lbs., which 9hall be worth £30 ? « SOLUTION. Let X =: number of lbs. at 4s. 3d. or 17 threepences per lb. Then 120 - x = number of lbs. at 5s. 9d. or 23 threepences per lb. I7x = worth in threepences of a; lbs. at 4s. 3d. per lb. 23(120 - x) = worth in threepences of 120 - x lbs. at 5s. 9d. per lb. 2400 = number of threepences in £30. Then llx + 23(120 - x) = 2400. l7« + 2t60 - 23x = 2400. • 17a: - 23x = "2400 - 2760. - 6x = - 360. a; = 60 = lbs. at 4s. 3d. per lb. 120 - 60 = 60 = lbs. at 5s. 9d. per'tb. "Ex. 9. Divide the number 90 into four parts such that the first increased by 2, the. second diminished by 2, the* third divided by 2, and the fourth multiplied by 2, shall all be equal to the saiiie quantity. SOLUTION. • Let X = the quantity to which the 1st part is equal when increased by 2. Then X - 2 = Ist part ; a; + 2 = 2nd,part ; a: X 2 = 3rd part ; a: -f 2 = 4th part. Then (x - 2) + (a: + 2) + 2a: + 1 = 90. x-2+x+2 + 2x + ^ = 90. 4x + «=90. 8x + x = 180. 9x = 180. x = 20. x-2 = 20-2 = 18 = Istpart; x4 2= 20-(-2 = 22 - 2nd part. 2x - 20 X 2 = 40 = 3rd part ; ^ = V =10 = 4th part. 9^ abt. m ] BDfPLB EQUATIONS. 95 Ex. 10. A workman is engaged for n days, at p cents per day, upon condition that for every day that he is idle instead of receiving anything he shall forfeit q cents. At the end of the time agreed upon he received' e cents. Required the number of days he worked, and the number of days he was idle. SOLUTION. Let X = the number of days he worked. Then n - « = the number of days on which he was idle. px = number of cents he received for x days work. q(n--x) s number of cents he forfeited for (n-«) days idleness. Then px - q(n - x) = c. px - gn + jx = c. jpx + jx = c + gn. (P + ?)* - c + qn. c + qn X = = number of working days. c + qn np + nq-'C-tiqnp-c ~ = number of idle days n - p + q p + q p + q EXIBOIBK XXXIII. 1. Required two numbers whose sum is 47 and difference 13. 2. There are two numbers, one of which is greater thain the other by 21, and the quotient of their sum by the less is 3 ; what are the numbers? 3. After paying away | and ^ of my money, I had $8'60 remaining ; how much had I at first? 4. Find a nuiiiber such that if 21 be taken from it, and the remainder divided by 8f, the quotient will be 6. 6. Divide 64 into three such parts that the first divided by ^, this Meond by 3, and the third by 4, shall all give the same I of my debts, and then paid f of the remainder, and owe $192 ; how much did I owe at first? >ve of cattle is disposed of as follows: ^ to A, ^ t» wad the remainder, which was 9, to D ; how manj there in the drove? . ■4I- Vi.'ji' -'■■'''■:■ SIMPLE BQUATIOlfS. [Stct. VI. >. 8. A farmer has two flocks of sheep, e«eh containing the same number; bat when he had sold 19 sheep from one flock and 91 from the other, the former no\^ contained twice as many as the latter. Required the number originally in each flock. 9. Find a number whose fourth part exceeds its seventh part bye. 10. What number is that the double of which exceeds f of its half by 26. 11. Find a number such that increased by one-half of itself the sum shall be 39. 12. What number is th*at which exceeds the sum of its half and its third parts by 17 ? 13. Find a number such that when 15 is taken f^om its double, and to half the remainder 7 is added, the sum is greater by 3 than f of the original number. 14. What number is that to which if 11 be added, two and *-half times the sum shall be 86. 15. Find a number such that one-half, two-thirds, and three- fburtiis of it added together, shall exceed 1| times the original number by 21. 16. A farmer sold a load containing a certain number of barrels of apples for $36, and he afterwards sold a second load at the same rate, but as it contained 6 barrels less than the formed, be only reoeired $21. What was the price, per barrel, and what was the number of barrels in each load t ' 17. A person starts to walk from Toronto to Bramptoa at ibe raite of 3| miles per hour; precisely 28 1 minutos aftbrwArds another person starts from Brampton to walk tor Toronto «t the rate of 4 miles per hoiir, and they meet one another exactly ludf- way between the two places. Required the diiitance from TArattto to Brampton. •' r. v^ 18. In ft certain grist-mill there are three runs of stoneif'; thCk first of which can empty the granary in 72 hours, the SfMioBd ip 84 ho«rs, and the third in 90 hours. Two teanns itreJirti gAdg x i di!*bwing whfiat and storing it in the granary, and of IhnirdiMi first ti«lt abt. m.] SIlfPLB IQUATIOI^S. 9T < K X 19. If from the nnmlMr of the year in which all the tlaret in Canada reeeiyed their freedom, tiw number 1T80 be takeni three times the remainder increased bj 1620, will give the year of the celebrated Indian massacre of Lachlne, and if the two dates be added together, one-half their sum increased by 116 will giro the year 1862. Required the date of the abolition of slarery in Canada, and also that of the massacre of LachineT 20. Divide $7400 among A, B, and 0, so thftt A shall hare $120 more than B ; and $106 less than A. 21. A pupil receires 24 music lessons and 32 drawing leesoi^ in the quarter, and tbe former cost her $3 more than the latter ; if, however, she had received 32 music lessons, and Only 24 drawing lessons, the latter would hsive cost her, at the same rate, $10 less tiian the former. Required the price per lemon for music and driiWing ? - 22. A library contains twice as mahy volumes on General Literature as on History, 1| tiiiies as many volmnes on ffistory as on Biography, as many volumes on Biogc^ihy as on Traveli, and three times as tMuny volumes on Travels as on the S^ciMoes, and the number of volumes on the Sciences is TO. Re quired the number of volumes in the library ? 23. The Rideau Ganal is six iniles less than four timet as long as the Niagara River, and their combined length doubled' and decreasecli tgr 100 miles, exceeds the length of the Great Western Railway by one mile. The G. W. R. being 229 miles loi^ required tbfli lenigth of the Rideau Oanal, and also that of the NiagarikRiv^r? 24. A o4n doia piece of work iu 12 days, which B can finli^ in 16 dafs^i^ O'^in 18 days. Now A and B work togethir sill it (br'l IliiqrfS «BfatidCI>woark togetlwr at it for two days : in whi^ ksee finish the worte remaining to be done t ii|ili|^»4o toe another T . ^-^ ; towns two fields and a horse, the latter being iM|^i|(i,' WiitOktB to sell the first field with the hociafn it for $25' time •Tlv 98 SIMPLB BQUAHONS. (Mot. VI. ?>.*■,...' more than he aiks for the second field alonei but for the seoond field with the horse in it he twks doable as mAii||m|flii pass in order that the fore-wheel shall ha»e;«i|d>j%pi|p|fii8 mora than the hiiidHfrheel? i ^ -./» rf ^^iv lt * ^>l;f^ 36. The hour and minute-hands of a ^Mi^'ta0if'fm0tiii' iit ABT. m.] smnJI IQVATIONS. noon. When mi4 how often will thej be together during the next twelve honn ? 37. Diride the namber 96 into two sueh parts that when the greater ie divided by 7 and the less mnltiplled by 3, the inOi of the quotient and prodnot shall be 30. 38. Divide $3560 among A, B, and 0, so that A shall have half as mnch again as B ; and that shall hare half as much again as A^ 39. A steaaner nutlcei th^tioim trip from the head of Lake Ontario to Montreal in 28 hoars, the conent being in its favor. When retoming it is found that in ascending the St. Lawrenoe (three-sevenths of the entire trip) the rate of sailing is 6 miles per hour less than the averatge rate in its downwfurd journey, but upon entering the lake It is enabled to increase its speed 2 miles per hour, . and again reaches Hamilton, at the head of the Lake, in H of the thne it would have required had the rate been uniformly ]^m same as when, ascending the river. Required the distance ^U^^ Montreal and Hamilton j and the rat^ of sailing 7 , 40. A gefitlea»aii bequeaths Ins property as follows :'^To his eldest chili' |l|ft>^ve8 ^1800 and ^ of the rest of his property ; to the secomi t^Cjd,fU^^0imd iof the part now remaining^; to the third tl^^,f|^f|ijL8^ and ^ of the part now remainifl^ and so ;^ ^ ,1^^, til|iUi;avr%iigemeut his property is divided eqiuiUy ainon|:^)^H|K|^ftt9. , |[ow many children were there, tad ? i wastiiat|i»#tiMi»vofeach? .\^y* 41. A pBtfl i ln iwmber consists of two digits, whose difihrence is 7fRi|«»,i^(|^'|;|i^>^ one be^lg the greater; ;lR)9n^e Btajfb^ ^jj|qi|ttin]ffite digits itgives^ ^ number. . . ; A, Bj'and 0, so that A shsitTl^ve ||pl^tti[»d,C«ilhares togel^erj and that © 'OuS^ : :^ " - ■■ • ■ ^ ■■* ^-.^ ■", im ^,,^tQhMA to plant witit' j| iitiii |;iti|iftt,when he has as mpdpty rowt tJhfire Are TiJlictfm'Zfmaining, but if he poto fi Wf and incKBiiiSjiHie number of roib by 6^^ie ^ees r^^inii^. 'pftatwas tfieqnation. 124. As Btated in Art 1.22, in order that the equation may be detenmnate, there must be aa many indepgtdmi oqnationa aa there are unknown quantities invohed. Now eqwxtiont are »aid to he indqpmdmt whm they exprete diff&rent reiationt between the unknown qmnHHee. Nora.— Tbat It, the two or three eq«attmw given aint not be derived flrom one another hjr niere miiltlplloatlon, or dMftoi, or mibtrae^ton, or addUkm. ThiMi If « -f* y » 8 be one of the eqnaMona, ft woidd bewwtew toaaweiKtewithUa» + 3|y = 16,or4«q>|ys$4^, 0» « + ^=±8 + y» y — 8« ss 8 — te, tie., beoaose theae eavations, thoogh tme in thenaelvM,. expreM no new retation between the vnknown giantMea, and aiie all redadble to the Jbrm of « + y = 8, having obvlOndy bMn derived .INnb it by mere addition, mbteaetlon, nniltipUealion, or divinon. 1^5. Simultaneous equatioils are solved by eUmingfion^ as it is termed, i. e., by so combining iSie ^Ven equations astogeliiid'Of one of the unknown qnantkies, and tkns to train ^Grn» thtei a new equation involying onfy one unknown. 129. (There are three methods of eliminating one of the unknown quantities, and thus of solving simultaneous eiqna* T^^^^ji9l»:mtigMmt$^ qftke^mmiitfwtdeiin ioOlmimUe Mfj^t^fke a>efifikiU$qftha$ fumHty dmilar, .M.^-'ip;mim|Miff ; ^ ' jfr^parcd: ^ a (HI) (iv) •OLUnOM. Here as the eoef. of x !• the lame In both eqnationi there it no neceieitj of mnltiplying, and we accordingly subtract at once. - (II) - (I). s (III) * 10. Thep 4x - 8y = 4x - 6 ^^ 6 (t) - (i) by snbstitating 3 for y. 4*«ia xs 8 Therefore ralaes are x s 3 and y s a. Ex. a. QiTcn 4x + 8y = 43-^ 8x + 8y = 43-> ^ f^^ ^^^ YtAuw of X and y. -aysin 4x + 3y s 48 3x - ay = 11 8x + 6y - 86 9x - 6y s 33 I7x ::119 (0 («) (ni) (IV) (V) SOLUTIOn. « (1) X 2. = (u) X 3. 3 (m) + (iv). We' add becanse - the signs are unlike. = (V) * It. s (i) with 7 substituted for x. x= 7 (VI) 4xf8y s a8+3y - 43 (vii) 3yr:16 y s 6 Therefore values are x s 7 and y » 5. iron.—We oaa always prepare ttaeequatioiis Ibradditfon w inlilraetion liy multiplying eaoh by that ooef. of the nnknown to he eliminated, which is given in the other equations. Sometimes, however, it is not neeessary to multiply tof/^tequations, but we can And by inspeetiea a mnttiplier Ibr one only, which will at once prepare the equation ibr elimination. Thus, '^ *^ ~ ^ ^ Jg } be the equation as given and we uMi to eUmi- nate x, we may multiply the lower equation by 4 and the upptt Vf% and then sabtnet, but we may obviously attain the same end, in the eitaina- tionof X, by simply multiplying the lower equation by 3, and then sab- triwting. Similarly if we wish to eliminate the y, bistead of mhltikdyliig the niqperequation t^ 9, and iiielowerby 8, we may prepare the^woeqnattons lor addition by stalky multiplyiag the upper oy 8. Awr.UB.] BIMPLB BQUATIOlfB. 108 . ' > to find the values of x and y. Ox — ay s n / a« + y s m 6«>ay a n o'x •!■ ay s am a'x 4- 6x = am + n (c^ 4- 6)x = am 4- n am+n *'"a»+T a^m-f Oft •OLunov. (I) (n) (in) (!▼) (V) (VI) 3 (i) X a. a (U) + (III). B (iv) factored, a (v) + a» + 6, (vn) s (i) with value of x lubsti- I tuted for x. y = a^m+frm-i^m-aii 6m-an 0^+6 a» + 6" ILDIINATION BT SUBSTITUTION. RULI. 128. I. Fund from one of the given eqwUiom the value of the unknown to he eliminated in term* of the other unknown quantity. II. Subiiitute thi$ value in the remaining equation for the same unknoum qiumtUyf and there vriU retult an equation containing only one unknown quantity. Ex. 4. Given 2x - Jf = 1 ? ^ find the values of x and y. . 7« + Oy = 16 > SOtDtlOH. ax-y= 1 (I) ■ ' . 7x+9y= 16 (n) (HI) • ya2*- 1 - (i) transposed. Vx+9(2x-I)ai6 (IV) ' (u) with 2x - 1 substituted for y 7«+18x-9 = 16 (V) = (IV) expanded. 26x = 25 (VI) » (v) transposed and collected. X- 1 (vn) = (vi) * 25. y=2x-l = 2-l= 1 f^Vui) = (m) with value of X substituted. 104 SaSFJM BQUATI0N8. [Sbot. yi. iy - *Ix Ex. 6. GiTen 6s - — - — = 8 4v t« -2v f *® ^°,* the values 7x - _i + 51 B3y-8) of' »»d y. 4y+7x 6* = 8 6 11 6 SOLUTIOl.' (0 4y 7x-2y 23x-4y =48 539x-244y=-ft38 48 + 4y X = 23 /48 + 4y\ 639 1 j - 244y = - 528 26872 + 2166y -_-Jl-244y=-628 3456y = 38016 y = 11 48 + 4y 48 + 44 X ^ = 4 00 (n) 6x + 7y - « = 63 ; . (III) (IV) 4x + 8y - 6« = 44 = (I) X 2. 10y-ll«=26 (V) (VI) = (IV) - (II). 6a: + 12y - 9« = 6ft. = (I) X 3. &y-8«= 3 (VII) (V) = (VI) - (III); 10y-ll2=26 > 5y - 8z- 3) (VII) . 10y-16«= 6 (VIII) = (vn) X i. 5« = 20 (IX) = (V) - (VIII). s= 4 (X) = (IX) ♦ 5. 5y - 8« = 6y - 32 = 3 (XI) = (vu) with 4 for «. 5y = 35 (XII) = (XI) transposed. y = t (XIU) = (XII) * 5. 23:+4y-3« = 2x + 28-12 = 22 (XIV) = (xiii) with 4 substituted for z and 7 for y. 2x'= 6 (XV) - (xiv) transposed. X = 3 (XVI) = (XV) T 2. 181. When there are more than three unknown quanti- ties, and consequently more than three equations, we pro- ceed in a similar manner, so that for solving a set of 91 equations involving n unknown quantities, we use the following:— Bulk. I. Combine one of the given n equations with each of the others separately^ eliminating the same unknown quantity, i there wUl result n - 1 equations^ involving n - 1 unknown quan^ tities. tl. Combine one of these equations with each of the others sepO' rately^ eliminating a second unknown quantity; there will result n - 2 equations inoolving only n - 2 unknovm quan* tUUs. til. Continue thus combining and eliminating untU an equation is obtained involving only one unknown quantity. l08 SIMPLB ^QVATiOird. Bitot. Vt. ly. Having solved this eqwUiwi and thw found the value of one unknown quantity^ substitute this value in one of two pre- ceding e^fluUionSf and thus obtain the value of a second unknown qwoUi^; then si/Astitute the values of these two unknown quantities in one of the three equations which involve only three unkn&wnSf and thus determine the value of another f and so on^ until tUl the values are found. Ex. Oiven »+ x+ y+ s=14 31, - 2a: + 4y - 3* = 6 y^^ g„^ ^^^ ^^j^^^ ^f 2i» - 6a: + 2y + 4s = 24 ( x, y, and *. 4i, + 3a: - 3y - 25 = 3 V + ic + y^r s-14 3v -2x + Ay -3z~ 6 2» - 6a: + 2y + 4a = 24 4w + 3x-3y-2s=: 3 SOLUTION. (0 (") (m) (IV) 3t, + 3x + 3y + 3« =s 42 2v + 2x + 2y ■<- 2s =: 28 4v + 4a; -t- 4y i- 4a =: 66 5x - y + 6« = 37 tx -2«=: 4 X + 7y + 62 = 53 36x - 7y + 42« - 269 36x + 48s = 312 3x + 4s = 26 14x-4«= 8 17x = 34 X = 2 3x + 4s = 6 + 4s = 26 s= 6 6x - y + 6s = lO-y + 30 = 37 y =: 3 (XX) »+x+y+s=^D+2 + 3 + 5 = 14 (xxi) v = 4 Therefore the required values are v - (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) (XIV) (XV) (XVI) (xvn) (xvra) (XIX) = (I) X 3. = (I) X 2. = (i)x4. = (T) - (Ti). = (VI) - (111). = (VII) - (IV). = (viii) X 7. = (x) + (XI). = (XII) T 12. = (IX) X 2. = (xiii) + (xiv). = (XV) T 2. = (xiu) with 2 for x. - (xvii) transp. and -f 4. = (viii) with 2 sufostit- uted for x and 6 for s = (xix) transposed. = (i) with values of x*, y, \ and s substituted. 4, X :: 2, y = 3, and s = 6, Abt. 181.] SIMPLE EQUATIONS. EXIROIBB XXXY. 109 Find the values of the unknown qnantities in the following eqttalions :— • 1. 2«~3y + 4«- 28 3x + 4y-tiz- 26 4ar - 6y - 6« = 16 3. ar+ y+ « = 2ar + 3y+4«=: -4 3x + 6y+1z- -6 } 5. x+ y+ z+ V- 2x-Sy- «-2«= 11 a: + 2y-3« + 6» = -l7 3« + 2y-4a- v = - 6 7. X + y = xy X + 2 = 2X2 2(y + «) = 3y« } 9. ax bx cy + by = c "^ + cs = a > + 08 = 6 ) * 11. X + y + zsa-\-b-\- c 6x + cy + «« = ex + ay + 6« - o'* + fc'** + c* 8. X + 3y + 2x = 6 "^ 3x + 5y - 22 = m >• 4x - -y + z=^n j 10. » + X + y = 13 » + x + 2= 17 » + y + 2= 18 x + y +2= 21 12. x + a(y + 2) = m'\ y + o(x ■>- 2) = n > 2 + o(x + y) = p 3 PROBLEMS Producing Simultankous Equations of the First Deorsb. Ex. 1. What fraction is that whose numerator being doubled and denominator decreased by unity, the value becomes |, but the denominator being doubled, and nnnferalor Increased by 5, the value becomes | ? 110 SIMPLE EQUATIONS. [SiOT. VI. SOLUTION. L^t — = the frAOtion ; then x = numerator andy = denominator. y 2x x+ 6 = 4 2y ex - 2» = - 2x - 2y = - 2 1 - 2y s - lOj (0. (u). (HI) (IV) IS (i) redaced. = (n) redaced. 4x = 8 x = 2 12 - 2y = - 2 -2y = -14 y=» Therefore the fraction. is f . Ex. 2. A certain field is rectangular in form, and its dimen- sions are such that if it were 4 chains longer and 3 chains wider its area would be 103 chains greater than at Jtfesent, but if it were 2 chains shorter and 7 chains wider, its area would be 119 chains greater than at present. Required its area. SOLUTION. Let X =: its length and y = its breadth ; hence xy s its present area. Then x + 4 = its length when increased by 4 chains. j^ •)- 3 = its breadth when increased by 3 chains. (x + AXy + 3) ~ its area, which is greater than xy by 103 chains. Also X - 2 = length when decreased by 2 chains. y + 7 = breadth when increased by 7 chains. Then (x - 2)(y + 7) = its area, which is greater than xy by }ld Qbains. Hence the two rec[uired equat)0Q9 ^fe Arts. 129-131.] SIMPLE EQUATIONS. lit (x + 4)(y + («-2)(y + 3) = 7) = xy + 103 ) • xy + 119 J xy+3x + 4y+12 = xy+7x-.2y-14 = - xy + 103 :xy+119 8x + 4y s c91 tx- . 2y: = 133 14x. -4y: = 266 ItXs s357 X : = 21 3x + 4y 1 = 63 + 4y == 91 4y = 28 y + l (0 («) (III) (IV) (V) (YI) (VII) (VIU) (IX) (X) 9 (i) expanded. 3 (n) expanded. - (hi) traniposed and col- leoted, = (iv) transposed and col- lected. ~ (vi) X 2. = (V) + (VIl). = (viii) * 17. =:(v)with 21 substituted for X. Hence the area =xy = 21x7- 147 chains. Bx. 3. Two plugs are opened in the bottom o^ a cistern con- taining 664 gallons of water ; after 6 hours one of them becomes stopped, and the cistern is emptied by the other in 20 honrs ; but had 8 hours elapsed before the stoppage occurred, it would only have required 16h. 36m. more to empty it. Assuming the discharge to be uniform, how many gallons did each plug hole discharge per hour 7 SOLUTION. Let X and y - rates of discharge per hour of the two plug holes. Then 6x -h 6y - No. of gals, discharged in 6 honrs. And 20y = No. of gals, discharged by second in 20 hours. Then 6x -l- 26y =: 664 (i). Also 8x -i- By s No. of gals, discharged in 8 hours by both. 78y ' And 15§y = -g- = No. of gals, discharged by 2nd in 16h. 36m. 78y Bx + 8y + -g- = 664 (") 40x+118y= 3320 120x + 620y = 13280 l?Qx + 354y= 9960 (III) (IV) (V) = (u) reduced. = (I) X 20. = (III) X 3, i -i L^ BDfPLB EQUATIONS. [Sbot.YI. 166y= 3820 y= 20 6« f 26y - 6« + 620 s 664 tfx a 144 (VI) (vn) (VIII) = (VI) * 166. s (I) with 20 lobftiiated tory. x» 24 Therefore ratei of discharge are 24 and 20 gals, per hoar. ABT8. EZIBOIBB XXXVI. 1. Find two numbers such that seven times their sam in- creased hj four times the less is equal to 60, and twice their difference increased by three times the greater, is equal to 16. 2. Find two numbers whose sum is equal to a, and such'that b times the greater is equal to c times the smaller. 3. Two tons of hay and 36 bushels of oats cost $44, but if oats were to fall in price 20 per cent, and hay were to rise in price 33} per ^nt.. they would cost $61*20. Bequired the price of hay and oats. 4. A rectangular garden is of such dimensions that were it 20 yards longer and 24 yards wider it would contain 4180 square yards more than its present area, but if it were 24 yards longer and 20 yards wider, its present area would be increased by only 3860 square yards. Required its present area. 6. Find two numbers such that the sum of one-half of the first and one-third of the second shall be 11 ; and one-third of the first shall be greater by unity than one-fifth of the second. 6. Divide the number 144 into two parts such that i of the greater shall exceed I of the less by 1^. 7. Divide the number 48 into two parts such that the greater contain 4 as divisor four times as often as it contains the as divisor. 8. Find three numbers such that the first is equal to ^ of the other two, the second exceeds half the sum of the other two by 6, while the third is less by 3 than | of the sum of the first and second. 9. In 4000 lbs. of gunpowder there are 3240 Ibs; less of sal» phur than of charcoal an4 saltpetre, and W^ lbs, less of char, £, throw dice upon condition that he who has the lowest throw shall give all tiie others the sum which they already have. Bach loses in turn, commencing with jSf and at the end of the fifth game each has the same sum, 983* How much had each at first 7 SECTION VII, INVOLUTION AND EVOLUTION. 182. Involution is the process of finding any proposed power of a quantity. 188. If the quantity to be involved have a negative sign, then the signs of all the even powers will he positive, and the signs of all the odd powers, negative. Thus,^ - «)* = - o X - o = + a^ ( -. o)' = ( - a)* X - a = + a* X - a = - o». ( - a)* = ( - a)» X ( - a)" = + a" X + a« = + a*. ( - a)» = ( - a)* X - a - + a* X - a = - a', Ac, 184. If the quantity to he involved have a positive sigp,^ then all its powers, both even a|)4 odd^ will have the positive sign, AJtn. 182-187.] INVOLUTIONi IW NoTB 1.— It fbllowi that no oven power of tmy qiumtltjr o*n be nejpttlro. ud that all odd powen will IwTe the •erne alga «s the quantity ftea which tbey are derired. Nora a.'-8inee(a~6)tsat — lob + ft* la a poalthre qnaatHj, K ibl* loiiathata*+&*>9oifr,aeolherwliea«+6*— Sobwonld be negathre. HfBoe the nun of the aqnaiee of any two qtMatltiea li greater than twloe their product. 186. Since (o^)** s a«* x a*» x a^ to n faoton, it follows (Art. 53) that (a**)* ^ a"^, and henoe we find a required power of the given power of a quantity by mnlti- jplying the exponent of the giten power by that of the required pQwer. 186i The Involution of algebraic quantities may be divided into three cases — the involution of monomials, of binomials, and of polynomials. Oabi I. INVOLUTION OF MONOMIALS. 187. RuLB.-- Jbiise the coefficient to the required power by actual mult^lictOion ; aUo raiee the different Uttere to the required power 6y mtM^lying the exponente they already have hy the exponent of the required power^ and connect the two parte thue obtained eoaito form one quantity, Noni.~A firaotlon is raised to any power by involving both naiherafor and denomlaJMkor separately to that powerr-> mixed member by involTing the equivalent improper fhictien. Ex. 1. (2a»ry»)* = 2* x {tflxf)* = 16 x a»x*v»« = 16o»«*yi«, Ex. 2. (- 30*")" = ( - 3)» X (ax*y = - 27 x a?*^ = - 2Ta''«®. ) Bxntoisa XXXYII. Write down the values ofw 1. (2<^)»; (3aft»)»; (4»i.»)»; (3o6*c»)»; 1'; (2a^y)o ; (3o»Ky3) 2. eo^*; (-2a'6c»)'; (-iaW»)9; (-i«»»)»; (-2«»y)». ^, (4»;p)»; (^ax^fs^r ; (Say-)"; (-3av»)»j (3fly>)«} <-*^^^ 116 INVOLUTION. [8SCT. VU. Gam II. . INTOLTTTION OV BINOMIALS. 1S8. By aotaal maltlplioation we find tha^— (o+c)» a a» + to»c + 21a»c" + 36o*c"+ 86a^c« - 8ac' + c*. We here obserre the following facts :— -I. I%efir»t term of the expannon ii found by ramng the ftr$t term of the binomial to the required power, II. 7%e literal part of the tecond term of the expaneion it obtained by prefixing the first term of the expaneion with exponent decreased by unity to the simple power of the second term of the biTiomial. in. In the succeeding terms of the expansion the exponent of the first term of the binomial constantly decreases^ whtie that of the second term of the binomial constantly increcues by unity. IV. If we take the coefficient of any term and multiply it by the exponent of the first letter of the same term and divide by the number of the termj the quotient is the coefficient of the next succeeding term. V. When the sigh of the binomial is + all the signs, of the expan- sion are +, but when the sign of the binomial is •'the signs of the expansion are + and - alternately, Ex. 1. )(36) + lB(iea*) (96») - 20(8a») (376») + 16(4a>)(8l6«) - 6(2a) (2436«) + I29b* ^ 64a0 - 676a'fr 4- aieOa^fr'' - 4320a"6' + 4860a^6« - 2916a&* + 7296«. THnomiaii may be involvid by writing thtm a$ binomialt and proceeding qfler a manner eimilar to the above. Ex. 3. (tt- 6- 2c)* = { (a-6) -2c j * - (a-6)* - 4(a -6)»(2c) + 6(o-6)» (2c)'' - 4(a-6) (2c)» + (2c)* . = (a*-4o»6 + 6tt''t» - 4tf6» + 6* - 4(2c)(a» - So^ft + 3ah* - 6») ^ 6(4ca)(o'* - 2a6 + 6«) - 4(8c») (o -6) + 16c* * = o* - 4tt»6 + eo^fti* - 4o6»+ 6*x (8o»c - 24a''6c + 24a6»c - 86*c) + (24ff'c»-48a6c» + 246V)-(32ac»-326c») + I6c* = o* - 4a»6 + 6a*6*- 4a6'* + 6* - 8a»c + 24o^6c - .24a6*c + 86*c + 24oV - 48a6c' + 246 V - 32oc» + 326c» + 16c*. ExRROiSR XXXVIIT. Write down the expansions of 1. (a -6)9. 2. (c + x)*. 3. (x-y)i". 4. (a+w)". 5. (2 -a)*. 6. (a;-3)». 7. (2a + 3)6. 8. (3-2/ft)». 9. (3a-2y)». 10. (26-5c)'». 11. (3x-4y)*. 12. (o6 + 3c)». 13. (2oc-xya)'». 14. (a+6-c)». 15. (2a- 6 -c)*. 16. (2a + 26-3c)». 17. (l + x-x")*- 18. (a-6 + 2c)«. Oasb III. lent of the INVOLUTION OP POLYNOMIALS. 189. No general method can be given for involving polynomials to a given power except by actual multiplica- tion. The second power of polynomials, however, may be expeditiously obtained by the following ; — 118 liTVOLtlTION. [Sio*. t*.h RvLM.— Write down the square of the first term and twice the product of the first term by each succeedi^ term of the polynomial. Under this set down the square of the second term cmd twice the product of the second term by each succeeding term. Similarly set down the square of the third term and twice the product of the third term by each succeeding term. And proceed thus through all the terms of the polynomial. Lastly, add the several results together for the complete square. Ex. 1. (a - c -d-/+ g' -*)=*= a2-2ac-2ad-2a/+2og-2aA + c* + 2cd + 2c/- 2cg + 2cA + «P+2d/-2djr + 2dA +/'-2/gr + 2/A + g^-2gh Here we cannot add the quantities together since they are all unlike. Ex. 2. (1 - X + X* '- ix" + 2x* - Jx» )=* -* 1 - 2x + 2x''* - x' + 4x* - x» + X* - 2x' + X* - 4xfi + X* + X* - x* + 4x® - xf + \x^ - 2x' - |x* + 4x»— ,2x9 + ix in - 1 * 2x + 3x* - 3x8 + Qx4 . 6a;» + ^^x^ - Sx' + ifcx* - 2x^ + ^x 10 BXBROIBB XXXIX. 1. (2 + Jx-3x»)2. 2, (x + x»-x»)2. 3. (2x-3x2-Jx7. 4. (l-ia+2o2-o')». 5. (l + x-Jr»-Jx»+x*)2» 6. (2o-ox + 2ax2)2. 7^ (l+ftx-cx*)". 8. (o-&x-cr«+rfr')2. 9. (l-o+6V-c»x«+(i*x*)''. 10. (a+6)». 11. (a-(r)^ 12. (ax- 2)*. is. (2 - 3x + 4x=' - Jx^ + ^x*y\ 14. (1 - 2x - x'-* t 2x» - x*)» AftTS. 140-144.] EVOLtrttOK. 119 EVOLUTION. 140. Evolution is the process of finding any required root of a quantity. * 141. Since ( + a)* = + a* and ( - of also = + o", the square root of a' may be either + a or -* a, and hence we always attach the double sign ± to the evm roots of a quantity. Thus, v«V = ± «y ; Vx^ = 1 «y ; &c. 142. Since all even powers are positive, whether the root be negative or positive, it follows that a negative quantity can have no even root. NOTB.— Expressions iadiccUing an even root of a negatire quantity, such asV-a*, V-16»t*, V-16a'», V-a^m'Ssi", ke., Me etiHe^ imaginary or impoaaUUe quantities. 148. The root of a complete odd power has the daMe sign as the powert Thus, ^-a» = -a; V-32aio6«» - -2a*6» ; 0Ta«w» = 3o=»/ft. Case I. EVOtUUlON OP MONOMIALS. 144. To extract any root of a monomial :— * ROLn.—-£xtract the required root of the numerical coefficient^ and then eitraci the toot of the literal part by dividing the exponent of each letter by the index of the root to be extracted, NoTB 1.— We extract a required root of a fraction by taking the rdoi of the numerator and denominator separately— of a mixed number by taking the root of the equivalent improper fractiOh. Ex. VlSa^ft" = Vl6 X a26» = 2o»6» ; ^64a»66 = ^^64 x o6« = 4flfc». Novk a.— When the exponent of the literal pArt is not ezaotiy divisiblS by the index of the root to be takeh, \re cannot obtain the root, aiid oonsequentiy we merely indicate it« extraction by using th^ riUUcM sigtt |i.'t.4l^:^)<.^. Now we endeavour to find some process by which we may use a, the first term of the root> as a divisor for finding the second term, and knowing that this second term is b, wo see at once that we must use 2a for a trial divisor, because 2ab -f- 2a gives b, the second term. Finally, as the divisor multiplied by the last term put in the root, must cancel the remain* ing part of the dividend, i.e., 2ab -\-bi,vre observe that we must ad^. b to the trial divisor in order to complete it. 149. The several steps of the above process give us the following: — 2ab-\-b* 2a6 + ds ABTB. 146-147.1 EVOLUTION. 121 I. Having properly arranged the given square, we take the square root of its first term for the first term of the root, and sub- tract its square from the given square. II. We double the part of the root already found for a trial divisor. III. We ask how often this trial divisor is contained in the first term of the remainder. J%is gives us the second term of the root. IV. We place the second term both in the root and also in the trial divisor to complete it. V. We muUipbf the complete divisor thus obtained by the second term of the root, and subtract. VI. If there be a remainder we again double the part of the root already found, for a new trial divisor ; again ask how often the first term of the trial divisor is contained in the first term of the remainder ; place the quantity answering this both in the root and in the divisor; multiply the divisor thus completed by the last term put in the root} and so on. 147. We are led to infer that the above rule will answer in all oases, from observing carefully the law by whioh any polynomial is raised to the second power, and that the given method for extracting the square root is just the reversal of this process. * Thus, (tt + 6)*^ = a" + 2ab + ¥. (o + 6 + c)' = a* + 2a6 + 6'* + 2(tt + b)c + c*. (a + 6 + c + 36 ) - 30ab + 96" - ZOab + 9b* Ex. 2. What is the square root of x* - 4x' + 8x + At OPBRATIOK. a* - 4x8 + 8« + 4 ( «' - 2a: - 2 = sq. root. 2a:» - 2« ) - 4x» + 8x + 4 - 4«» + 4a:* 2x»-4«-2) -4*» + 8a: + 4 - 4«* + 8x + 4 Bz. 3. What is the square root of 4x-a:) -4a!*-2x»+tx' -4x*-6x»+ x» 4x»+6x'-2x + 1 ) 4x»+6x»-2x + l 4x'+6x"-2x + l Nora 1.— If the given quantity is not an exact square, it is an irrational quaati^, and of course its exact square root cannot be extracted. Nora 2.— In tbe above exaioples, and in all others where an even root is estracted, all the terns of the root mi^ have their signs changed, and ihii rMUUing expression will still be the root required. (See Art. 141) . BziROIBB XLI. BxtTMt the square roots of :-<- 1. 4a" + 12a6 + 96* ; fi^iax + 4x»; 4o^' - 28acx -f 49c . 2. dahn^ + SOomxy + 26xV ; 16a»x* - SoftVx' + 6*c«. 3. 6x» + 1 - 6x + 12x» + 4x*. 4. X* - 2xy - ax» + y* + 2y» + 1. 6. a* + 2ab - 2ac + 6* - 26c + e*. 6. 12o» + 9o* + 34a« + 20a + 26. 7. tt' + 2a6 + 6* + 2oc + 26c + c* + 2ad + 2bd + 2cd + , the trial divisor, by Sab (i.e., three times the product of what was in the root by the term last put in), and b* (1. e., the square of the term last put in the root). Upon multiplying the complete divisor 8a' + Sab 4- b* by b, and subtracting, we And that there is no remainder. 161. The above prooess enables us to extract the eube root in this particular case, and as it holds good in everjr case in which it is tested, we conclude that it holds univer- sally. Thus for the extraction of the cube root we get the following : — Rdlk. I. Arrange the given cube according to some letter of reference. II. Take the cube root of the 1st term of the arranged cube^ and place it as the 1st term of the root. III. Subtract the cube of the 1st term of the root from the given cube. IV. Take three times the square of the part of the root already found as a trial divisor. y. Divide the 1st term of the remainder by the 1st term of the trial divisor^ and place the quotient as the 2nd term of the root, VI. Complete the trial divisor by adding to it, Istf^Three times the prodwt of what was.m the root bV J, ^ the term last put in the root; and 2ndf The square of the term last put in the root. „ ippt. Multiply the divisor thus completed by the last term put in the root, and subtract the product from the part of tkt given cube remaining. 126 BTOLUTTON. tsioT. vn. Tin. Again find a hial dtotior, u$ in (iv) ; divide the Ut term nf' Uut remainder by the Ut term of thie trial divitoTf and place the quotient ok Brd term of the root. Again complete the trial divieor a* in vi| by moArtngr ^^e two additiont there detcribed ; multiply the complete divieor by the last tefm put in the root, cuifrarf,— and go on. 162. We may be led to infer this rule for extracting the cube root of a polynomial by reversing the process by which a polynomial is raised to the third power, as may be seen by an attentive examination of the following : — (a + 6)» = 0^ + 3o»6 + 306" + 6'. (a + 6 + c)' = o" + 3a:*b + 3ab' + 6' + 3(a + 6)»c + 3(o + 6)c« + c*. (o +6 + c +rf)» = a" 4- 3o»6 + 3o6» + 6» + 3(c + 6)«c + 3(o + 6)c» + c*. + 3(a +. * + c)'<^ + 3(0 + 6 + c)d* + «P. . Whence it appears that :-— The cube of any polynomial is equal to the cube of the first term, plus three times the square of the first term multiplied by the second, plus three times the first term multiplied by the square of the second, plus the cube of the second term, plus three times the sqtiare of the sum of the first two terms rAuUiplied by the third, plus three times the sum of the first two terms multiplied by the square of the third, plus the cube of the third term, plus three tim£s the square of the sum of the first three terms multiplied by the fdurth, plus three timnthe sum of the first three terms multiplied by ^^e square of the fourth, plus the cube of the fourth term; and so on. Ex. 1. Find the cube root of Sa" - 84o»« + 2940*" - 343«". * OPIRATION. 8a? - 84o»a: + 294fl«» - 343ii^(2a - 1x 80? 3(2a)» = 1202 3(2a)(- 7x) = • - 42a« l-1xy- +49«» - 84a»» + 294o«» - 343x* - 84o»« + 29402;' - 843x* 12a'>42ox4*49at' Sz. 2. What is the cube root of 270^ - 540^ + 63a* - 44a' + 21a« - 6a + 1 ? Art. 1m.] iSVOLtJTION. isf + « M 1 •M •2 + ^ S ^ 1 Is s I *§ + + > A s • 1 a \m^ 8 f>4 •o + II $ r^ 1 + 5 1 + -4Alf I I CO + I + I % + I I ,1 CO "5 « i I O) + I + s % CO I ' + s 1 % "«^ + + + + Is 1 • 1 1 1 1 04 5 C4 + I + CO I 1» 9 I 5,5 "^ 4:*^ ^ !U^d.m^ 128 kVOLUTtON. [SaoT. Vit ^--^ EzPLAMATioN.— The foregoing Is a aecond method of oxtnoting the cube root, known m Homer'i method. Upon onreftd exnmlantlon it will be aeen that the Mune trhd diviion and complete dlTlion are lued u In the other method, bnt that they are obtained eomewhat dlilbrentlf. The •eyeral ttepe are as follows :— 1st, Take the onbe root of the first term and plaoe it as flrst term of the root, also place it to the left of the arranged cube, under the head First Column. 2nd, Multiply the flrst term ot the first column by the first term of the root, ard plaoe the product as flrst term of the second column ; also multiply the flrst term of the second column by the flrst term of the root, and place it in the third column, i.e., under the given cube, and subtract, trd, To the flrst term of the flrst column add the flrst term of the root, multiply the sum by the flrst term of the root, and place the pic* duct as the second term of the second column. 4th, Again add to the flrst colunjn the flrst term of the root. 6th, Add the flrst and second terms of the second column together tbr a trial divisor. Ascertain how often this goes in the flrst term of the dividend, and place the quotient (-3a) in the root, and a)|o attach it *A} the 9a* in the flrst column, eth, Multiply the 9a« - Sa in the flrst column by - 2a, the last term put in the root, and place the product - 18a3 4-^* under the 37ai In the second column and add ; this gives 27a4 - 18f>- ^ -|- ^' foe <>om- plete divisor. 7th, Multiply the complete divisor by -8a, the term |fwt put in the root, and place the product in the third column. 8th, Subtract and go again through the whole process as before. EXBROISS XLII. Extract the cube root of each of the following quantities : — 1. 8x» + 36x» + 54» + 27. 2. o6 - 40a? + 6a» + 96a - 64. 3. 1 - 6fl + 12a» - 8af. 4. a6 - 6o« + 15a* - 20a* + I5a^ - 6a + 1. 5. 8aV - 84a'*6x* + 294a6*a;» - 3436''a:8. 6. 8a:« - 36aa:» + 102a''x* - l7laV + 204a*x2 - 144a»x + e4a6. 7. x« - 3x» + ex* - Tx" + 6x» - 3x + 1. 8. o^ + 30^6 + 306* + 63 + 3 (o + 6)2c + 3(a + 6)c» + c' + 3(fl + 6 + c)=*// 4- 8(a -i- 6 + c)rf2 + d" + 3(a + 6 + c + rf)«e + 3(o -»• 6 + c + d)e* + e». NoTS.— In Ex. 8 endeavour to keep the quantities in brackets, and ttie labor of extracting the cube ^oot will be materially lightened. J iiiM.i«t.m.] Kjcituonott «v jMvis. m 168. TnoRiM. — In the extraction of the cube root of a nunUfer when n+ 2 figurei have been found by the ordinary ruUf n ftguree mare may be found by dividing the remainder by the laet trial divisor. DncoimRATioN.— Let N repreaent the number whoie enbe root !• required ; let a repreaent the n + 3 figures Atready fbund, and let « repre- sent the n renudnlng figures. ThenV^=:o + a?, .-. A^= o3 +8a«a? + 8oa?« +aj3. ' N—a*^z the remainder after n -f 2 figures of the root have been found , and 8a* k the trial divisor. ^ — a» da* 9a*x 4- 9ax* + ars , x* 8o* ^ a Now if we can show that + -^-^ is a proper fhiction, we shall have proved that, neglecting the remainder arising from the division, we msiy obtain the next n figures of the root by dividing by the trial divisor. By hypotheslsx oontalnsonlyn digits, while it is manlftst that 10* oontalns n ■{- 1 digits; hence x < 10* and .*. x* < 10*". And since a oontains the left hand n + 2 digits of the root, taking into aocount the podtien of these with refbrenoe to the decimal point, a must contain 2n + 2 figures. And therefore a is not loss than 10^ ^^ Hence --- < --!!_. that Is, — a SU + 1 a <:^, SimUarly a>3 Jo* < 10' m 8X10 ,4n ♦ a ' that IR < M*lt Hence X* a ^ da* < l^tf + at 10 >«*a and 8X10 < unity. fix. Required the cube root of'10973936866941016122086048. Herie since there are 26 figures in the cube there are 9 In the root, and we proceed to obtain the first 5 of these by the ordinary rule. The five digits thus obtained are 22222, with a remainder 829181890015122066048. and a trial divisor 148146186200. Then 329181898016122086048 -i- 148146188200 = 2222 •{• zs remaining four digits of the root, which Is therefore :=z 222222222. jSxtraotion of roots in general. By observing the mode of writing the square, cube, ft«.,^c»r polynomials, we can deduce the following general ^e for the extraction of any root of a polynomial: >^p,f| 180 tHBORT OF INDICES. [Sim. ynt RULI. , I. Arrange ih€ given polynomial aeeording to a Mttr ofr^fwmct. II. Extract the required root of the ftret temit thit vHll be tk« first term of the root. III. Subtract the power of thii firtt term of the root from the . given polynomial. IV. Divide thefiret term of the remainder by twice the firtt term ^fthe root for the tquare root, three timet itt tquarefor the cube root, four timet itt cube for the fourth root, five timei itt fourth power for the fifth root, ftud so on ; the quotient wUl be the tecond term of the root. V. Involve the w^le of the root nowfoutul to the tpee\fied power, and ttdftraet it from the given polynomUU. VI. Divide the let term of the remainder by the tame divitor a$ btfare, and the quotient wiU be the third term of the root, .Again involve the whole of the root now found to the epeei- ^ fled power; eubtraet, and so on. Kora.—It it maiilftit that the rale yeriflM itself. Bz. What is the fourth root of 16^;' - 32x^ + 88x^ - 104x'' + 146x* - 104a;» + 88*'- 32* +16 7 OPIBATIOM. (roots 2*'- « + 2) 16x»- 32x'^+88as«-104x»+146a;*- I04x»+88x^32x+16 (2x»)* = 16x» 32x^) -32x^= Ist term, ofrem. (2x»- x)* = 16x" - 32xT +24x« -8x» +x*. 32x^) 64x* s 1st term, of rem. (2x«-x+2)* = 16x»-32x'+ 88x8-104x''+146x*-i04x»+88x»-32x+16 Rem. ="0^ Hence 2x' - x + 2 is the fourth root required. SECTION VIII. THEORY OF INDICES. 166. It has been stated (Art. 17) tliat when a ftwh tional index is emfdoyed; the numetator of tl^firaetl^ Awr. 16^-168.1 TRBORT OF INDIOM. 181 denotes the power to be taken, and the denominator indi- cates the root to be extracted. We have now to add that a negative exponent is sometimes employed for the purpose of denoting the reciprocal of a quantity with Jthe tame exponent taken poeitively. m 1 Thua, a ' is niiod to denote — whether m be fr«otion»l or integral. 156. Throrkm l.Ifm and n be any po$Uive inttgral qaanlitit»f then a"» X a" = »'" ♦ «». DiMONSTRATiON. a^ = a X a X a , , , , to m factors, and a* - a yaxa,.*, to n factors. Therefore o"* x a* = a x a x a .... to m factors yaxaxa»,», to n factors. - a X a ... . to m + n factors - a^*^^ which was to be proved. 157. TniORiif II. ^ m and n be any pottitive integral quantitietf then (a">)» = a"" = (a")". DsxoirsTBATioir. (a^)^ s 0*^ x a*^ x a*^ to n factors => tf im^m + m to •* terms - a"***. (o*)** = o* X a* X a" .... to m factors = o'**'**** •••• *•♦* »««* But win = nm .*. a""* = o*"*, and since (a*)** and (o"*)'* are each = d^^ .'. (o*)" = o"*" = (tt**)" which was to be proved. » 168. Thiorim III. I/m and n be anypoaitive integerif thtn the mth root of the nth power of a, ii equal to the nth power of the mth root of a. That m, )(|'(o«) = (Jlja)^ 9ini0|i8TBATioN. Let ^(a^) = x^ ; raising both to the mth power ^'9'M0. r^ = That is:— a-***** — = o-^i-" -m (I) Powers of the same quarUity are multiplied together by adding their indices. (II) One power of a guantity is divided by another power of the same by subtracting the index of the divisor from that of the dividend. (Iliy ji power of a given power, or a root of a root, is obtained by multiplymg together the two indices. (TV) Powers having unlike fractional indices may be reduced to equivalent expressions hamuli fractioncd in^et vdth a common denominator. ^ (Y) ji factor may be removed from one term of a fraction to the other by changing the sign of its exponent. (Yl) Hie product of the same root or power of two or more dis- similar quantities is equivalent to tke same root or power of their product, and vice vered. iLLUSTBATlVfi ExAMPLlg. . 4m 4«i 1 4m 4 Ex. 1. _ • = - J ma" *, or -r-j- = — ;; — 7—. 3V 5(mn»>* .6m*n« "4 -I -i I 4 '^ da" '6 ■ c 'i»*?j* ABT. lee.] ■ THEOBT OF INBIGBS. 135 reduced to ee$ with a Ex. 3. ar ^b'^ m^c^ _ (mc*)^ V(»»«*) Ex. 4. a-"'xo-» = o-«4 a''x = a 3*7 _ = a*. 1. ahaKa^'i.J'i = a*: a'^ * a'i = a^-(ri) Ex. = a = a* Ex. 8 (o*)»=:a>''8 = a* ; (a-»)» = a -a»2- = a' -•va=:a-»»-« («■•) = a« J (a*) = a :ir"= .-*•"= a (a-^)-':=a, Ex.9. K«VM""= («*''"*)'"=(«'*)■•=«'*'*"• -at Ex. 10. ^V(«''6''V{«*«* V(alfe»)})"* ^^i(^^abc\a-H'»r')^^y* = [a»6»{a6c*o~V*c"J}*]^t-(aW»aVcia"'A6-Ac~*)+* '-a^ + 2oi^-1-a~^ Ex.11. Dividea»-a» + ao*-a-a » + a-»byo«+a»-o I + a* - o ■ » ..-». OPBBATION. x« + a* - a" » - o ~ » )a? - a^ + 2o» - 2 - a" t+o-»(o»-o«+a'"»-o"* o' + a -J- -a^ -0^ + 0^ + 20*^- 1 -o'''-»%-« -a-*- 1 + a '"^ -i.a-U.-V •fa- fa P 186 THEORY OF INDICES. rSBpT. VIU. BziRCisli XLIU. ^ 1. SxpresB Vtt ; -V**; V«* ; V(«ft'«*) ; -yc^ftO*; V(o'*c»«)» and i|y(a**6V)* with fractional indices. 2. BxpreSBO^; 6" ; c^ ; aW ; (o6c)» ; a^6^ ; (a«6»c)« ; (o%*c«»i')*, and (a^6^c« )« with the radical sign. 2a 2 3a m^ 3a6m ia'm^ 3. Express ^ ; " i ~ 5 ^ J 4^? i "5^^ J 3o*6V(m») 1 «*6^c* — -srrr; — J ~"r"i ^^^ — TTT ^i*'* negative indices, 80 as to remove all tiie literal factors into the.numerators. 2a i. 3aa:y' 4ac Bt/fabafl\ ' ^ negative indices so as to remove all the literal fikctors into the denominators. 3oi6-» 6-8 2-ia"6-a „ b* Sam 4. Express 2a ; — ; 1 1 fc"^ y r ^*'** positive indices. 6. Simplify (a " * x a " ^) and (a^ x a "" ^ x j)' * and (a-»x Vax^ox^a)'^. 7. Singify (^{V(«-"x^)«c})" and |V(Vf#i)xai}■'• 8. Simplify the following expression : r -/• Aam. 167, 168.] SUBDS. 187 10. Mttltiplj a^ - 806^4- 8a^6 - 6^ by a^ - bK 11. MttlUply •'^ - o»«» + »' by a* + aW + xV • la. Multiply U - Jx^y"* + a«^«^ - » + y" *** by U^+y"*-*^ 13. Divide 9ar»y - 4«- '^y " > by - 8« - *y - J« -•. 14. Divide a + aU* * -a*t" ^-*6 -» by a*+ o * 6 " * + a ' 6 " * 4.aVt + aV*+6"* 15. Dlixideaf» + »""*-l + «»4-«by «"** + «' + l. 16. Square o» - o + a» + 1 - o" * -a - * + a" ». 1 7. Extract the square root of o* + 2o* -l-aa-' + o"**. 18. Extract the square root of x>->4x-fl0»* - Ifix' -f 19 - l'C«~^ + 10«"5-4«-» + «"^. 19. Extract the cube root of « "* y" - 8*"" 'y +8»'y'*-afy'*. 20. Extract the cube root of x* - 6x'y' -f aix'y' - 44cy* + 63»*y' - 64a;'y* + 27y. "*»».<■ SURDS. « 167. A Mrd or an irrcttianal quanti^f is a quantity which cannot be represented without the aid of a fraetionid exponent or the radical sign. Thus, V3, V«» V2, Va? or a*, V(o + 6) or (o + fr)^, Ac, atf surds or irrational quantities. 168. A rational qwmtiijf is one which does not necet' san'^ involve the use of a radical sign or a fraotiQi^al exponent'. Thus,(i>o?6>3aiiH(rf)*» (Btfi)K (32«i««»o)^, 4c., are rational quMitities. iSmm 1. The last three of the quuittties gtren above are written in the forp of Mirdii but, the power being aaoh that the root4ndioated in eadi «NI ea|i.be extraeted, tlM qnaatltiee are really nttoaal, Thusi a>)i m^\ <8(i»)* = la J (8>»»»» J* w jpajf , las SltKDlS. tSiiol*. Viti. NoTS 2. Thd tohUM Mtidnal ind Irnttiontl aM nwd limply to exprcM SI fluit that the qoantitjr hM or has not Bome determinahle rtUio to nnity. ua, V2 is irrational, bMaoae, since it is equal to 1 + a deoisud which i^either repeats nor terminates, we cannot compare it with unity so as to siir that it contains unity, or that nnity contains it any definite nnmher of times. ^ 169. Surds are either entire or mixed. An entire twrd IB one in which the whole expression is affected by the radical sign or fractional index. A mixed surd is one composed of two or more fiictors, one of 'which is not affected by the radical sign or fractional index. ThuB, '/SB] Vt ; (o + 6 - ley] (oftV)* are entire sards. 21^] 4V5; 3(aby] AyTf] a&(ac'x^)* are mixed sards< 170. In mixed surds the part not affected by the Midical sign or fractional index is called the coefficient 0/ the aurdf and the part affected by the radical sign or fractional index is called the 8urd factor, 171. Surds are either similar or diesimila/r* Similai^ •urds are such as havC) or may be made to have, the samei Burd factor i all others are dissimilar surds. Thos, V^i W^) (<"' "^ ^)V^»_V8) which is equal to 2^/tf &c.| ard similar surds. So also ^ab] mJ^ab\ (a + m)(a&)^, 17x(a6)i^ a&d paUi are similar surds. « 172. ^ surd is said to be reduced to its simplest form When tlie surd factor is made as small aS pOssibld without putting it in the form of a fraction* KoTB.— A quadratio surd is one in which the fractional indta ^ is em« ployed ; a cu^ic surd is One in which the index i is employed, ftc. 173. To express a rational quantity in the form of a imrd :— RuLS.-— itdjitf it to thi^ot^if Mie i^ooi ihi sUtd ixpreiiesi M fUue it beneath the radical Bign, Juy. imii] ktitCDB, m Ex. 1. 2a = (2o)^ = { (Ja)»} * s (4o»)^ = V(ia'). Ex. 2. a% = (a%)* = (tt«m»)^ = y(o«»*). ., 174. To reduce ^ mixed surd to an Entire surd:-' RuLi.«— JZaue the coefficient to the power indicated by thi denoUi^ inator of the surd»indeXf and place beneath the radical eign ihe product of this power and the given surd f actor * Ex. 3. 4V2 = Vr6x V2 = Vl6x2 = ^ ; a^m = V^V"^ = V«^« Ex. 4. 2^7 = ^8 X -^7 = V (8 X t) = ^56 ; c»o'i»* = ^c* x-y(oni) a '^(acfim). Ex. 6. ea V^~^ = {/(216a») x (/^^^ :. ^^216a» x ~^ = ^(72a»»i). 176. To reduce an entire surd to a mixed surd :-— B.VLK.'^Resolve the quantity under the radical sign into two factors^ one of which is the greatest possible perfect power of the root indicated. Extract the root of this factor^ and place it as coefficient of the remaining surd factor. Ex. 6. V^ = V36ir2 = '^Mx V2 = 6V2 ; V20a^ = V^^^^^o = 2a^5a, Ex. 1. ^135 - y2T"x"5 = 07 X ^5 = 3^5; ^o^^ - a^sifl = 176. To reduce surds to their simplest form :^ l^DLK.— -i{e(2t4ce the entire surd to a mixed surd by last rule^ and if the remaining surd factor be fractional^ multiply both its numer-i rotor and denominator by such a quantity as will enable us to tmoie the latter from under the radical sign, Ex. 8. ^432 = -^216^2 a ^216 x ^2 s 6-^2. 6x6 146 (8>oi^. Viti. 177. To oompare dissimilar surds so as to determine which is the greater : — RULL—Iir mi«<(I iurdif reduet tkim to enHn turdt^ then reduce their mdkee to a common denominatorf and raiee each ewrd to the power indicated by the numerator of ite eurd4nde* when thue reduced, Ex. 11. Oompare 3^3, 4V6, and ^326 with one another, that is, '^ ; V8b and ^SB; that is, (81)^, (80)^and (326)^, that is, (81)^, (80)^ and (326)^; that is (81')^, (80')^ and (326)i, that is, (6S61)^, (612000)^ and (326)^, whence it is erident that 4V6 is the greatest and ^325 is the least. 178. To add or subtract surds :— B.vhMt^^Reduce them to the eame surd factor^ %ohen eimilar^ and then add or subtract their cot^fficieniei Dieiimilar eurde are unlike fttonttftcf, and we can only indicate their addition or subtraction by connecting them by their proper signst Ex. 12. 4V24 + 2V&4 - V« + 3 V96 - 6V160 ' = 8V6 •!• 6V6 - V6 ■>■ 12V6 - 26V6 > (8 + 6 + 12)V6 - (1 -I- 25)V6 = 26V6 - 26V6 = 0V6 - 0. Bx._13. 3V|_- 2VTVJ; Vi = fVI* - 2V^ + V^^ 179i To multiply two or more simple surds :^- 13Lvb».-^Reduee them to the same surd indeXf then midtiply the coefficients togdher for anew coifficiint and the surd factorn together for a new surd factor* Ex. 14. 4^7 X 3V14 = 8 x 4 x yTxTi a 12V49x"i =« 84/2. Ex. 15. 2V5 X 3^2 s 2(6)* x 3(2)^ s 2(5)* x 3(2)* ^ 2V12B K.3V4a6y600r Ann. 177-188.] 8tnu>8. 141 180. To difide one simple surd by another : — HxruM.'^Redvet both to tht tam§ nard index. T%en ditfide cotffim eieat by eoeffieitnt and nwdftutor. by iurd/aetor, Bx. 16. 4vn + aVft "> t V7" Wis ' f VfiS' Bx.lT.(2Va.3V8 + 7V5)*6Va = 5^--^ + -^ (I X i via) + a X 1 V40) » i yaa . ^^^ yia + t^^ V40. 181. To find a multiplier which shall rationalize a binomial quadratic surd, and hence to rationalize the denominator of a fraction when it consists of a binomial quadratic surd. RuiiB -^Oumge the conneeting tign of the given binamiai quad* ratic ewrdf and the retulting expreuion wiU be the mvUiplier r«* Bz. 18. What maltiplier will sationalise 2^/2 - 3^3 ? Jne. aV2 + 3V3. pBOOr. (aV2 - 8v3) x (2Va + 3V3) = 8 - 6V6 -I- 6V6 - 27 « 8 - 27 = - 19. 6V2-V7 Bx. 19. Bationaliie the denominator of the fraction ojmtjg* Here the maltiplier iB 3V6 > V6. ' 5V2-V7 (6V2iiV7)(3V5..V6) '^^^^ 3V5 + V« "^ (3V6 + V6)(3Vft - V6) ISyiO - 3V85 - 10V3 + ^42 46.-6 182. To find a multiplier which shall rationalize a trinomial quadratic surd: — Bj3La.~~Fir»t ute ae multiplier the given trinomial quadnitic furd ufUh one of Ue connecting tignt ^angedf the retult triU be n binomial turd tohieh can be rationalized by the lati rule, 1 Ex. 30, RatiQQaliiie the 4euoml«ator of ^5 ^2 + 8V8 ' 142 SURDS, [BioT. vm. Here the firat multiplier :; VS "^ V " 3V3 or VB + V.^ + 3V3> Un either, say the former. 1 yS - V2 - 3V3 V5 ~V«+3V3 ' !(V5 - V2) + 3V3 H(V6-Va) -3V3} « ys - ya " 3V3 _ ys-ya-sys ^ y s - yi - sy s " (ys - y3)»- 27 ' (5-2yio + 5)-a7 " * - 20 - 2 yio" ^ y2 - ys + 3y 3 2o + 2yxo Xext multiply both terms of this by 20 - 2yIo. y2 - ys + 3/3 (y2 '^/& + sys) (20 - 2y 10) Then 20 + 2yio (20 + 2yi0)(20 - 2yio) 3oy2 -^ 24ys + 6oy3 - eyso sya - 4y6 -i- loys - yso 400 - 40 60 ExiROiBi XLIY. 1. Express 2^; 7^; 2*; (10* i (3*)"^; 3^; (yo»)"', as equivalent surds with indices whose numerator is in each case + 1. 2. Reduce a ; 3 ; 4i ; 2a ; 3a*b ; 4xV} to equivalent surds baring indices ii - i, and {. 3. Reduce a» ; y3j 2o»6» ; ac» ; 4| ; 3"* ; (If) " » and (x-» y-* »»)-i to>equiTalent surds with indices - i and i. 4. Reduce 4y3 ; sys ; 2y31 ; 4ya; |(|)' ; and rll) to entire surds. :^r©^ S. Reduce |(fV;if|V; 1(30* ;l(lA and -I 3a(i&) to their simplest form. 6. Reduce 3^4; 2^0; 3(|)*ra(c>^; iHW^hWrn)^ and (am + pq\-i ) to entire surds, am'-pq/ 7. Reduce ^^135 ; y 162 ; ^SO ; 7^324 ; ly? ; IWd (0^6 ^ aSfli* ^ ^K- no \-^ 704mc to their simplest totTS{, i ABt. in.) SVBDS. 148 8. Bedttoe V(«-c^)>rV(^)'^<-->"* l^< -—■ ' V to their limpleit fonni. 9. Oompare ai to their magnitude SVa and 8^3 ; S^Hi ^^^ and 8^7. 10. SimpUfy 4V18 4- 3V32 » V^ * ^8 -I- 6V98 ; alio 8Vi + V<{0 11. Simplify V5* + ^11 + aVS'S - 2(^14 ; also St'Crf'c)* 4. :-(a..o»-cg)'. 12. Simplify ifS'^eFfF'^W^^ + np^gW^w^WH+Hiiiiif . ^^t^mi 13. l£altipl7.5V6b7 8Vt; 3Vio bj 2V5; ^V^ bySVlO; aivd 3V6by4y60. 14. Multipl,' jyre by V8 ; ia^hjlJ^ 2V3 by t^; an4 (V4 X 7^6) by J^6. ox 6y eV 16. ICnltiply together -r- V*"*! "T ^h wd — yc« ; aisp * - V^^ + y by V* + VV' 16. Multiply 4V3 +. 3V7 by 2V2 - 4/6; and 2V3 + |Vi by 3V2r- 4V3. l7.^iTide 3V2 by 4V3; 6V7 oy 3/8 ; 2VS by V2; and h/i^ by 3V3i. 18. Divide 6VT2 by 3^7 ; 3{^4 by 2V6 ; 4Vi by a^i and 4^i^ by 3Vax. 19. Diride V2 + 3Vi by iVi i H^ ' ^V* + ^^ by 2^3; and V55*^by^(^^. 20. Bationaliae V^ + « > V3 - V2 i W^ - OVaT ; Wi -** }V2 «nd 2 V2 + V* 144 SURDS. [Smt. TIU. aa. lUtioiialiie the denominAton of Mkd 3-f a/l iV4 - 1 VI as. Rfttionalise the denominator of 34. Bationftlise the denominators of and a-i-svs 8 gym - myo V«*+«+i + Vx*-*-i' I i- sya V3-V2 + V6' 1+3V2-V3 1+2V3-V2 THEOREMS. 188. Tbiouk I.— -T^c product of two dutimilar quadratic surdt cannot bt a rational quantity. DmovBTBATiOM. Let Vo And V^ be any two dissimilar surds. Then 's/a'x'^b cannot be equal to r, a rational quantity. For if it be possible let ^a X'^b a*r. Then, squaring, we get ab = r* ' r" r*a f* ,',h s — s ~j s -^a. Hence extracting the square root we get r V& = —^a ; that is, ^b may be made to haye the same surd factor as Vo, and therefore V« and ^/b are similar surds (Art. ITI), but by hypothesis they are dissimilar, therefore they are both similar and, dissimilar, which is impossible. Hence Vo x V^ cannot be equal to a rational quantity. 184. Thiobiii II.— .i quadratic mrd cannot be equal to the turn or difference of a rational quantUy and a quadratic eurd. DiMOHBTBATiOH. For if it be possible let V^i ft quadratic surd, be equal to the sum or difference of r, a rational quantity, and V^ i another quadratic surd, i.e., let ^/a=ir± V&. Then a = H f 2rV6 a — r* — 6 ' + 6 .'. ± 2r^/b - o- r"— & or f V* = — , that is, a quad-' ^r ratio surd equals a rational quantity, which is impossible from the definition of a surd. 186. Thiosim III.— ytf quadratic turd cannot be equal to the turn or difference of two disnmilar quadratic turds, DiMOXBTiUTiov. For if it be possible let Vo = V^ 1 V"^ wbci'o yo, V^ w dissimilar quadratic surds. V* ABM. 188-188.] SURDS. 146 -=T lo tht Then a s b ± %'^b x i/m ■¥ m .: f 2^^ n^/msh + iii'-a or .. . 6 -f m - a Jbx Jm s: . That is, the prodaot of two dissimilar inrdi equali a rational quantity, whioh ii impossible by Theor. I. ise. Tbiobm IV.— /n any equation eorui^ting of rational quantitiet and ^tadratie $urdi the rational parte on each eide are equalf and to aUo are the quiadratie eurde. ** Dbmomstbation. Let a 4- V^ ^ x 4- Vv* then a = x and V^ ^ ^V- For since a + V^"= ' + Vv* then V^ '- (J^ - o) '^ Vv t hence if x~a does not = 0, that is, if « does not = a then we hare V^ ~ the sum of a rational quantity and a surd, which (Theor. II) is impossible. Therefore x = a and consequently V^ - Vv* Cor. 1. Hence if a + V^ = « + Vv then also a • V^ ~ ' - Vv* Cor, 2. Hence also if a + V^ = 0, then a s o and also V^ * 0, as otherwise we should have V& - " o» i< e., a tnrd f a rational quantity, whioh is impossible, 187. Thiorim Y.—If the tquare root o/" a + Vb = « + Vji '*•* the square root o/ a - Vb =^ « - Vj* . • DiMOMSTRATiON. Siuoe by hypothesis 'J(a ■¥ ^/b) ^ x + VVt squaring these equals we get a -t- V^ = ^ + ^Wv "^ Vi ^'^^ •*• (Theor. IV ) a = x' + y and ^/b = 2xVy. Then, subtracting equals from equals,, we have a - V6 = x' - 2xVy + y*, whence V(o -* V^) = X - Vy. • I Cor. Hence if V(V<* + V^) = V^ ^ VVt then also V(V^ " V^) = Va: - Vy« 188. Suppose it is required to extract the square roo# of a binomial, one of whose terms is rational and the other a quadratic surd, we may proceed as follows : — Let the given binomial whose square root is to be extracted be 9 4- 4V6| and let V + Vv = the required square root. Then V(9 + W*) = V* + VV •*• 9 + 4V6 = x + 2^/3ty + y. Henee (Theor. iv) x + y =: 9, and 2V^ » 4V6 or 4xy = 80. Then (x •»■ y)> =: x' -i- 2«y + y* ^ 81. Snbtracting the eqaafai 146 SURDS. JSiWT. VUI, 9 4xy and 80 fropi tl^ese equals, we get «? -^ a«y 4- y* ? ), wh«n«e « - y = 1, bat X + y = 9 .*. 2x s 10 and x-ti. Alio 2y ss 8 and y = 4. ^ Hence V> '(- Vv ~ V^ **" V^ = V^ '*' ^ ^ aqoare root required. 180. Instead, however, of working out the queition ihns in fhll, we oan easily deduce a general rule for ezr traoting lilie square root of certain binomials of the kin4 filluded to. Thug, let a + ^/b represent the given binomial, and let V^ ■¥ yy ff the required square root. Thus we hare V(a + V*) = V* + Vv } t^®'* ^7 ^^* Theor. v, Vl40. 5. 10-V57. 8. 43 - 15V8. 11. 8 + VS5* 8. 32 + ^. v^^"^ 6. 42 + Z^iUX 9. a^l^a-X. 10. 20+^V*'^*'' 180. It appears from Art 189 that when a' - & is not a perfect square, ^/x and V;y will be oomplez sards, and the expression ^/x + Vy will be more complex than the ^yen expression ^/(a-\-^b). Sometimes, however, the square root may be similarly found of a binomial oonnsting of the sum or difference of two quadratic surds, i.e., a binominal hoik of whose terms are quadratic surds. This is evidttnt from the fact that ^/€?c ± ^b may be written Vc (« ± V~)> and th#n, as above, if a' - - be a peiiS^t squaxe, the square root of a ± V— may be represented by V^ ± Vy* Ex. Bxtraot the sqnare root of V27 + 2V6. OPIRATION. V2'r + 2V« = V9\^3 + 2V2V3 = V3(V» + 2V2) s V3(3 + $^%), Hence y(V27 +2V6) = v{V3(3 + 2V2)} = vayrrwi: LetV3 + 2V^2 = V* + Vyi then V3 - 2V2 = V*"Vy« And V?;— 8 = « - y .*. * - y = I. But 3 + 2V2 = « + 2V«y +'y .*. « + y = 3. Henoe 2« - 4 and x s 2 ; 2y s 2 and y = 1. ' Therefore VST575 = V^ + 1, and V3 (V2 + 1) = V3 (V* + VO BV12 + V3. : Exnoiu XLTI. 9fa|iil ^sqnure roots of :— ^'^^yai. 2. sys+yio. d.syo+a^ii. 4.yM^^4t, /y-ivf^ Ufi IMAaillfAET QVAMTmBS.. CSwv.imi. IMAQINART QtTANTITIBS. 191; An im^glBaiy quanti^ is an ezprewioBi whfoh represents an even root of a negative quantity. (See Art. 142). _ _ _ _ T}ii», V^ ; V - *; V*-^ ; V"*; V^i ^^*% *f® imaginary qaaiitf Ues. We can approximate to the yalae of sord ^antitieB, but we cannot even indicate an approximation to tlie value of an imaginary quantity, which mutt therefore be regarded as a pureljr symbolical expression. Such expressions, however, often occur in practice, and so tu from being useless have lent their aid in the so^tion of questions requiring the most'skillfiil and delicate analysis. . 183. Inuiginaiy quantities may be added, snbtraotad, mnJtipUed, divided, &o., like ordinary snrds, attention being paid to the few simple prinoiples given in next para- graph. . 198. I. Any imaginary quantity may be redaoed s o as to inv olve on ly the imaginary expression V- 1 > because V '- a' a V<**><-1 =V<**V-1 = ±»V-1» So al8o/^=V<*V~l; II. (V - «)* - — «i tiiat is V, - « X y - o s - a. For though H is true that V-axV-aa:V-«x-a=»Va"»±«, we say here that ^a*s-a because we know ti^at the a' has arisen from squaring - a. We only use the> doable ' sign ± where we wish to indicate that a* might have arisen from squaring dther + a or - a» iiL QfTiy :=jrri ; (vTi)«= -jj (v:ri).^^i)» xv-i^-ij^v-i =-V-i; (V-i)* = i(V-i)*P = (- 1)* = 4-1, and, since every whole number may be ex- pressed by one of the four expressions 4n, 4fi -f 1, in -h 2, 4n + 3, aocording as when divided by 4 it leaves a remain- ^dwf Pf 0,l,?,or3, and (V-ir*»^V-lr(V-D**"' im. ^-m MkmiM (^njMtm^. e m> = -1; (V-1)*"*' = -V2land (V-1)*" =. + 1, it fol- lows that the fonn ulaa V - 1, - 1, <- V **1| = ei^6(V^)* s(6V6)x-l = -6V6. Bx. 4. (1 + V^l)' = 1 + 2V~1 + (V^)* = 1 + 2V'^- 1 = 2 V^ Bx. 6. (E /■-T)(6-^r7) s (6)»-(V'^)» = 26 -(-7) « 36+7 = 32. Bx. 6. 2V8-/:T?)*-V^ r ^ - ^-v V6V-2 4V2 -V-2 -V-2 V6(-V'^) -V-3 J^V-2 — V-'2 V6V-2 -V6V-2 4V-2V-1 -v-2 -7-2 -V-2 Bx. T Pind the square root 2 + 4V - ^2. Let V« + 4V-« = V* + Vy. V2-4V-42 = v«-Vy. V4-16X-42S V^ -i- 672 = ^S7h = 26 = « '^ y. Also 2 + 4V-42 3 X + y + 2V«y .*. 2 = « + y. Heace « = 14 and y = - 12 and V* + Vy-Vl.4 + V-12 « ^14 + 2^-" 3* .........^ BxtBOisi XLYII. Find the ralne of t*-^ 1. (4V^nr7) - (2V^^) and also of)• < 9. The cube of V2 ^ V-"** 10. The square root of - 2 - 2V - 16. -■■*-. ll. The square root ofV" 1 and of- V- l* 12. The square root of 31 + 42 V-2. 13. (4 + V^) divided by <2 - V"^- 14. 14 - vis - ( V3 + 2V6)V"^ divided by 7 r V"^ 1». (« + b V"^) multipUed by (o - 6 -/-!)•* SECTION IX. QUADRATIC EQUATIONS. 184. A quadratb (aquation is one whioh invoives the Second power of the unknown quantity, but no higher power than the seeond. ' ' Noxa.— Quadratlo eqUattoni, like equatioiu of the flrit degree, may in- Y<^e only one unknown quantlty» or they may Involve two or mord tmknowa quantitiei. In the Utter eaae they are oaUedtfimuJtoiieoiugiiad* 186. Quadratic equation^ are of two kincUi :— I. Pure Quadratic Equations ; and ll. Adfeoted Quadratic Equationst 186. A Pmre Quadratk Hquatian iB one which involves) When reduced) only the second power of the unknoWn qumtity. , « l9ll|linn|ieiiidi«at«iamodetff rMolvibga* >d* taitblMtdnt -»■* .f,- kitk-U-m QOASEAXtO AiOAtlO^B. IM Thus, a:»ao;a:» = 0;*^ = (x*)* = 16} «*^ = «* = («^)' = 4 j tix> + fr s cx> - m, ftc.| are pore qaadMtici. 107. An Ad/acted QuaditaHc HqUaiion Is one which involves the^rt^ poWet tm well as the second power of thd unknown quantity. Thus, «■ + e« * 2t; ox«^- 6x a c, 4a:* - 3* :i 2« - x* + a, &e<) are adfected quadratic equaiionB. 198. Any equation may be solved as a quadratie if^ when reduced by transposition, &o., the unknown quantity appears in but two terms and its exponent in one, is double that in the other. Thus ai* + x^ = 3, » - 6>/x = 50 ; Vas + 3 Va: ^ 9, afi* •« 2a^ s 8, &c., may be solved as quad^ raticS) but they are not properly speaking quadratic equa^ tions. lOOi Equations involving surds are generally capable of being solved only by the methods employed for quadratic equations, but they are frequently reducible to simple equations by the following :-^ - lB,viM.'*jirrange the sufd ierms (moneoit both ndti of the tqua* fion, a» appeare nu>it ContttnieiUf equare both nde$ of the equaiion^ irantpoee and rednee ; agttm tquare ^neceuary^ cmd so onk Ix. 1. QiVen VT + V^ + V* - 3 to find the value of x* OPUunoN. V7 + V6 + V* * ^ *l + V6 + V* = d V6 + V* * 2 e + V« = 4 Vx = - a » a i (I) (°) (m) (IV) (t) s (t) squared. » 2 (n) transposed and reduced, £ (m) squared. = (iv) transposed and redticedi s (v) squared^ 152 QUADIIATIO BQtJATtONd. 'tSaot. IX. Ex. 2. Given V {' + 2V(<» +<>')} ^ ^* - y/a to find t|ie rtXw of «. OPBBAlftOir. JT 4* 2 V(> power. = (n) squared. s (m) transp. and reduced. = (it) + 3a. V9s-4 U + Jdz Ex. 4. Given . ^„ = i ^;.^ to find the value of*, yx 4- 2 y X + 40 OPBBATIOH. Vox** 4 ^ 15 + V9x Vx + 2 " V* + 40 8x-4V« + 40V9X - 160 _ ) = 15V« + 3x + 30 + 2V9x > - 4V«+120V«-16V*- 6V* = 30+160 96V* = 190 V* = a ' -■ • ■' « = 4 (0 («) (ni) (ly) (V) (VI) - (i) cleared of fractions. F (ii) tr«n^[>. and fed. = (iXi) collected. s(i|)*1)5. <-s(v|) squared. . •MMM tSaot. IX. tUe ralae then * a. len T 16. alue of ». jm*"* power* id redacQd. of 2. k fractions. id fed. »d. jUT, aoo.] QUADRATIC BQUATIOirS. Bmoin 158 XLvni. Find tb« Talno of « in the following eqnntioni : 6. 8. 10. Va + « H* V<> - ' - V<^»« 11. a + « a 7<^ + «V^?^ 18. ft + « •!• VC*" + a» + a») =s a. 13. V« + 2a 4a + V« ^+Va V« + 3i' 14. v«+ V2o+* = a» (1 +.«) " *• i«. V«^^^= !• - V^ 16. \«+*y * \«-v i f 4Jbc \i 7* 11, V«+V« - V«-V« 8 i 18. V«+ « = «-V«^« i9.«'H-r ' RnLi»**4SivJAjr r«itw«d[ tSe ejua/ion to the form of «* s «i^ iximet th* $q»iare root of tack iUUf and prefix th» dotAtt^H^ i lo tkiright'kaiawmiibtry^jrmtl^ equation. Is. 1. CKyeaif »i|tO(ftt|dJttie Tohies of *, ¥ - la I (n) 4 HA with square iP(K)t •Jft^r^cted 154 QUADRATIO BQUATIOHS. [SaoT. IX. Nora.— The young itadent in Algebn it ■onMtimM at a lo« to lau>w why the dontde lign t is not alao prefixed to the lefUhuid member, linoe ex- traotlng the eqniure root of eaoh Mde doee r^y giro i « ^ 1 a inetead otx ^±a. The Ibrmer of theie expreMlone ii, however, earily jdudbletothe Utter. ThuB, It ±x = ±a, then + « = + -. ox + x = —a, or — » =s + a, or — « s -.- a, but the iMt two of these expreeflfons ere eqniralent to ibe first two transposed. So that on the whole « =r a or « ss — a, tAit is, « =: i a. It appears Irom this that when we extract the square root of the two members of an equation it is snfllcient to put thei double sign before the root of one of the members. Ex. 2. Given 4x' •(• 11 = x' -f 14, to find the values of «. OPIBATIOH. 4«»+ll = «»+14 x»=l x = il 0) (n) (in) (IV) s (i) transpoBed and ooUected. = (n) * 8. = (m) with V of each member taken, X* + 2 Ex. 3, Qiven 3»*- 4 = ,_» to find the yalues of », «" + 2 J6*««»a0 = «*+2 14c' ^'22 6x0 OPBBATIOK. 01 (n) = (i) X 6x«>, 1. e. x 5 since «» = 1, (in) s (n) transposed. (IV) = (ra) * 14 Iz. 4. Given x + V<»' + *" * ,4 ^ ^ find the values of «, OPIBATION. (0 2a> xysrr? + o" + X* = 2a« 3aV=a* 3 (n) (m) (IV) (V) (VI) = (i) X Va* +*" = (n) transposed, s (m) squared. - (iv) transposed = (V) * 3o». « = ±«V*»laVasiW8. ,^s SSOT. IX. abt. aoo^ QUADRATIC BQUATIONS. 155 know why r, linoe ex- [iiit««dofa9 olble to the , or — » = lulTtlentto -.a,tftitto. MO root of double Ag^ )r taken, fx. jO = 1, Ex. 6. Olren Vg'-g* - V^+g ^ * ^ fi^d the value of ». OPIBATinM. c» + X* " (6 - d)" o«'- x» (6 + d)« (0 (n) (in) (IV) =: (i) taken as in Art. 106 (vn). - (ii) oanoelled and then squared, s (m) taken as in Art. 106 (vin). = (IV) X (o« + c*). a^ + t^ " 2(6» + tP) (b + d)* «^*'=2(PTi^<'»'^|<^> a .a Jll^/^ax^s 2a»(6'.l.d')-(6 + d)\i»-(6-t-d)V * =*-2(6'' + rf') («' + «')= 2(6» + d«) fl a(26H 2d'-6'' + 2M~cfO-c»(6 +(0« tt»(6»» 26d + (P)- c»(6 ■«- x = -p, we may regard the left-liaad member as being composed of the ftrst two teniBSof the square of a binomial, and that we may make the Ist member a complete square by adding to it the square of half the coefficient of its second term, and of course adding this to one side we must also add it to the other, in order to preserve the equality of the members. Thus we get x*+px+ r = -tf + T- 4 4 The first member of this equation is npw a complete square, and we observe that by extracting the sqetare root of each side we shall get rid of the second power of the unknown quantity, and thus reduce the quadratic to a simple equation. Thus, Whence by ^jN^psposition * = - J l» ± L "9 Tbatis,x^i . ^ Ruu.— ^ trantpotUion and r§dveHon arrangi th* equaiicn in nieA a mmuur that tht two ttrnu inaoMng tki unknown ^uantUiu tkall be atom on th$ Irft-hani tidif and thi eoeffieient ofj^thaSl ho + 1. II. Add ' iodi iidi of the equation the tquare of haJf the wejfi^ dent of X, ni. Extract the equare root of both tidee of the equation^ and thenee by tranepoeition Jind the valuee ofx. Ex. 1. Qiren «* •»> lOx r . 24 to find the raluei ofx. OPIHATIOV. (0 (u) «»+io« = -a4 3fl+lOx + %6s 1 « = ii-6=-4or-e (ra) (nr) = (X) with (V)* s 5* a 25 added fd each fide. = (n) with iqiiare root taken. = (ni) transposed. WoTB.-— Whenw e lolved tbe geneni eqattion a*-fp«4^]»0, we ehtslaed « ss |( i Vp* —49 — p). Now we may me this •• a fbrnmla Ibr llndiBf the valve of « In a qaadratlc equation. Thus, in the last ezanvie jp » 10 and g as M; then » « s i(i Vp^ 4? - p) = K± VIOO - 96 - 10) = 1 (i V4 - 10) = K±a-10)= yor^ =-4.or-6. But although qnadiatie equations may thus he solved hy fbnnula^ tUs method should he resorted to only hy the advanoed student, ss the Junior student requires all the praetioe he oan get in the solution of quadra- tios hy oom^etlng the square, fco. X x + 1 18 Bx. a. Given — — r + = -— to find the valnes of «. ■ >jp> X'I'l il+ 1 X 6 OPIBATIOir. 13 6 « + 1 X ■ 6«»+6(« + l)"= 13«(« + 1) 6«»+6»' + iax + e = 13x»+ I3x a? + x=: 6 x«+x + i = 6 + i = »f V x+4=t| «r;{.{-|s2or-8 (I) (n) (m) (IV) <▼) (VI) (vn) = (i) cleared of fractions. = (n) expanded. = (ni) transp. and red.. = (IV) with i = (i)« added, s (v) with aq." root taken. 3 (vi) transposed imd red. m. QUADBAttO EQUAtfONS; (Sio^. it. Bz. S» Qiren — r— ^ + -r—r-z = 8 + — rr— to find the Yalues of Xk 9 2c-l'9 . 4«-8 8X-16 9 4x-f3 *8+- 18 4x + 8 OPIBATION* (0 18 , ,^ 72* -64 4» + 18 + -j^^^Pg- X 64 + 3» - 16 72X-.64 4^T8"=*<^-*- 72x . 64 s 80x + 60 - 4x' - 3x •4x*-.6x=114 «afy+f = 6or-4J (n) (in) (V) (VI) (vii) (vm) (a) a (1) X 18. 3 (u) transp. and red. = (in) X (4x -I- 8). s (ly) trantp. and red. = (V) + 4. = (Ti)with(|)*added. - (vu) with square root of eaoh lide taken. s (vm) transp. and red. a'x' 2ax — m^ Bz. 4. Given — r = — ^- to find the value of x. OPHBATION. 2iii" m* . ' 2m* m* m* ' jc - — = ae m« X a A ae ' (I) (u) (ni) (IV) (V) = I i oV added. = m with aq. root not taken. ^ IV transposed. NoTR.— In this example we may oonolude that the two roots of the equation anp equal. BZBROISI L. Pind the valnes of x in the following equations :— * 1. 2x« + 8x - 20 = 70. 2. x» - 19 = 8x - 10. 3. x« - 8x = 20. 4. x« - 29 = 16 - 12x. Aiilk«M.906.] QUADRAtlO ^QtimoNd. 159 •5. 2x*4-x-16-70-x-x>. T. I18x-|af« = a» + a8|. X a 2 a ^ X a 11. 8x X'l 13rar*-jra:|«-a. 16. ^ - = «-ai. « - V* " * * ,- * 8 s 4 11 17. r- + - = - + -. 18. 8 X 4 X la 6. k* - 4x + 16 = lOx.- 2x«. 8. 4x* - 3x - ao X 6x •!- 300* 10. x>-l-3«-7asa01-x-4x«.' x»+ia la. — r — * 4x 4- Ix a 0. a 14. aea^-¥bcx^adx'¥bdt 16. x«-x-40 = 170. x-a x-3 x + 4 at-^a x+a x+3 " x-4 x^a* 10. (7x + 8)(3 + 7x) = 10| a(x - 1)(8 + x) - (8 + ax)(x - 3)}. 80. ox* * 6x 4- tf ayi" + ex - 6. aii (o-m + x)-* = o-»-m-»+x-». aa. a6x* - ax(a + 6) Va2^ s (a - A)*. 204» Many of the foregoing eqoations when reduoei'^ assume the general form oob' 4- &» + c s 0, where a, h and e may be any quantities whatever; now when we father reduce this to bring it under the rule (Art. 203) we get a?+ -^ Si — , and consequently we have the inconvenienoe of dealing with fractions throughout the entire process^ To, obviate this difficulty we may proceed as follows :— Taking the equation ax> 4- &x s c, let U8 multiply every term by 40, and then add 6' to each side of the rei^alting c':aation, and weget 4a'x' + 4afrx •«■ 6* = > 4c(c 4- 6". The left hand mem- ber is n^w a complete square, and e xtractin g the square root of each member we get aax + 6 = l 'Jh^-Aac whence x -6±Vft'-i<^ » 2a ' 1105. This operation translated gives us the following :<-^ RuLK.—JHimnj: reduced the efuution to the form ax' + bx = C| fHuttiply every term by four timte the coefficient ofi^^ and to each number of the reeutting equaiion add the equare of the eoejffkient of the eeeohd term. 'then extract the equare rod of both term»ftrani^t9 ankd rtduei and thw obtain the valuee of X, 160 QUADRAflO BQUATIONS. [Siov. tX. Bz. 1. Oiyen Ss* - 3x = 65, to find the Tallies of x. OPIBAnOM. 3«" - 2« a 66 (I) 86x«-24«=780 (n) 3 (I) X 12 i. e. 4 timet 3, the coef. of x*. ^ 86x*-24c-l-4s7M (m) 3 (n) with (2)" s 4 added to each side. 6«-aaf28 (!▼) s (m) with square root extracted. 6x = ail8 (V) 3 (IV) transposed. 6ars30or-a6 W 3 (v) reduced. x = 6or-4* (vn) 3 (V) + 6. 8jb - T 4c - 10 Ex. a. GiTin — - — + -z-rr « 8J to find the valnes of*. « « + 1 OPUUTIOVt 1 8«-T 4C-10 1 * +Trr=»* (X) ' , 1 T«*-89x = 70' (n) * (I) X 2x (x -f 5) and red. 1 106c'-169axcl960 (in) 3(n)x28Le.4time8 7. K 196x*- 109ax-t- (39)*s 1960 •!• 1S21 (IV) = (m)+(39)« m 14»-39=:V3481»i69 (V) 3Viv P 14x = 89i59s98or-20 (VI) 3 (v) transposed. 1 .% X ■ 7 or - If (vn) 3 (VI) + 14. Bx. 3. Given (3aF + 6»)(x« - x + 1) s (36> 4- a^)(x« + x + 1) to find the values of X. OPIBAnOV^ (3a»+60(«"-* + O (I) 3(3ft»+«^(x«+x + l) x»»x-H 36* + o* «"+x+l " 3o* + i" 2x« + 2 4y + 4o* -ax " 26"-2o« x'-l-l 26»+2o» (6«-o«)x»+6«-o»3 -2(6*+<0* (6« - a^x« + 2(6*+ a«)x = a» - 6« (n) (m) (IV) (T) (VI) 3(i)*(8«"+^(x«+x+l). 3 j^n) as in Art. 106 (vn). 3 (m) rediioedi - 3 (iv) cilear^dof flraoUons. 3. (r) transposed'. 4(5"-o«)^x*f 8(t*-a»)x+4(W«^)" = 4(4-a^)=:i3a& (6" * 0^)* = -(6» + a^)±aa6=?-a?f2a6-6« (o - 6)« (a + d)» * * -yrpr or ^^:T5r • « — X s g -6 o +6 or (yn) s (yi) x 4 times ooef. otx\ i. e. x 4(6* - a*) and then eaeh lide increased by the sq. of 2(6' + a'), the ooef. of the 2nd term, (mi) = (tm) with right-hand member reduced, (ix) = Vvm. BZIBOIBK LI. Find the value of x in the following equations :•*- 1. 8««-9=76-.2«. 2. a:»-x = 210. X' 6 8. 4x*-3x = 86. 6. 4x"+«x = 2x-x«+273. 6. 8x» + 8x+ 11 = 32 -x«. 8. a'x* ^ ahx = ocx 4- 6c. 2 7 9. lx«4'6s}xitATtC BQitA^^Olld. i^i^. tx. Now from this it appears that I. The two roots are real and different in value if b* >4ac. n. The two roots are real and equal in value if &' = 4ac. ni. The two roots are impo8sible,or imaginary if ^ < 4ac. Hence if any equation be expreteed in the form q^ ax* + bx 4- c = 0, U$ roots are bbal and DiFriBiNf, rial and iqual, or iiiAdivABt, according a« b' >, = or < 4ao ; and nmilarly if the egwUion be of the form x^ -t* px 4- q = a, its roota are bbal and DiFraaiitt) biAl and wquAL, or imaoihabt, according at p* >, = , or < 4q« 207. Thbobim 1.— vtf quadratic equation cannot have more than two root*. DiMONBTBATioN. For if it be possible let the quadratic equa- tion aai^ + bx + c hare three roots as j3, 7 and S. Then afi^ + bfi + c =0 »/^ + by + c =0 08" + 65 + c - (0 («) (UI) (IV) (V) (VI) VII) (vra) * tt(jB»-7») + *(i8-7) = rt(/8"-B»)+6(/5-5)-0 = (i)-(n). = (i)-(m). a(/B^y) + 6 = tt(/8 - 5) + 6 = = (IV) T (3 - 7) which is not - 0, ••• by hypothesis /3 is not = y. - (v) T (jS - 8) which is not - 0, •/ by hyp. /3 is not = 8. « (vn) - (vt). a(7-8) = Now a is not - 0, otherwise aafi + bx-¥e = would become 6x -f e = 0, which ''3 not a quadratic equation ; therefore (y - 8) mnit = Of and therefore y-S] but by hypothesis y is not - 8, which is absurd. Hence a quadratic equation cannot have three roots. 208. TmOBiM n.-~/n any quadratic equation reduced to the form qf x* 4- px + q = 0, the coefficient of the 2nd term it equal, mftM it» tign itchangedf to the sumoftheroottf and the Brd term <0 equal to the product of the roote. ititim-Mt ^VASBA^o ^cAttoire. m DiMOHsnuLTiON. Let the two voots of the equation x* •(■ px +9 = be 3 and y. Then - iy + V(J J»* - ?) = /8 ^ And - J p - V(i|>» - 3) =* 7 By addition -pis fi-^ys: sum of the roots. By multiplication {-ip+V(il^-fl)}f-iP-V(il'*-9)| - That i8, Ip^'Cip"-?) Which iB^^'q' fiy- product of roots. Cor. If jS and y are the roots of the equation ax> + 6x -t^ e » 0, 6 ■ c then j8 +7 s - — and J87 = — » ' 209. THiORvif III.— j[^/3 and 7 a/e Me roo^t o/Me equation f -f I* px + q = 0, *A«» (x - ^) (1 - 7) = X* + px + q. DiMONBTBATION. (X -/3)(x •» 7) =: ac' •* (/8 'f 7) ^^ + /S7. But O + 7) = - p and 0y^q. (By Art. 208.) ;•. (x - /9)(x - y) = x» - ( -p)«-i*9 = «■ +px + q. Cor. If j8, 7 are the roots ?f the equation ox* + frx + — x + — ) « 0. Then we have . ox" + tx + c = '= o (X - /8)(x - 7). Cor. 2. If ox" + &x^ -i- ex + d = be a cubic equation, and if its roots be j3, 7, 8 ; then (x - /8)(x - 7)(x - 8) = ox* + 6x* + ex + d. ILLDBTBATIVI BXAMPLIS. ' Ex. 1. Form the equation whose roots are -* 3 and 4. OPIBATtON. Since x--3, x + 3=:0, and since x = 4, x- 4 = 0. Then (X + 3)(x ~ 4) = 0, that is X* - X >- 12 = 0. &x. 2. Form the equation whose roots are 2, - 2, 3 and 0. OPIBATIOM. X - 2 = 0, X + 2 =t 0, X - 3 = 0, X = 0. Then we have (*-2)(x + 2)(x-3)x = (x«-4)(x«-. 3x)= x* - 3x»-4x«+ 12x = 0. Ex. 3. Form the equation whose roots are 1, - 1, 3, ^ 2, and OPIBATIOM. x-l=:0, x + i=iO, x-3 = 0, x4-2sO, x-2-V7 = 0, and *-2+yT=:o. Then (x - l)(x + l)(x - 3)(x + 2)(x - 2 - VT)(» - 2 + V?) * 0, that is, (x" - l)(x« - X - 6)(x«^ - 4x + 4 - T) x 0, that is, «« - 6x» * 6x* + 32x» + 23x« - 27x - 18 » 0. . 164 QtrADBAKlO tSBOV. DC. Bx. 4. Find, withoat solving Uie eqtiatidn, the sum, diftrenoe, »ad prodnot of the rooti of «* -- 42« 4* lit s 0. OPUUiTIOV. Let fi «ad y be the roots, then Art. 208 /S -f > = 42 Mid fiy= 117. Then bj inipectimi find two nmnbere whqie sum = 42 and prodnot s 117, and they are eridently 3 and 39, and hence the diftrenoe of the roots s 36. Bz. 6. For what yalne of e^m will the equation Sc" -F 7« + c% s hare equal roots T OPntJLTION. From Art 206 it appears that in the equation oc' + 6« 4- c = the roots will be real and equal when 6* = 4ae, that is, in this equation when t* = 4 x 3 x e^, or when 12c% = 48, or <^m = A^, Bx. 6. If /B and y be the roots of the equation a^'-px + q- 0, By find the ralue in terms of j> and got— •¥ ->, and of jB" •«• y. y P OPIBATIOV. Art 208. fi-^ysp and fiy^q. Then — + - a - ~ — s + 2 -2 = -2 ^y .J 2 And/8'-l>>'8 a" + 8/BV + 8/»y» + 7»-(3/8«y + 3/8>«) - (fi + y)' - 3/87(/8 + 7) «!»•- 8«p = 1>(1>» - 3g). BxnoiSi LII. 1. Form the equation whose roots are - 2, and - 7. 2. Form the equation Whose roots are 4, - 2, 1, and 0. 8i Form the equation whose roots are 2, - ij 8, - 8, and 0. 4. Form the equation whose roots are 5, - 6, 2, - 2, and 9±^/2. 6. Form the equation whose roots are 1, 2, 3, 4, and 6 i V^* 6. Form the equation whose roots are 5, 4, 1, 0, andll i •y^^ 7. Qiren 6 and - 2, two roots pt the equation ,x* p^ + 6x' •!• 12« s 60, to fiad the other roots. '' 8. Qiyen 1 4 V""*i two roots of the equation ar* - 4ap» + 8x* - 8x 3 21, to find tiie other roots. v;. AST. no,] q^UAD&AnO EQUATIONS. 165 9. OWen 14, one root Of tho aqvfttioii x* -f 8x* - 3920 s 0, to find the other rooti. 10. OiTMK 2, one root of the e(|ia»tion x* - ex" •(• ISx^ - lOx « to find the lather roots. 11. Oiven 8 and - 4, two roota of the equation x* - 2«* - 2Sx* •I- 26x* + l2Qx 9 0, to find the other roote. 12. GlreniV- 2/ two rooti of the equation x* -x^Tlix* - 4x = 0, to find :' "* other roots. 13. For what valv of e will the equation 2x< -i- 4x •(> e s o have equal rooti. 14. If /3 and y he the roots of the equation ox* -f 6x -f* c s 0, form the equation whose roots are the reoiprocals of these. 15. If /B and y be the roots of tl^ equation x* + f» + f = 0, find the value of /B* + 7*, of (3 - 7)"; of /9" - >*; of ;r + - and of p y EQUATIONS WHICH MAT BE SOLVED LIKE QUADRATICS. 210. There are many equations which though not quad- ratios in reality may be solved by the rules for qiiadratios. Such, among others, are equations which come under one or other of ihe general forms aaB^ + &aG^+ c= or aafl + ha^ + c = 0, in wUch n is tny integral numher, and a, (, c, positive or negatiye, integral or fimotionij. Bx. 1. Clven x -i- 6x' =^ 8 to find the values of x. x+6x" + 0=l x* + 8 = il x^'»il*-3 s-rllor-* («) (m) (IV) (V) (VI) OPIBATIOV. (i) with square completed by adding A toeadiside. • (n) with squttre root eitrapt^, (m) tranqK>|Md. (nr) redu^. (y) squared. h:iite -'liMliiku^' >i-«^' -„.l,i kj.. 166 QUABRATIO BQUATIONS. [BioT.IX Bz. 2. Oiven ^ifl ^ 22^x = 23 to find the values of x. OPIBATION. .! i- x' + 2aa:' = 23 «* +22** + 121 = 144 «* + ll=:±12 x*= lor -23 « = 1 or - 1216» (0 (n) (in) (nr) (V) = (I) with (11)' added to each side. = (u) with square root extracted, s (ui) transposed and reduced. = (it) cubed. Ex. 3. Given V* + 12 + i^* + 12 = 6 to find the values of x. OPBBATIOM. (x + 12)U(x+12)*=*6 (x + 12)*+(x+12)*+i=af (x+ia)* + i = ±| (X + 12)^ B 2 or - 3 X + 12 = 16 or 81 X = 4 or 69 (0 (n) (w) (D (v> (VI) = (i) with i added to each side = (n) with sq. root taken. s (m) transposed and reduced. = (iv) raised to 4th power. = (v) transposed and reduced. Bx. 4. Given x^ - 36x^ = - 216 to find the values of x. OPIBATION. x«-36x* = -216 4x«-140x*+ 1225 = 361 2x" - 36 = ± 19 2x» = 64or.l6 x" = 27 or 8 X = 3 or 2 (0 (n) (ra) (IV) l(v) "(VI) = (I) i^A and (36)> added. = (n) with sq. root taken. s (m) transposed and reduced. = (IV) * 2. = (v) with ^ taken. Ex. 6. Given 5V(x' -l- 5x + 28) = x' + 5x + 4 to find the values | ofx. . OPERATION. x»+5x + 4-6V(x»+6x+28) = (x»+ 6x + 28) - 6(x« + 6x +28)'= 24 («*f64H-28)-6(x»+5x+a8)*+V=^S^ (x» + »« + 28)*-|»±V^ (0 (n) (m) (IV) = (i) with 24 added to| each side. = (n) with ay added. = (m) T7ith y taken, Abt. 210.] QUAPJIATIO EQUATIONS, 167 (x» + 6* + 38)* = 8or-3 xi-f 5x-(-28=>64or 9 x'+6xs36or- 19 X + f » ± V or i iV-5i X = 4 or - 9 ; or J(- 5 ± V-61) (V) (yi) (vn) (vm) («) (X) = (it) tranap. and red. = (y) squared, s (yi) tramp, and re4. =: (yn) with (|)' added to s (yiii)with sq. root taken = (ix) transpi and red. ^OTli«-^I|i thin eiwmple we ebould find by trial that only the lint two roots, i. e. 4 and - 9 are roots of the proposed equation, the other two bein|p roots of the equation «* + (to + 4 + fiV(»* + 6a; + 28) = 0. « - «, (6x* + 10x»+l)(6a*+10a»+l) , ^ ^ ,^ ralues of x. OPIBATION. (5x*f 10x^1) (5o*+10o»4-l) (x*fl0x*^6)(oHl0a»+ 6) " "* 6x*+10x»-fl o» + 10a» + 5a ?» + 10x»+ 6x ~ 6a* + 10a* + 1 x» + 6«* + 10x» + 10«* + 6x + 1 x» - 6x* + 10x» - 10x» + 6x^1 1 + 6a 4- 10a? + 10a? + 6o* + o» ' 1 - 6a + 10a» - lOo? + 6a* - a» (x+iy (1 + fiy (x-l)» " (1- a)« x+ 1 1 + a x-1 ~ 1-0 2x 2 2 ~ 2a 1 a (I) (n) (in) (nr) (V) (VI) (yn) Ex. 7. Qiyen x^ - 1 a to find the yalues of x. 1 o* + i0a» + S "'^^^'^x ^ 6a*+10a'» + l = (u) taken thus : Den. + Num. Den. + Nun, DenT^Num," Den. - Nwm, s (m) bracketed. =: (ly) with i^ taken. = (y ) taken as in(m)aboye s (yi) cjincelled. OPRRATION. X.6-l = (x»+l)(x*-l) = x»+l-0 x»-l = (0 (n) (ra) (w) = (i) factored. |. 1 Equation (n) is satisfied by taj,^ either x?- 1 s or «'+ 1 » 0, and there- fore we consider x"- 1 s one root and x'4-1 = other root, and we get sepa- rately x*-f 1=0 and X*- 1 s 0. 168 QUADBATIO BQTJATI0N8. [Sbot. el (at+l)(x^4-l)xO (V) s (ni) fitotored. (*-l)(a5*f«+l)»0 (VI) 3 (it) fkotored. X+lsO (yn) a one fkotor of (t). «"-» + laO (vm) s other flMtor of (▼). S-lsO (a) = one fkctor of (ti). «?+«+laO (X) 3 other factor of (vi). .'. « 3 1, ar 3 - 1, « s 1(1 f V- 8) and X 3 |(- 1 i V- 8). Nova.— 17m. (yn) end (ix) give lu by tnuuqpoiitlon » sa -l and « b 1, end MlTlng the qwulntic equatloiiB (vin)jmd (z) we get the other fbur IOO«i«=:i(llV-8)Mld«=:i(.l±V-8). The ehore la of coone eqilTelent to finding the Ox, «izth roots of nnitj . Bi. 8. Oiren x^ + a^-fix'-l-x + lsOto find the ralnes of x. x*+x»-4x"+x + ls0 2 1 X x*+x-4+— + ;^ = <>. OPIBATIOir. (0 (n) . 1 1 6 x + -- = 2or-3 X (in) (IV) (V) (Tl) (yn) (vm) = (I) * x». 3 (n) transposed and arranged. 3 (m) yrith 2 added to each side. 3 (ly) diflforently expressed. 3 (y) with S4* completed by adding i to each side. 3 (yi) with V taken. 3 (yn) transposed and reduced. Thnf we get two distinct quadratic equations :— * I. x + — a2orx«-2x = -l whence x = 1 j ^|I. x + — = -3orx* + 3xs-.i whence x = i(>3 i ^6), !^f . 9. Giyei) x' f 3x = ^4 to fin4 tbe.ya|ue8 of ^^t Abt. 310.] QUADRATIC EQUATIONS. 169 OPIIATIOV. «*+7ar'a4x«+14« x*+7«» + ^a4«'+l4a: + V (I) (n) (in) (17) (V) -(i)x«. B (u), ix^ added to eiich tide. s (m) with aq. completed by adding V ^ ^ch side. > (iv) with V taken. This gireg U8 two separate quadratic equations :— I. «* + }=» 2«+l or aj'- 2x = whence x = 2 or ; and II. x'+is-2x- J orr' + 2x = -7 whence xa-1 iV'^* 49x' 48 6 Ex. 10. Oiyea — ; — + -a-40 = 9+ -to find the values of x. 4 X* • X 49x> 48 6 „ + >.. 49 = 9 4-- OPBRATION. (0 49x3 48 Q -^--49 + ;^= - + 9 49x> 49 1 6 -4— *»+?*-^+¥ + » 7x 7/1 \ • (n) (m) (IV) = (i) arraQgAd. • 1 a (u) with 3- added. = (m) with V taken. This also gives us two distinct quadratic equations : — 7x 7 1 I. -X- - -— - --7 + 3 or7x*-6x=16whencexs2or-l|;and II. 2 7x 2 -—■ = --— -3 or 7x*+ Gxa 12 whencex=|(-3iVS3). EzsBci«i LIU Find the values of x in the following equations :-^ 1. X -6V* = 16. 2. x" - 4x* X - 3, 3. X* + 20 a 14x» - 20. 4. x» + 7Vx' = 1 107 - 7x^ 6. x-3Vx + 6 = 2-Vx + 6. 6. 2x*-x»=4»6. 7. x« - 8x» a 613. 8. X + 5 a 6 + V« +^. w ■ 110 QUADRAHC EQVATIOl^S. [fltooT. IX * *;f'^' Art. hi.] I SIMULTANEOUS QUADRATICS. 171 37. (* - IX* - 2)(* - 3)(x - 4) 1 8. 38. (« - 1) (« - 2) (« - 3)(x - 4)^z - ft)(x - 6) (X - 7>(* - 8) ::(«>- 9x) (I7x> - lS3x + 230) + 401. * ^fllfe 39. (X -!)(«- 2) (X - 3) = (X ^- 1) (X + 2) (x + 3). ^W 40. (VxT 1 - 2)(V«T 1 - 3) + 5V{V*+i(V* + i- 6) + V«+i - M = 0. 4 1 . 8x*- 16x*f 4x'-lr - 2 ( 2x2 - 2x + l)V4x* -8x»- 4x»+ 3x^1= . 42. abx , 2(a + x)(o"c-ix'''-^) , . 6cx ox 43. 8x' + 22r» + 24x + 9 = 0. 44. 3x* - 4x» + nx" - 6x = - 5. x' + 2x(V3 - V6) - W135 + 8 xMx(V3rV5W2(V30-V32) X + V3 - V*^ *** X-V3 + V6 = x» - 8 - V^. SIMULTAl^OUS EQUATIONS OF THE SECOND DEGREE. 211: No general rule can be given for the solution ^of quadratlb e<|uations inyolving more than one unknown quantity. In dealing with these therefore the student must bo left Very much to%i8 own Ifi^nuity. Very often by attentively considering (he question an artifice will. suggest itself, by means of which the roots may be easily found. The following solutions afford illustrations of the employment of artifio4j^which are very frequently used with much advantagQi ^ Ex. I. Gitenx2-y = X +y = find the values of x and y. •••■; x2-»2 = 61 i x + y = 17 X-J^ = '3 , , 2x = 20 X s 10 2y = u y* 7 OPERATION. (0 (") (in) (IV) (v) (VI) (vn) = (i)-(n). = (II) Hr(m)^ = (iv)V2. = (n) - (m). = (v)V2. ^ 172 \»- «. 8IMULTANB0US QUADKATOC(|._.^. Mv / }J ■ Art. 21 Ex. 2. Oiven x* + j^m 741 V to find the valuof of x and y. X + y s 12J OPIBATIOll. x» + y»=74 (I) • X +> = 12 (n) (HI) . x*+2xy + y'ai44 e (ii) sqtfared. 2xy m 70 (IV) = (III) - (1). . x' - 2xy + y* = 4 (V) = (I) - (IV). x-y a 2 (VI) » (t) with V taken. 2x=14.-. x=7 (vii) = (II) + (VI). 2y = 10 .'. y = 5 (viu) = (U)-(VI). Or ihm :— xHy^s 74 (0 x + y » 12 (") (III) ' x = l2-y = (ii) transposed. x" = (12 - y)» (IV) = (m) squared. (l2-y)» + y»=74 (V) = (I) with (12-y)» subs, for x\ 144-24y+y'+y'a74 (VI) =:(v) expanded. 2y»-24ys-70 (yu)j p (vi) transposed. y='-12y = -36 (VI^ = (vu) T 2. y='-12y + 36=l («) s (vm) with sq. completed bjr adding 36 to each side. y-6=il (X) = (ix) with V taken. y = 7 or 6 (3 «) s (x) transposed. Then x = 12 - y = 12 - 7 or 12 •♦ 6 «= 6 or 7. Ex. 3. Oiven x -i- y - 33 xy = 266 0|BIU.TION h find the values of x and y. X + y = 33 xy - 266 x» + 2xy + y2 = 1089 4xy =.1064 x" - 2xy + y* = 25 x-y = ± 6 2x = 38 or 28 .*. x = 19 or 14 2y B 28 or 38 .*. y = 14 or 19 (0 («) (m) (IV) (V) (VI) (vif) = (i) squared. = (u) X 4. = (m) - (IV). = (v) with V. taken. = (I) + (VI). « (I) - (VI), r ART. all.] SIMULTANEOUS QUADRATICS. 178 Orthut 4y r»- « + 3f a 83 xya 266 X s 38 - y y(33-y)s266 y - 33y a - 266 132y 4* (33)' « 26 2y - 33 = ± 6 2y = 38 or 28 y = 19 or 14 Ex. 4. Given 2ar» + 3*y+y» 6x» (0 (") (III) (IV) (VI) (vii) (VIll) (JX) y+y»3 20*l + 4y»=:41J a (i) tnin8|ioBed. s (n) with 33 - y Bub. for or. ' (nr) expanded and x - 1. 3(v) X 4 and witli 1089 added to each side. = (vi) with V taken. 3 (vii) transposed. m (vm) + 2. to find the values of x. OPIRATION. In equations like this, in which either or both of the equations are homogeneow in all those terms which involve these quan- tities, put X = vy, then x* = v*y*, and xy = vy', and the solution will lie much facilitated. 2x«+3xy + y»=20 j 6x« + 4y» = 41 ) 2»V + 3tly* + y* = 20 HvY + 4y' = 41 (2»« + 3r + l)y» = 20 (Bv' + 4)y» = 41 20 * " 2»'' + 3»+l y 6t» + 4 20 41 2«'' + 3t>+l " 6»* + 4 6»2-41t; = -13 « a i or »^ „ 41 41 ^"^ 6i>« + 4 " (n) (in) (IV) (V) (VI) (VII) (vni) («) (X) (XI) 41 or; = (i) with vy written for x. = (II) with vy subs, for x. = (III) factored. = (IV) factored. ~ (v) ^ (2r2 + 3d + 1). \ a (VI) T (5»» + 4). a right hand members of (vn) and (vin) equated to one another (Ax. xr).^ a (ix) reduced. - (x)8olved by ordinary rule = 9 or -jV* Hence y a 3 oi^ " 6(i)^4"'^6(V)»t4 ViT^^Tl. _ __ x a t>y = i X 3 or V >« AV21 * 1 or ifV21. 174 SIMULTANEOUS QUADRATICS. [Sect. IX. Ex. 5. Given x^ + y^- 189) , . „ , J to find the values of x and y. x^y + xy^ = 180 J OPERATION. In order to show that several different plans may generally be adopted in dealing with simultaneous quadratics, so as to evolve the values of x and y, we shall give two or three different solu- tions of this problem. 1st Method. T^ + j/a = 189 (I) (") (III) x^y + xy'^ = 180 ^x^y + Zxy^ = 540 x" + Zx'hi + Zxy'^ + y** = 729 a: + y = 9 ^ xy(x + y) = 180 xy = 20 (IV) (V) (VII) :- (11) X 3, = (I) + (m). = (iv) with ^ taken. = (ii) factored. = (VI) V (v). Hence z = 9 - y ; xy = y(9 -y) = 20 or y^ - 9y = - 20, whence y = 5 or 4 and .r = 4 or 5. 2nd Method. u:» + y» = 189 x^y + xy^ = 180 xy (a; + y) - 180 180 X + y = — - ^ xy 180" x' + 3x^ + Bxy'^ + y^ ^ -^ 3xy(x+y) = xy(x^ry) 180=* 5832000-1 89a; Y 3x'y + 3x1/=* = ^ - 189 x'^y^ 1944000- 63a; V 3.,a ^iiy'i 180 = 1944000 - 63a;V xY ISOa-y = 1944000 - 63aY 243a:V= 1944000 x'y^ = 8000 xy = 20 0) (") (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) = (ii) factored. = (III) -r xy. - (iv) raised to 3rd power. = (v)-(i). = (vi) simplified. = (vii) T 3. ' = (viii)with 180 substituted; for xy (x + y). = (ix) cleared of fractions. = (x) transposed. = (xi) V 243. = (xii) with ^ taken. Abt. 211.1 SIMULTANEOUS QUADRATICS. 175 Then, as before, since xy(x + y)~ 180 and xy = 20 .-. x f y = 9 and a; ■• 9 - y, whence y(9 - y) = 20 or y* - 9y =• - 20, wherefore y = 5 or 4 and x = 4 or 5. 3kd Method. «» + y* = 189 (I) a*y + xy2 = 180 (ii) (V + zf + (v- zy = 189 (III) 2v(»2 - -!«) a 180 2v' + 6vz^ =189 2t»'» - 2vz' ^ 180 6r' - Qvz' = 540 8i>V729 or2i;=9or» = f Sws** 9or8*^| = 9ors = + i Hence x = v + z = ^±^ = 5otA. (IV) (V) (VI) (VII) (VIII) (IX) = (n) with (t> 4- «) written for X and (v - z) for y. = (ui) written thus, xy(x-{^) and then (v + z) and v-z substituted for x and y. = (III) expanded and red. = (iv) expanded. = (vi) X 3. = (V) + (VII). = (V) - (VI). yr:v-s = ^^-(f.i) = |+i=4or &. 4th Method. ar' + y» =r 189 x'^ + xy"^ =180 xy(x + y) = 180 180 a? + y = » xy 189a:y x^- xy ■\-y'' - v^y^ - wy^ + y'' = 180 2891^ "TscT 180t»»y^l80»y'»+180y2= 189i>y'' 20»2 - 41i> + 20 = 20w2 - 41v = - 20 180 v'Y + «y" = 180ory'= D-' + i; (0 (") (HI) (IV) (V) (VI) (VII) (VIII) which (IX) = (ii) factored. - (in) -f xy. - (0 - (IV). = (v) with vy subs, for x. =■ (IV) X 180. = (vii) trans, and v 9y"^. is a quadratic equation, whence v = f or |^. = (ii) with vy subs, for x. 180 180 Hence y* = ^s . 6 or tt -— r = 64 or 125 whence y = 4 or 5 and X « 6 or 4. 176 SIMULTANEOUS QUADRATICS. MXi In order to save figures, the second method is better at>plied by letting x + y = $ and xy = p, then x" + y» = 189 xhf + xy^ - 180 «* - 3sp = 189 «p= 180 180 s = P 180» P" 180^ 3sp = — ,- - 189 p 180' 540 = -189 729 = 180» p3 180 9 = P ;> = 20 «p= 180 .'. s = 9 0) (u) (m) (IV) (VI) (vii) (VIII) (IX) (X) (XI) (XII) •.• x^+V--(ar+y)'- Zxy(xMf). '.' x^ + xy^ = xy(x + y). = (m)4-^. = (v) cubed. = (V) - (III). = (iv) X 3 and subs, for left- hand member. = (vm) transposed. = (ix) with ^ taken. = (x) X p and v 9. = (iv) vsrith value of p. subs. = R . X ^y = D 4. a:2 + y''«= 113 X -y = '15 7. x^ + 3y^ = U8 2x + y= 24 10. a;3 - j/« = 26 X -y=: 2 13. w + 4y = 14 2/^ + 40: = 22/4-11 9a; + 5y Hence p = xy = 20, and s = x + y = 9, &c. EXBRCISB LIV. Find the values of x and y in the following equations : — 1. x«-y2= 45^ 2. x^-y^=lQ5) 3.x^ + y^ = 4l X + y - 21) x + y = 9 5. a;"^ + y^= 89 ) 6. a;^ - i/2 = 55 a:y = . 40 J 3a;y = 72 8. 3x^~2y^=yttl^^4:xH3y^=5U 2x -3y = Sf 3ta? + 2y = 27 11. a: + ^B4 ) 12. VaJ + Vy = 3 x'+f=(x+yy\ HJxy 14. 2«"+ary-5y'= 20^ 'a:y = 2j K 2x - By = 1 ) 15., X - V = 2 J 16. .'rY + 4xy = 96 /■ Ixy = 96 ) + y= 6*) AiW. 211.] SIMULTANEOUS QUADRATICS. 177 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 38. 4x \ y ^ X -y = 2 } x^ + xy = 66 x^-^= 11 ar" + y« = 3368 ) x+y=8 j x* + y*= 97 X -y = fl x + y _ 3 xy ~ 4 X + y- 13 = 13 18. x^ + jpy = 77 a;y - y=* = 12 20. ~ + — -_ 18 ; y X L x + y= 12) 22. x3 + y3= 133) x+y= 7 j 24. x^'h- y« = 91 x'^y + xy'-* = 84 X + y X - y 26. ^ ^ X + y = X* 7y - 2x = 36 x^ + 2y''' = 74 - xy 2xy + y2 = 73 - 2x2 3x2 ^ 2xy - 4y2 = 108 x^ - 3xy - 7y''* = - 81 -V x + y x-y 'v 7 26. ^ + = ^H > x-y x+y > -x^-yV x2 + y2=52; . X* + y* = 14xV ) X + y = 7/1 J ^ 28 30. X* - x^ + y4 - y2 = 84 ; x* + 2xV + y'"* = 85 J ^ 32. y2 - x2 - y - X = 12 (y-x)2(y+x) = 48 2x + y y y2+XN = 20-^^ / y \ = 41/ J X + 8 = 4y VCy'H- 1)+ 1 V(x+9)+3 ] ar(y+l)2 =36(y»+15)J (x6 + l)y = (y2 + l)x» (y« + l)x = 9(x2+l)y3' 34. x8 + y3 = 36 ) x^ + y2 =: 13 j 36. x* + y* ^ s\ x» + y" - 1 ♦, X X 27 y 4 X x-y = 2 V(5V^ + 5Vy) + Vy = 10 - Vx" + Vy" = x* + y" = X - y ) } 39 40 '41. xy + fl(x 275 X* + y2 = axy j + fl(x - y) = a* ) X + y2 + a' - 5 178 SIMULTANEOUS QUADRATICS. [Sbot. IX. 42. x'-* + y^ + o^ - x^ + y* + a* + x\3y'' + a=*) = J 43. a:2 + 3y + a8 _ ) x6 _ 3j^ ^. a6 + a;Zy(3x'2 -, y) = a^x\x^ + 2) ) 44. a; - y = a > X* + y* = 6* ) 45. X* - xy + y2 = o^ X* - x*y2 + y* = 6* 46. 3x6 _ 12x4 + xax'-* = 2y6 - lly* + 52y*'' + 27Ato find x and y X* - y* - 3 + 2x2(a-l) - 2a(y=*-l) + 2y2(x='-l) ] independent of a 47. (y2- x'')(y2-x2+4)+5 = 2VT(yS-x6) -(5x^+T2 xy-5y'')(y'^^^") y* - Sy'' - 1 3.r^ - 8x(l - Vx^ 2x + 5) + 4 48. (a;2 - y-^)(x2 + y^ _ 4) = 4(x'^ - 3) | x2y2 + 7(x''' - y-*) = 6xyVy^ - x'^" ) , PROBLEMS PRODUCING QUADRATIC EQUATIONS. 1. What two numbers are those whose difference is 5 and the product of whose sum by the greater is 228 ? SOLUTION. Let X = the greater, then x - 5 = the less. x + x-5 = 2x-5 = their sum. Then x(2x - 5) = 228 (i) 2x''* - 5x = 228 (II) 16x2-40x + 25 = 1849(111) 4x - 5 = ± 43 (IV) 4x = 48 or - 38 .'. X = 12 or - 9J = the greater. X - 5 = 7 or - 14J = the less. 2. A poulterer bought 15 ducks and 12 turkeys for 105 shillings, at the rate of 2 ducks more for 18 shillings than of turkeys for 20 shillings. What was the pr-'ce of each ? = (11) X 8, then sq. completed. = (ni) with V taken. Art. 211.1 PROBLEMS IN QUADRATICS. 179 SOLUTION. Let X = price of a duck in shillings and y = price of a turkey. (0 (") Then 15x + Uy = 105 ♦ 18 20 — = — + 2 X y 5x + 4y = 35 9y - lOx = xy 10a: + 8y = 70 I7y = xy + 70 35 - Ay X = ny-2/| 35 - 4y )■ 70 (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) = (I) reduced. = (ii) reduced. = (III) X 2. = (IV) + (V). • = (in) transposed and reduced. 35 -4v - (vi) with — ^ subs, for x. 2f + 25y=ll5 \6y'^ + 2001/ + 625 = 2025 4y + 25 = + 45. 4y = 20 or - 70 whence y = 5s 35 • - Ay _ 35-20 = (viii) reduced. = (ix) X 8 and sq. complete. X - = 3s. NoTK,— The negative value - ITs. 6d. lor tho price of a turkey is not taken into account here, as although - 17} is undoubtudly a root of the equation 2//'' + 25j/ = 175, yet - 17s. Gd. as the price of a turkey does not satisfy the conditions of the problem as given and must therefore he neglected. 3. Find a I'Viinber such that the sum of its square and its cube shall be nine times the next higher number. SOLUTION. Let X - the number, then x^ = its square, and .r" - its cube ; also a: + 1 = the next higher nuuber. Then x^ ■{■ x^ ^ 9(i + 1) (0 x\x i- 1) = 9(x + 1) (") - (i) factored. a;^ = 9 'n) = (ll) -f X -1- 1. a; = i3 iiv) = (III) with V taken Verification. Take + 3 ; then 27 1 J = 36 = 9(3 + 1). Take -3; then-:^7 f9 -- 18 - 9(-3+l)-9x-2. 180 PROBLEMS IN QUADRATICS. [Sect. IX. 4. A person at pley won, at the first game, as much money as lie had in his pocket ; at the second game he won 6 shillings more than the squara root of what he then had ; at the third game he won the square of all that he then had, and he found that he then possessed J£112 16s. What had he at first 7 SOLUTION. Let X = the shillings he had at first. Then 2x = the shillings he had at the end of the 1st game. '/'ix + 5 = sum won at the 2nd game. 2x + V2a: + 5 = sum at end of 2nd game. (2x + V2-r + 5)'' = sum won at 3rd game. (2x + V2x + 5)'* + 2x + '\/2x + 5) = sum at the end of the 3rd game. Then (2a:+V2a; + 5)2.-H(2ar+V2x-|-5)r- i256 (2ar+V2a;+6)''' + (2a:+V2^+5) + ] = ^V'* (2.T + V2x + 5) + J = i %^ 2x + V2i = 42 or - 53 Rejecting the negative result we (2a:) + V^I- = 42 (2x-) + V2^ + i ^ "-f "* V 2^ f J = + V ^/2x = 6 or - 7 2x = 36 or 49 X = 18s. (0 I (li) = (i) with i added. (Ill) ] = (ii) with V taken, (iv) [ = (in) transposed. i have (V) (VI) (vii) (vni) (IX) = (v) with sq. comp. - (vi) with V taken = (vii) transposed. = (viii) squared. - (IX) V 2. Note.— The 24^ whivjh ve get hero as ono value of a; is not admissible as an answer to the problt^An, simply bccai " it does not answer the conditions of the problem as ^.vcit, aud it obviouHiy arises ft'om the fact that the VZx may be either ±. It becomes an answer of the problem if we iinder- st&nd that at the 2nd game he /os£ a sum which was 5 shillings less than the square root of what he then had. • 5. What number is that which being divided by the product of its digits, the quotient is 2, and if 27 be added to the number, the di»its will be inverted ? AuT. 211.] PROBLEMS IN QUADRATICS. 181 SOLUTION. Let X and y = the digits, x being the left-hand one. Then lOx + y = the number, and xy = the product of the digits lOx + y xy 10x + y + 2l = 10y + x) x = y-3 lOx + y = 2xy lG(y-3)+y = 2y(y-3) 2y^- I7y = -30 W/-136y + (liy=49 4y-17 = ±7 (0 (") (111) (IV) (V) (VI) (VII) (VIII) = (ii) reduced and transposed. = (I) X xy. = (IV) wiih y - 3 subs, for x. = (v) reduced and transposed. = (vi) X 8 and with sq. complete. = (vii) with V taken. 4y = 24; y=6; a; = y-3 = 6-3 = 3 Hence the required number is 36. XoTi:.— The second value of y is obviously not admissible hero. 6. A and B travelled on the same road and at the same rate to London. At the 50th milestone from London A overtook a flock of geese, which travelled at the rate of 3 miles in 2 hours, and 2 hours afterwards he met a waggon which travelled at the rate of 9 miles in 4 hours. B overtook the flock of geese at the 45th milestone from London, and met the waggon 40 minutes before he came to tlva 3l8t milestone. Where was B when A reached London ? SOLUTION. A and B travel in the same direcUon, at the same rate, and on the same road, and consequently the distance between them is always the same. Let x = rate per hour of travelling. The places where A and B overtook the geese are 5 miles apart, and as the geese travel at the rate of ^ of a mile per hour, to travel over 5 miles they would require 5 ^ § = ^^ hours. But in lOx •3^ hours A has moved on - -- miles, while the geese have moved on only 5 miles. 10a; Therefore distance in miles between .4 and £ - — - o 5. 182 PROBLEMS IN QUADRATICS. ISkct. IX. Again, A met the waggon 50 - 2x miles from London, 2x while B met it 31 + — miles from London, consequently as the waggon was travelling frofin London, the distance in miles / 2x\ travelled by the waggon between the two meeting was ( 31 + — ) 8JJ-57 \ ^J - (50 - 2x) = — - — miles. And since the waggon travelled at ., 8a; -57 32a; - 228 the rate of '} miles per hour, — ^ — -^ 4 = — 07 ~ ^*™® ^^ hours which elapsed between the meeting. „ . 32a: -228 But in — ^^ hours J has moved toward London 32a: - 228\ 1 X miles while the waggon has gone in the opposite ( 27 /8a; -57 direction 1 -^-^ ) miles. Therefore distance in miles between j1 and B - 8a; -57 32a;2 - 228x- 2f~ And since distance between j1 and B is always the same, 32.t3_228x 8X-57 10a; + -5 0) (") (ni) (IV) = (i) reduced. = (11) X 64 and with sq. then completed. ^ = (III) with V taken. 27 ' 3 16x2 - 123x = 189 1024x^-7872a;+ (123)^= 27225 32x - 123 = 165 165 + 123 X = ^ — = 9 = rate per hour of travelling. lOx Distance of B from A = —- — 5 = -'f -5=25 miles - distance 3 •* of B from London when ji arrives there. Exercise LV. 1. Divide the number 19 into two pajrts such that their pro- duct shall be 84. 2. What two numbers are those whose sum = 1 7, and the product of who5'j difference by the greater is 30. ;t. IX. Art. 211.] PROBLEMS IN QUADRATICS. 183 as the 1 miles 2x\ jlled at time in London opposite 8 _ 228x- "27 imc, with sq. id, len. distance their pro- and the ' 3. There is a rectangular field whose area is 2080 rods, ajid its length exceeds its breadth by 12 rods. Required its dimen- sions. , 4. What two numbers are those whose difference is 9, and the sum of whose squares is 353 ? 5. Divide the 16 into two parts such that their product added to the sum of their squares shall be 208. 6. A commission merchant sold a quantity of wheat for $171, and gained as mnch per cent, as the wheat cost him. What was the price of the wheat ? 7. A person bought a number of sheep for $80, and found that if he had bought 4 more for the same sum, they would have each cost $1 less. How many did he buy ? 8. A certain number consisting of three digits is such that the sum of the squares of the digits, without considering their position, is 104, and the square of the middle digit exceeds twice the product of the other two by 4 ; also if 594 be subtracted from the jiumber its digits will be inverted. Required the number. 9. A farmer paid $240 for a certain number of sheep, out of which he reserved 15, and sold the remainder for $216, gaining 40 cents a-head on those he sold. How many sheep did he buy, and what was the price of each ? 10. What two numbers are those whose sum is 10, and the sum of whose cubes is 280 ? 11. What are the two parts of 24 whose product is equal to 35 times their difference. 12. Find two numbers such that their sum, their product, and the difference of their squares are all equal to one another. 13. This fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 129 yards, but if the circumference of each had been increased one yard, the fore-wheel would have made only 4 revolutions more than the hind-wheel in going the same distance. What is the circumference of each wheel ? 14. The sum of two fractions is 1 ]■} and the sum of their reciprocals is 2^**^^. What are thTj two fractions ? 15 A person dies leaving $46800 to be divided equally among his children. It chances, ho\vever, that immediately after the ) 184 , PROBLEMS IN QUADRATICS. [8bot. IX. ^ T death of the father two of his children also die, and in conse- quence of this each remaining diild receives $1960 iriore than it was entitled to by the father's will. How many cLUdren were there? 16. During the time that the shadow of a sun-dial which shows true time, moves from one o'clock to five, a clock which is too fast by a certain number of hours and minutes, strikes a number of strokes, which is equal to that number of hours and minutes, and it is observed that the number of minntes is less by 41 than the square of the number which the clock strikes at the last time of striking. The clock does not strike 12 during the time. How much is it tco fast ? 17. Two locomotives commeoce running at the same time from the two extremities of a railroad 324 miles in length ; one travelling 3 miles an hour faster than the other, and they meet after having travelled as many hours as the slower travelled miles per hour. Required the distance travelled by each. 18. A person ordered $144 to be distributed among some poor people ; but, before the money was divided there came in two claimants mere by which means the share of each was $1 below what it would otherwise have been. What was the number at first? 19. Find a number such that, being divided by the product of its. two digits the quotient is 2 ; and 27 being ;^4^d,. to the numbj^r its digits are inverted. ; > >;!.;i?;; 2Q; A grocer sold 60 lbs. of coffee and 804^. pf^sugar for $25, but he sold 24 lbs. more of sugar for $8 tha^ hj^did of coffee for $10. What was the price of a lb. of each ? ; • 21. A and B engage to cradle a field of grain for $3(6, and as A alone could cradle it in 18 days, they promise to complete it ,, in 10 days. They found however that they were obliged to call in G, an inferior workman, to assist them for the last foiir days, in consequence of which B received $1'50 less tha^ h,e would otherwise _ have done. In what time could E ox separately r^apjtl^ field? , < . 22. A reotf^gular vat 5 feet deep,holds, when fillea to the depth of 4 fee|;r, less than when completely filled by a number of cubic feet equ^,;tO;80, together with half the , nuigalbi^' of feet in Sfcv ^:: amt. ai2.] RATIOS. 185 the perimeter of the botie. It is also observed that the length of a pole, which reaches from one of the corners of the top to the opposite corner of the bottom of the rat, is equal to -4^^ of the number of feet in the 8<)|iare inscribed on the diagonal of the bottom. Required the dimensions of the vat. 23. Two persons set out at the same time to travel on foot, A from Toronto to Gobourg, and B from Cobourg to Toronto. When they meet it is found that A has travelled 15 miles more than B, and that A will reach Cobourg in 2 hours ; anfl B, Toronto in 4} hours after they have met. Find the tlistance , een Toronto and Oobourg and the rate of travelling of eacl 24. Find two numbers such that their product shall be equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes. 26. Bacchus caught Silenus asleep by the side of a full cask, and seized the opportunity of drinking, which he continued for § of the time that Silenus would have taken to empty the whole cask. Silenus then awoke and drank what Bacchus had left. Had they- drank both together it would have been emptied two hours sooner, and Bacchus w^uld have drank only half what he left Silenus. How long would it have taken each to empty the cask separately ? SECTION X. mTIO, PROPORTION^ AND VARIATION. RATIO. 212. Batio is the relation one quantity bears to another in regard to magnitude, the comparison being made by considering what multiple or fraction the first is of the second. • NOTV.— It will he seep from this definition thut tb6 terra ratio is equiva< lent td the ooninon arithmetical term gttof tent. N ' ■ IMAGE EVALUATION TEST TARGET (MT-3) ^^^^.^ 1.0 I.I 1.25 £ 1^ 12.0 ■ Ht'A ^> ^ ^V.-^ s ^^^* ^ Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. I4S80 (716)a72-4S03 186 RATIOS. [SaOT. X.I abto. 21 213. The ratio of one quantity to another is expressedl by placing a colon between them or by writing them in thc| form of a fraction. Thus, tho ratio of a to 6 is written a : b or more oonunonly -^ 214. Ratio can exist, of course, only between quantities! of the same kind, because it is only between such quantif ties that any comparison as to magnitude can be instituted. 216. Quantities are of the same kind when one can \x\ multiplied so as to exceed the other. Thus, a ratio can exist between a cent and £100, or between a aquartl noh and an acre, or between a grain troy and a owt., because in each C8a«| the one can be multiplied so as to exceed the other, or, in other words th quantities entering into the ratio are of the same kind ; but no ratio cul exSst between a linear inch and an acre, because the former cannot h| mnltiplied so as to exceed the latter. 216. The term of the ratio which precedes the sign; or which is written as numerator of the fraction is calle the antecedent of the ratio, the remaining term, the am««9U€}t/.| 217. A ratio is said to be a ratio of greater inequalim a ratio of equality y or a ratio of less inequality ^ accordin as the antecedent is > , =, or < the consequent. • « 218. If the antecedents of any ratios be multipled gether and also the consequents, there is formed a neij ratio which is said to be compounded of the former ratio Thus, the ratio ace : bcy^ said to be compounded of the ratios a: b, c:i and : /. 219. A ratio compounded of two ratios is ci^ed sum of these ratios, thus, when the ratio a: & is con pounded with itself the resulting ratio a* : I? is called •double of the ratio a : h or more commonly the duplic ratio of a : & ; also the ratio a^ : h^ is called the triple ISaoT. X.I Arts. 218-222.] EATIOS. I8t the ratio a : 6 or more commonly the triplicate, ratio of a : 6. NoTB.— Similarly the ratio '^n -. Vfr is called the t/ubdupliccUe, the ratio ^a : 3^, the mihtriplioate;a«~: d*, the seequiplioateof the ratio a : (, &o. 220. Problems upon ratios are solved by writing the I ratios as fractions and treating these 'fyRjtions by the ordi- nary rules. Ratios are compared with* one another as to [magnitude by writing them as fractions, reducing these fractions to a common denominator and comparing the 1 numerators. 221. Thiorbu I. — A ratio of greater inequality ir diminishedt land a ratio of lest ineqiuUity increased by adding the same quantity I to both its terms. m I Demonstbation.— Lot a : 6 be a ratio of inequality, and let x be added I to each term. Then* — ^ Mab4-ax < & then oa; > 6a? and oft + oo? > a6 + 6a; and _ > — - — ; but it 6 6 + a; a aJL. X |o < 6 then oa; < &ar and a6 + oo; < o6 -f 6* and — < j— — ■* » 1 • 1 I 1 ^1 '* • o -T- a> lultipled tOH "' Bead — is greater than qr leas than ' according as, &c. 222. Thsorbu II. — A ratio of greater inequality is increased^ land a ratio of less inequality diminished by subtracting the same \(Mntityfrom both its terms.* DaxoNBTBATiON.— Let a : 6 be a ratio of inequality, and let ar be mib. ^racted from each term. Then ■— ^ ~ , a8a6-aa!^a6-&a;;oras6a;':i-aa:ora86^a. h<. b -X ^ ^ ^ M *v? I *Tbe quantity subtracted must howerer be less than either of the terms. 188 PROPORTION. [Sftcrr. ll VI 223. j1 ratio is increasetl or dimini^ai by being compound with another ratio according as the latter is a ratio of greater i less inequality^ DEMOKSTBATioir.— Let the ratio a : b he compounded with the rati m : n, the latter being a ratio of inequality. a ^ am ^ ^ Then — ^ -7—, according as oftn o abm, or as n <^ m, or as m : n iiJ o *^ on --^ ^ I ratio of greater or less inequality. Exercise LVI. c : a' ; and ah \ 1. Find the ratio compounded of a : & ; 2. Compound together the ratios a^ - b^ anda2-fl* + 62 ; (a-bf. 3. Compound together the ratios x'-'- 22;- 15 : x^-3x-\ti\ x^ + x-^2 : a;"'* + 8x + 15 and x' + 12x + 35 : x'^-l. 4. Which is the greater ratio that of o* + 6=* : (i' + bU a' + b^ : a + b. 6. Which is the greater ratio that of x^ + y=* : x^ -fi (x + y)* : a:* - X* y + xi' y^ - xy^ + y* ; x ^5 being > y^*I. 6. What quantity must be subtracted from each term of tlj ratio a : 6 in order to make it equal to the ratio c : d. 7. What quantity must be added to each term of the is^ m : n in order to convert it into a ratio of equality. 8. If a : 6 be a ratio of greater inequality, what is the ral] compounded of the ratio of a + 6 : a -6, the difiference of] duplicate ratios of a : a and a : 6, and the triplicate ratio] b : a + b. ^ 9. Prove that th^ ratio a : 6 is the duplicate of the ratiof a -t- c to 6 + c, if c be a mean proportional between a and 6. 10. Prove that a^-b^ : a^ + 6''' is greater or less than the 1 of a -6 : a + 6 according as a : 6 is a ratio of greater or in«quality. PROPORTION. ^ 224. Proportion consists in un equality beiween ratios, the two equal ratios being connected by the sigt j or by the ordinary sign of equality. ISl;' i.. ^ ^-,.. &i£-i. ISIWT. lB^„g. 228-227.1 PROPORTION. 189 4 with the rati or as m : n at For exampid, if a, b, e, and d be four proportional quantitien, the pro- jportion existing between them is expressed by writing them thus, b : : c : d. . NoTK 1.— The first and fourth of such proportional quantities arc called ' I extrtmes ; and the second and third, the means. NOTB 2.— When three quantities a, b and e, are proportionals, so that b : xb '. e ', the second term, b is said to be a mean proportional etweon the other two, and the third term c is called a third proportional |to the other two. • ■ 225. Thbobbk 1.^-1/ four quantities be proportionals, the product Wthe extremes is equal to the product of the means. Dkmonbtration.— Let a : b : : c : d, then ad = bc. a c » For — = — and multiplying each of these by bd wo have ad = bc. b d Coa Hence if three terms of a proportion are given, the fourth may be bo ad ad be oadily found. Thui,a= — ; 6 = — ; c=— ; d = — d c b a 226. Thborkm ll.^If the product of any two quantities be ^qud to the product of any two others, the four are praportUnuds % -the factors of either product being made the extremes, andtk$ ^factors of the other product the means. Demonstration.— Lot ad — be, then dividing each of tbesidby bdvad we a c ave — = —that is a : 6 b d d. 227. Since the two ratios composing a proportion may be written as two eqaal fractions, it follows that all the results ob^ ined in Art. 106 may be applieil to proportional qaantitief, or In other words, we may combine together in any manner whatn |rer by addition or subtraction the first and second terms of a proportion, provided we similarly combine the third and fourth erma. So also we may proceed with an^( \ ■I & ■I 190 PROPORTION. Sect. X. ■ Art. 228. In solving ' problems in proportion the student must carefully ]feav the' last proposition (22 1) in mind, and also that :— I. Any proportion may be converted into an equation by taking the product of the extremes equal to the pro- duct of the means. II. Any proportion may be converted into an equation, by | vtrriting the fifst term divided by the second = the third term divided by the fourth. b Ex. 1. If o : 6 :: c : d prove that (o + 6) (c + d) = -^(c+df b ^ a c be , ad Here -r =-;.•.«= -r and c = — - odd b be In the expression (a +6)(c + d) substitute "^ for a, and we /be \ /be+bd\ have (a + 6) (c + d) = ( "^ + * ) (c + d)= I — - — 1 (e + d) = 6 6 -(c + d) (c + d)=-(c + d)3 ' a d ad Similarly in the expression (a + b)(e + d) substitute — for c,\ /ad .\^ i This gives us (a + 6) (c + d) = (a + 6) f— + d j = (a + ftn /ad-{-bd\ d d (j-j-yid + b) ia+b)- = -^(a+by. .. Ex. 2.— Given «^ + y** : ar*-^^. : : 559 : 127 and «*y = 294 to| find the values of x and y. OPERATION. 127r« + 121 f = 559a:»- 559y» or Q86f = 432x3 q^ 343^8 _ 2II or 7y = 6x .*. y = fx. Substitute this value of . y in the secoitl equation and we hcive 6*» x5 = 294 or x2 X ^ X = 294 or — = 294, or ~ = 49 j ^ if .«^43 J or X = 7, whence y = 6. ■!fcHrf3&--f^Vi; Sect. X ■ Art. 228.] PROPORTION. 191 Ex. 3.—- If a: b : : e : d and also m : n : : p : q. Prove that mo^ nb :*ma •'ttb : : pe + qd : pc-qd. % a c Since a : b : : e : d and m : n : : p : q^ then , = j- and <» - a — . Multiplying these equals together, we have .x — « ? on -r n ma pe - o' -r^S-- • Then, Art. 106 (vii), q nb qd . ma + nb pe + qd ma " nb pc — qd j that is ma +nb : ma-rib : : pc + qd : pc~qd. ExiBciSB LYII. 1. If (I, 5, r, d be any four quantities whatever, 'find what I quantity added to each will make them proportionals. 2. If four numbers be proportionals show that there is no I number which, being^added to each will leave the resulting four I numbers proportionals. 3. If a : b :[: c :d andm : n : : p: g prove that ma' > 2n^ : \pc^-2q

229. Variation is an, abridged method of indicating pn^rtion, and is oonvenientty used in investigating the relation Ifrhich varying hut dependent quantities bear to one another. The two ternu of a variation are the ikwo antecedents of the oorrespoud- ing proportion— the consequents not being expressed. , Thus, when we say tiie interest varies iu the prineipal,' we mean thai if P and p be any two pifncipals and /and i, the corresponding interests at a given rate and time, then / : < : : P : p or briefly, omitting the consequents, Jk P. 230. The sign cc is called the sign of variation and is read «(:(n6« a«. VtuUffTkPiBretA, IvariesaaP. . , 21^1. Onbe quantity is said to vdrif directljf ai^ an^Jtlitif when the two quantities depend upon eaoh^otiber, ^ ^ if one be changed in any manii|ir/,1ibe <#er.i||i|f^ changted in the sanle proportion^ '1 ~! ABT8. 22a-2M.] VABIAIION. 193 Thus, iMving time Mid rate per cent, out of consideration, the interest (7) vules directly m the principal {F), fbr if /is changed to i, P most also Jkeohangedtopinsvohainannerthat/: < : : P :/>. * Nonv.— When we simply say that one quantity vaiHe$ as another, we are always understood to mean that the one varies directly as the other. 232. One quantity is said to vary inversely 9b another when the first cannot he changed in any manner, hut the recijorocal of the second is changed in the same pro- portion. Thus, Ate -^(A varies inversely as B), if, when A is changed to a, B •" 11 must be changed to b, so that A : a i :— : — :: b : B. B For example, if the area of a triangle be given the base varies inversely wi the altitude, for ISA and a be the altitudes and B and b the bases of two equal triangles, thenwlB=a6 .\ A '. a : ib i Box A : a x i -^ i ~ or A B b *5 ^ 233. One quantity is said to vary as two dthers Jointly^ if when the first is changed in any manner the prodtict of the other two is changed in the same proportion. That iBA A A \ A DKMONSTBATioir.— By Tboorom I,A= mUC, then B = — - s= — . I tqwil B then ernately A—mB an equa- theorem [ a JB, i. e bC. ntly, m andti loc Cand vyk'l B oe and C A^ mB m B C oc .4 C mB ~ m B ' ~ " Ji 289. Thiomm \.--If 4 oc B and C ac D, then AC oc BD. DxxoNSTRATiOH.— By Theorem I, A^^mB and C aenD; then .^Css mn^Bi) and .*. ACae BD. 240. Thiobim VI.— ^ Ace B then -4* oc JJ*. DmcoKBTRATioir.— By Theorem l,Az= mB, then -4" = nC^B*, but m is a constant quantity .'.A^ds B*> Note.— So also if ^ oc J3 then ^A « I^B. 241. Thiobim VII.— J/ A cc B and P be any other quantity A B then AP oc BP and p oc p DmiOKSTRATiON.— By Theorem J, A = mB hence PA = wiPB .'. PAx PB. Also ^ = toJ5 .•. — = -—- = m —.•. — oc — P P , P P P A Note.— Hence — Is constant, for Vi Ax B dividing both by B, we have A B ^ ^ — oc — oc 1. B B 242. ToBOREu VIII. — When three quantities are so related that the increase or decrease of one depends upon the increase or decrease of the other ttoOj in such a way that if either of these latter be invaritd)le the first varies as the other, then when both vary the first varies as their product. That is, if Ace B when C is constant and Ace C when B is constant, then A oc BC when both B and C are varitAle. Demonstbation .— The variations of A depends upon the variations of two other quantities B and C; let the variations of these take place separ- ately^and when B is changed to 6 let ^ be changed to a, and whou C is changed to c lot a be changed to a'. Then At a: : B : b; and a : a' : : C : c and by compounding these we have 4 : a' : : BCi be .'. (Art.229) A x BC. Note.— In a similar way it may bo shown that when there is any num* ber of quantities, A, B, C, D tfc, such that A varies as each of tiie others when thii niif^ are constant— then, when they are all changed* Jk'wAeti as their] L^MlfilJfMri^ ■ 'J-.'fc*^*-* «i ..x^.~a»A^:t.i:. fltr^'WW''". vflj-.W! ■ i 196 VARIATION. [SlOT. Z Ex. 1. l(x ccyx* and 2, 3 aud 6 be contemporaneous yaluet of x, y and «, express x in terms of vs* OPBRATIOM. Since x oc yt' .*. x s myx* and when « a 2, y k 3 and s & 5, then eubslituting. those values we have 2b}||x3x5's 75ai .•. m » ^. Then X " myz* or x s ^^ yz^. Ex. 2. Given that aazb and that when a s 2, fr s i, 0Dd the value of a when 6^6. OPIRATIOM. Since a oc 6 .\ a = mb or 2 ^ m. because a s 2 and 6^1. Then when A s 6 we have asm6s2x5sl0. Ex. 3. Given that x ex: yx, and that x s 2 when y ai s s 2, find the value of x when y s x s 3. OPKBATION. Since x oc ys .'. x = myz^ that is 2 = m x 2 x 2 = 4oi .*. ms^ Then x = mys six3x3B|s=4^ when y s s s 3. Ex. 4. If 4y -I- 32 oc 5y + 4x, shew that y oc 2. OPIBATION. 4y + 3x oc 6y + 42 or 4y + 3z = m (5y + 4«) = 5my + 4m* /4»/i-3\ .*. 4y - 6j»y = 4»»x - 3« or (4 - 5m)y = (4m - 3)r or y = I ■■ ] « 4m - 3 \*-o»»/ air#is X multiplied by the constant quantity - — -— .*. y oc 2. '♦"' 4 — 6m • ' Ex. 6. If y = the sum of three quantities vf which the first OC 3^, the second oc X, and the third is coiHtant, and when X 6 1, 2, 3, y = 6, 11, 18 respectively, express y in terms of x. OPBBATION. The first quantity oc x' and is .'. = mx^, similarly the second quantity oC x and is therefore ~ nx, and the third quantity is constant, and is .*. = p^ say. Then y being = the sum of these we have y = mx^ + nx +p, and taking x = 1, 2, 3 and y = 6,11, 18j we get the three equations : — 6-m+n+p ^ 11 = 4m + 2n + p > J • 18 = 9m + 3« + p) which when solved give m s 1 • ns 2,andp - 3, and substifeiitiiig these la the equation y s mx* -l- nx H- p we have y ~ x^ •¥ 2x <¥ $:. ▲bv. 118.] VARIATION. 19T ' X. ExiROiu LVIII. ,p^ • 1. If mx* ^y cccx'^dy ihaw that x oc y. 2. Oiren that x ex: y and that when x = 7, y ^ 3 find the equation between x and y. 3. Oiren that x 3 the sum of two quantities whereof one is constant and the other varies inrerselT-as y, and when y = 3,. X =: 1 when y 3 1, x 3 2, find the value of x when y s 15. 4. Qiven that x' oc y' and x = 2 when y 3 4 find the equa- tion between x and y. 5. If X 3 the sum of two quantities whereof one is^constant and the other oc xy, and when x'= 3, y 3 3, when x = 3, y = - 3,. express x in terms of y. 6. If y 3 the sum of three quantities, of which the first is- constant, the second oc x, and the third ocrx' ; and when x 3 3^ 5, 7^ y 3 0, > 12 - 32 respectively ; find the equation between* X and y. 7. Given that y 3 the sum of two quantities one of which varies as the square of x, while the other varies as x inverselj^ and that when x 3 6, j/ = 7 and when x 3 9, y 3 5 find the equa^ tion between x and y. 8. Given that y oc (6" + x"), and when x = '\/(a*-'b% y 3 -. find the equation between x and y. 9. If X, y, 2 be all variable quantities such that z-x -y i0 constant, and (x + y + «)(x - y - 2) oc ys, prove that x - y + » oc yz. 10. A locomotive engine without a train, can go 24 miles per hour, aud jts speed is diminished by a quantity which varies a» the square root of the number of cars attached. With 4 cav» its speed is 20 miles per hour. Find the greatest number of cav» the enginf» can move. .,r « = {20 + (n - l)rf| - = {8a + (50- l)2a} ^^ = (ga + 49 x 2a)25 s (8a + 98a)25 » I06a x 25 = 2650a. Ex. 2. Qiven 3, the first term, and 55, the last term, of a series consisting of 27 terms, to find the common difference. OPIBATION. / a a + (n - l)d or (n - l)rf = / - a .•. rf = n- I 55-3 52 27-1 " 26 " ^' SOT. XI, Ai»T. 249.] ARITHM£TIOAL PROGRESSION. 201 vhich the , Thus, 4 y three of a fourth, Lany prob- er even im- tiilly study Iries 4a + 6(1 L9 X 2a)25 1, of a series a T Sx. 3. iDsert 5 arithmetical means between 1 and 33. . - OPIBATION. Since there are five means and two extremesj there are in all 7 terms, and we mast find the common.diflference of an arith- metical series of 7 terms whose first term is 1 and last term 23. / - a 23 - 1 22 ' Hence the series is 1, 4|, 8i, 12, 15|, 19), 23. Ex. 4. How many terms of the series 6 + 8i + 10§, &c., make up 3796? OPERATION, »=*{2o+(n-l)d}^; 3796 = {12 + (n-l)2i|- " 2 2 7590-12»+(n"-n)2J; 22770 = 36n + 7n«-7n; 77t«+29n= 22770 n = Note,— The negative value -57^ does not satisfy the conditions of the question, and is therefore inadmissible. Ex. 5. The sum of four numbers in arithmetical progression is 32, and the sum of their squares is 276. Required the numbers.- OPERATION. Let X = the second number and y = the com. diff. Then x-y,XfX+y, and x + 2y ia the series. .•. » - V + X + a; + y + X + 2y = 4x + 2y = 32 or 2x + y = 16. Also (X - y)* + x« + (X + y)* + (x + 2y)» = 4xy + 4x« + 6y* = 276 or 2x2 ^ ^xy + 3y^ = 138. And y = 16 - 2x .-. 2x» + 2x(16 - 2x) + 3(16 - 2x)* = 138. That is, 2x'* + 32x - 4x''* + 768 - 192x + 12x' = 138. That is, lOx* - 160x = - 630 ; x" - l6x = - 63 ; x»- 16x + 64 = 1. x-8 = ilorx = 9or7. y = 16 - 2x = 16 - 18 = - 2, or 16 - 14 = 2. Hence taking x ^ 9 and y = - 2 we have the series 11)9, 7, 6 ; taking x = 7 and y s 2 we have 6, 7, 9, 11. 202 AKrCBlfBtlOAL PROGRESSION. [Sicft. ZI. Otherwise, let x - 3y, a; -> y, a + y, and x +3y repfesent the number, where 2y = the common dURnrence. Then x -r>3y + a;-y + x + y + af + 3y = 4x = 32.*. x = 8. (X - 3y)' + (X - y)=' + (x + y)* ^ (x + 3y)» = 4x!» + 20y» = 2t6 or 20y» = 276 - 266 = 20. y* = 1» y = i 1. Hence x - 3y = 8 7 3 s 5 or 11, kc. EXBRCISK LtX. Sum the following series : 1. 63, 6&, 67, &c., to 31 terms and also to n terms. 2. - 200, - 188, - 176, - 164, - &c., to 22 terms and to n termiC 3. ^,3}, 6, &c., to 17 terms and tilso to 2in +jp terms. ^. }, 0, - }, - lif tc,, to 11 terms. » FUid the l7th and 28th and nth terms of the series : f .* 2, 5, 8, &o. 6. 3, - 2, - 7, kc. ». 2i, 3A, 3|i, &c. 8. Ifsert 3*arithmetlcal means between 3 and 33. '&. Insert 4 arithmetical means between 9 and - 66. 10. Insert 7 arithmetical meains between - 1 and 100. , 11. Find the sum of 73 terms of the series 1, 2, 3, 4, ko. 12. What is the nth term of the series, 1, 3, 5, 7, kc. 13. ProTe that the sum of n terms of the series 1, 3, 6, 7, Ac, is equal to n^ 14. If a body falling to the earth descends a feet the first second, 3a feet tiie secoud, 6a feet the thirds and so on ; hov far will it fall in t seconds 7 ' 16. How far will the body (Question 14) fall during the 20tb second and during the t th second. 16. Tliere are four numbers in arithmetical progression, of I which the gum of the squares of the extremes is 200„and the | . lorn of the squares & the means is 136. Find the numbers. 17. There are four numbers in arithmetical progression whose j continued product is 1680 and common differsnoe 4. What ml thiSnQinbers? 15. There are five numbers in arithmetical progiqessii^ irlnpM j iQia is 26 and continued product 946. What are the itiadMmtl EOT. XI. Abt. 848.1 AKITHMETICAL PROGRESSION. 20B «iit fbe f s 216 ^ntonn^ 18 )0. 19. A man borrowed $60 at 6 per cent simple interest, per year of 360 days. How much must he pay daily to cancel the debt, principal, and interest, in 60 days 7 20. Prove that the sum of n terms of the natural numbers 1, 2, . n(n + 1\ • 3, &c., IS - \ • A * 21. Prove that the sum of the squares of the first n natural / n(n + l)(2n +1) numbers is 6 22. How many terms of the series 2, 11, 20, &c., are required to make up 617? 23. Find the arithmetical series the last three terms of which amount to 96, and the preceding four terms of which added together make up 86. 24. Find the arithmetical series of which the 6th and 7th terms are respectively 7 and 5. ^ 26. Given « the sum of an arithmetical series = frn 4- en' for all values of n, find the t th term of the series. 26. Prove that the sum of the (m - n)th and (m + n)th terms of an arithmetical series is double the mXh. term. 27. In an arithmetical progression if the (p + 9)th term s m, and the (p - 9)th term - n, prove that the 9th term of the — P 5^7j^c.,isB series is = m - (m - n)^-. 28. SucH to n terms the arithmetical progression whose pth p )t the fint Igo on •, how [ing the 20th 1 igression, of |200„and the | mmbete. Ission whowj ]. WhatiWJ ttMdwnt term is 7 - 29. There are three numbers in arithmetical progression, such that the square of the first added to the product of the other two is 16 ; the square of the second added to the product oi^ the other two is 14. What are the numbers ?. • :J$0. f he sum of four whole numbers in arithmetical profrression ||/$^j and the sum of their reciprocalj! is 'j^. Reqniredf th$ lumbers. .'Ji'l: 204 GEOMETBtOAL PROGRESSION. [8>ot. XI. GEOMBTRIOAL PROGRESSION. 260. Quantities are said to be in geometrical progres- sion when they increase or decrease by a common multiplier. Thus, 2, 4, 8, 16, 82, fco., are in geometrical progj^ession, the oommon multiplier being 2. 6a, - 16a^ 46a", - 186a^ &o., are in geometrical progression tlie common multiplier being - 8a. 251. In geometrical proffremon there are five things to be considered : 1. The firit term. 2. TA« last term. 3. Tkt common ratio. 4. The number of terms. 5. TTie sum of the series. As in arithmetical progression, these five quantities are so related that any three of them being giren the other tyocan be found, and hence there are 20 distinct cases arising fi-om their combination. 262. Bepresenting these five quantities by letters, thus, a = the first term, I = the last term, r = the common ratio, n = the number of terms, s = the sum of the series, the general expression for a geometrical series becomes o + or + ar* + ar* + or* + ar'^ + kc, whetre the index of r is always one less than the number of the term. Thus, in the third term the index of r is 2, which is one less than the number of the term ; in the fifth term the index of r is A, which is one less than the number of the term, &c. Hence I = ar^ - ^ ; that is, the last term is equal to the first term multiplied by the common ratio raised to that power which is indicated by one less than the number of terms. 263. Since the sum of the series is equal to the sum of all the terms, Awrn. 280-954.] dBOMETRICAL PBOQRESSION. 205 «s $'^B-har + ar'+,,.. •\-ar'*'*+ar*~\ multiplying by r, we get Hence 9r-$-ar^-a] or « (r - 1) = a(r* - 1), and therefore o(r»-l) r-1 254. From the formula obtained in Art. 252 we get by transposing the terms, &c., log. I -log. a I as .n-i n=: log. r + 1 And substituting these values of /, a, r, n, in the formnla obtained in Art. 254, we find ^^ « = « = rl-a r-1 Z • - o Z(r«-1) *- I I r-' - a"-' (r-i)r»-i and these together with the two formulas obtained in Arts. 262 and 253, a(r'*- 1) are the fundamental formulas of geometrical progression from which the other fifteen are derived by reduction. Thus, ' rl — a ' « =: gives formulas for s, r. I. and a, = 4 r — ^1 Z(r«-l) is «s (r-l)r»-i a K 1 1 |« - 1 - a» - 1 ss a(r**-l) r-1 / K or" ■ * (C • 8, r, Z, and n, s 4 (1 s, Z, n, and a, s 4 M «, r, a, and n, s 4 ( Z, a, r, and n, s 4 •>■ Total 20 m GEOlfETRIOAL PB0GRE8SI0N. [SsoT. XL 266. When the oommon ratio of a geometrical series is* a proper fraction, the series is a descending one, and if the numher of terms is infinitely great, r" becomes infinitely small; i. e., r^ becomes = 0; hence ot^ in formula ^_. becomes equal to zero, and the formula for finding the sum becomes -^^ = — ^. The expression r- 1 1 - r '^ 1-r properly speaking, however, represents the limit of the sum of the infinite series rather than the sum itself. 266. By means of these formulas many problems in geometrical progression may be solved, but as a rule ques- tions in which the value of n is sought are incapable of solution except by the h|per analysis^ Ex. 1. Find the last term and the sum of the series 3, 6, 12, &c., to 11 terms. OFBRATIOM. Z = ar»-i = 3 X 2" = 3 X 1024 = 30'72 8 ■=■ a(r'' - 1) 3(2" - 1) r- 1 2-1 = 3(2048 - 1) = 3 X 2047 = 6141. Ex. 2. Find the limit to the sum of the series 8 + 4 + 2+1 + Ssc.j ad infinitum. OPSBATION. 8 8 «,= = - = 16. 1-r 1-i . i Ex.. 3. Find the 7th tenn and the sum of 8 terms of the series fli 9» XT' OPERATION. The common ratio is always = 2nd term -r 1st term. Hence in this question »'=f-r^=f=! / = ar"-i = (^)(D« = ^ X T^V = Vi^sS- fl(r"- 1) uar-i) Mh^ -i) wi]-;;'' ) »- r-1 - §-1 ~ -i " J ■^JW MTs. 266.966.] GBOMBTBIOAL SBOaRBSSION. m Bz. 4. Insert three geometrical means betweeh 4 and 324. QffMUTIOI. And since tliere are here 3 means and 2 eitremes there are in all 5 terms, then r' ~ ^ s i\i.^ r* s 81, whence r is evidently - 3, and the series is 4, 12, 36, 108, 324, Ex. 6. Find six numbers i^ geometrical progression sach that the sum of the extremes is 99, and the sum of the other four terms, 90. opibation. The sam of the six terms is evidently 99 + 90 s 189. Let X s the first term and y~ the common ratio. Then x, xy, xy>, xy", xy*, xy', represent the terms Ir-a xy'-x * x(j/* -r 1) « s 189 s r = r- = ^ y-1 y-^ W^ 99 .*. X = — ,' . . But xy» + X = x(y» 4- 1) = 99 .•. x - -g- 99 21(y»-l) 11 ~ y*-y' + y"-y + 1 r-l 1 89(y - 1) y«-l • 189(y-l) y«-l 21 y« + l ' 11 ^•-1 ••y*+y»+l " y*-y» + y»-y+l .-. 21y* -. 21y» + 21y» - 21y + 21 = lly* + lly» + 11 lOy* + 10y» + 10 * 21y» + 21y 10(y* + y»+l) = 21y(y»+l) 10(y* + 2y» + 1 - y") =: 21y(y2 + 1) 10(y" + 1)" - 10y» a 21y&c. to 8 terms. Find the limit to the sum of the infinite series : T. -li + 5-if + &c. 8. 1- + iV + A + Ac. Oi.7-3i + lJ-&c. 10. 64 - 32 + 16 - Ac. li;**623. 12. -7. 13.. -976. 14. '86232. ^^ Sum the following series : 16. 1 + 3 + 9 + &c. to » terms. 16. 2 - I + /(f - &c. to n terms. 17. 2 + V8 + 4 + &c. to 10 terms. 18. aP +«P+« + aP*« + Ac. to n terms. 19. Insert three geometrical means between 1 and if. 20. Insert seven geometrical means between 2 and 13122. 21. Insert thre^ geometrical means between 9 and ^. 22. The sum of the first and third of four numbers in G. P. is 148, and the sum of fhe second and fourth is 8^8. What are the numbers? V , * Amr. 967.] HARMOirrOAL PROGRESSION. 209 23. The Bum of the first iind second of four numbers in 0. P. is 15, and the aum of the third and fourth is 60. Required the nnmbert. 24. The sum of$316 was divided among three persons in such a way that the first received $135 more than the last. The three shares being in Q. P., required what they were. Interpret the negative result obtained in the solution. 25. There are five whole numbers, the first three of which are in Q. P. ; the last three in' A. P. ; the second number being the common difference of these three terms. The sum of the last four is 40, and the product of the second and last is 64. Required the numbers. 26. Prove that the sum of n terms of the series a + (a + b)r + (a + 2ft)r» + (a + 36)r» + &c., o-{o-t-(n- l)6}r" 6r(l-%-^) l_r~ "*■ (l-r)» 27. If a, hf c, d, are four quantities in G. P., prove that a* + h* + c» > (a - 6 + c)2, and that (o + 6 + c + d)» = (a + 6)» + (c + 1/)» + 2(6 + c)2. . 28. In a G. P. if the (p + q)th term - m, and the (p - q)ih term - n, show that the pth term = ^^mnf and also that the qth term = »©"• ■ £9. The sum of three numbers in G. P. is 35, and the m«an term is tq the difference of the extremes as 2 : 3. Required the numbers, v 30. There^ a number consisting of three digits, the first of which is to the second as the second is to the third ; the number itself is to the sum of its digits as 124 : 7, and if 584 be added to it, its digits will be inverted. Required the number. HARMONIC AL PROGRESSION. Quantities aire said to be in harmonical progression ^ir reciprocals are in arithmetical progression, or three consecutive terms the first is to tlie tUrd 2ib fiABltONIOAt mOOBS88ION. (SlOT. It. as the diiferenoe between the first and second is to the diiferenoe between the second and third. Thiu. a, b, and o are laid to be in H. P. when a:c::a-6:&-0. •Inoe 8, 7, 11, he, are A. P., their redprooale ii ft nVt *e„ are in H. P. 268. It may be easily proved that the reciprocals of a series of quantities in H. P. are in A. P., as follows: — Let 0, 6, c be in H. P. Then a : c : : a - 6 : 6 - c or a(fr - c) = c(a - 6), or a6 - oc = oc - 6c, and diridiug each of these by ahc we hare r = t • But when the difference between c b a the first and second is the same as the difference between the second and third, the three quantities are said to be in A. P. 250. No general rule can be given for finding the snm of a series of terms iigH. P., but, by inverting the given terms so as to form a series in A. P., many useful problems may be -solved. Ex. 1. Oontinae the H. series 2), 1|, 1^, three terms each way. OPBBATION. Since f, ^,^, are in H. P., their reciprocals, f , f , f, are in A. P., and their common difference - |. Hence - i, ){, i, |, \t |, h f , If is the continued A. series, and these terms inverted give us for the required H. series - 6, oc, 6, 2], If, 1^, 1, ^, ^. NoTK.— The second term of the A. P. is %, which inverted gives ns % which = ae. (See Art. 66.) Ex. 2. Insert four H. means between 2 and 6. 0PIBA.TI0N. I Insert four A. means Jbetween \ and ^. Here d s - — - i 6-1 = - g- = - tV- Hence the A series is i, iJ, ^, ^j, aV» Tft»» .'. H. series is 2, 2^^^, 2^, 3|-, 4f, 6. Ex. 3. Insert three H. means between 10 and 30. OPHBATION. Here d = ^jp'/ Insert 3 A. means between -^ and ^. - - ifc» and the A. series is tV, A, «Vi u' h- Hence series - 10, 12, 15; 20, 30. [BaoT. Xt. I Asm. 268-908.] BARIIOKIOAL PR0ORB8SI0N. 211 to the IB ib-e. ire In H. F. rooals of a Hows: — J or a(h - c) of these by aoe between between the I in A. P. ing the sam ig ihe given ful problems le terms each I, are in A. Ii if h li i» ^» certpd give ub Irted gives na ^ 6- 1 Ex. 4. Find the nth term of the H. series 1), 1, ), Ac. OPIRATIOH. The nth term of the A. series §, 1, J, *o., = a+(n-l)// = § +(n-l)J 1 1 n 3"*"3~3~3''"3~ n + I \ the nth term of the given H series is 3 n + V 260. Let a and h bo any two quantities, and let A be their arithmetical mean, O their geometrical mean, and B their harmonical mean. Then, I. J'-a^b-A or 2.i = a + 6 .-. Jl= i{a + b). Art. 243. II. a : G :: O : bor 0'* = ab .: Q-*Jab, Arts. 224 and 260. I 2ab III. o:6::o~H:fl^-6oroff+6/r=2a6.-. ffs — r Art. 267. a + 6 261. Henct the A. mean between two quantitiet is equal to half I their «Mm, the O. mean between two quantitiee it equal to the tquare root of their jtroductf and the H. mean between two quantitiee it I tqual to ttoice their product divided by their sum. 262. Thsorbm l.—Takit^ A, G, and E, as Hi last article^ G is I the geometrical mean between A and H. 2ab Dkmonstratiok. Since .i = i(a + 6) and H^ .'. AH -^ a +b \a-\- b 2ab -r— X r! oft, but G^ a oft .*. CP = AH. Extracting the 2 a+ 0^ Isqiiare root of both, we have O - '/AHf that is, O is the geomet* pical mean between A and H 263. Thborbm ll.-^Taking A, G, and H as in Art. 260, then of ^he three A is the greatest and H the least in magnitude. Dbhonstratiom. Because, (Art. 134) a" + 6* > 2 06, a' + 2a6 + 6' 4a6 ,0^6^ 2a6. a+ b 4«6, and o + 6 ^ r, and — -— > r>but — -— a A, and 2bJ a + 6' 2 a + 6' 2 -7-T = Hf .'. A> H. And G being the geometrical mean be- jtween A and H is of intermediate mi^nitnde, t. e., is greater than Iff and less than ^, .'. wi > 6 > IT. v ' 212 HARMONIOAL PROGRESSION. [Scot. XI. E Ann 364. Throkbm lU. — Three quantities^ a, b, C, are in A. P. wr a •> b a a a H, P., or O. P., according b-c — or -r- or — . a b c a— h a DlKONBTBATIOM I. ; = — = 1 .'. a-6 =6 - C OF 6 = \{a-{-c), b-c a 11. III. a- b r^~c a- b a = -J- .*. ab - b' = ab - ac or 6' = ac .•. 6 = V*^* .•. a : c :: a -b : b - c. b a c b-c Ex. 5. Find the A. G. and H. means between 1 ^ and 10. OPBBATION. A = i(a^+ 6) =^i + 10) = i X lU = J X «g' = f 5 = 5,V. G = ^ab = VU X 10 = Vl6 - 4- 2a& 2x11x10 32 H = a+ b ~ 1^ + 10 ~ IH = 2^. Ex. 6. The difference of the A. and H. means between two numbers is l|- ; find the numbers, one being four times as great &8 the other. OPIBATION. 2ab a+ b 2a& A^Ua + b) and H= — — r .-. A-H- —z tTT. 'v' a-\- b 2 a+o g" + 2a6 4- ft' - 4aft (a ~ ft)^ ~ 2 (o + 6) (46 - by _ (Bby 2(46 + 6) ~ J5x66 a = 46 = 8. 2(tt + 6) 96 » 96 9 106 "" 10 = ^ .*. since a = 46 we have 6 5 ••• 13 = i or 6 = 2 and EXBROISB LXI. 1. Continue three terms each way the H. series, (i) |, J^, i Cn) iSr, ^, -h ; (ni) i, i ^ (iv) 14, If, |^ j (v) A, li, - 1? (vi)-i, oc, i. 2. Insert three H. means between 2 and 3 ; betwieen 6 and 7 between 11 and 3 ; between 2i and 3|; between 6 and r §. 3. Find the 5tfaf 11th, and nth terms of the H. aivles SI, l, I 4. 5. 8. 8. i-; fi. Abtb. a64,.9BB[.i PERMUTATIONS. 213 4. Find the 6th, 10th, aQd last term of the H. series 4i, 6|, 13. 5. Find the 4th and 8th terms of the H. series -^, Vir, ^. '8. Find the unknown terms of a H. series whose first term is 4 and fourth term 1. 7. Find the 8th term and the nth terms of a H. series whose first term is a and second term b. 1 1 8. Find the H. mean between r— 7— and ~ — - . 9. Find the A. G. and H. means between 4 and 9. 10. Find the A. G. and H. means between 6 and 4^. 11. If 0, 6, c, be three quantities in H. P., prove that d^ + c* > 2b^, if a and c are both positive or both negative. 12. If a, 6, c, are in A. P., and o, 7/16, c, in G. P., prove that a, m?b^ c, are in H. P^ 13. From each of three quantities in H. P. what quantity must be taken away in order that the three resulting quantities may be in G. P. ? 14. The sum and difference of the A. and G. means betWeen two quantities are 16 and 4 respectively. Required the numbers, 15. The A. mean between two numbers is 2'<^ of the H. mean, And one of the numbers is 2. Required the other. 16. Find two numbers whose sum is 30 and H. mean 13^. 17. Find two numbers whose difference is 16^ and the G.mean between the H. and A. means of which is 9. ^ PERMUTATIONS, VARIATIONS, COMBINATIONS. . 265. The different orders in which any given number of quantities can be arranged arc called their perwMta 2) variations of n things taken three and three together. Similarly the number of variations of n things taken four and four together, may be shown to be n(n - l)(n - 2)(n - 3), and five and five together, n(n - l)(n - 2)(n - 3)(n - 4), and ao on. Now it has been shown that variations of n things taken 2 togetl^er = n(n-l) orn(n>2 + l) 3 " =n(n-l)(n-2) orn(n- 1)(«-3 + 1) 4 "• =n(n-l)(n-2)(n-3) orn(n- lX»~2)(n-4 + 1) and so on. Hence the variations of n things taken p together = »(ii-r- l)(n - 2). . . . (n - j> + 1). 75^^««S^!st'- ABT8. 966-268.] PERMUTATIONS. 215 Cor. 1. If J) - n, that is, if the quantities are taken all together, the variations or permutations of n things is n(n- 1) (n> 2). . . . (n - n + 1) s n(n - l)(n - 2) . . . . 3.2.1, or, revetsing the order of these terms we have permutations of n things = 1.2>3«4*>«*n. Cor. 2. Hence denoting the variations of n things taken 1, 2, 3, 4, &c., p together by F,, F,, Fj, F^, Ac, F^ we have ri=n; r, = n(»- 1); F, = n(n - l)(n-2); r4=n(n-l) (n-2)(n-3);&c.; F, = n(»- l)(tt- 2)(n-3)....(n-p+l). NoTB.— For the sake of brevity n(n-l)(n - 2) — 3.2.1 is frequently indi« oated by \n (re^d factorial n.) accordingly, \n denotes the continued pro» duct of the natural numbers from 1 to n inclusive. • 268. Thborbm II. — The number of permutations of n things taken aU together^ whereof p are !('«, q are b'<, and r are c'«, is lE la l£ * DiMOMBTBATiOM.— Let N denote the number of permutations under the given conditions. Then if we suppose that in any one of these .y permutations we change the p a's into letters differ- ing from all of the rest, we could from this single permutation produce \p diflferent permutatidns, and as the same would be true for eaxh of the N permutations, it appears that if the p a's are changed to letters differing from all the others, there will be JH \p permutations of n letters, whereof there are still 9 &'s and r c's. If now the q &'s were changed to letters differing from all the rest, it may be shown by similar reasoning that we should have 2V [£ |£ variations of n things, whereof there still remjain r c's. Similarly, if the r c's are changed to letters differing from all the rest, we shall find that the number of permutations of n differ- ent things -'S\'p_ |£_|r. But the permutations of n different things is In. Hence N |£ |£|r^= |n^ and dividing both sides of the equation In by |jp [£ jr we have xV = if is ll 216 PERMUTATIONS. [SsoT. 21. Ex. 1. How many variations can be made of 10 things taken 3, 5, 8, and 10 at a time ? OPERATION. r3=n(n-l)(«-2) = 10.9.8 = Y20 Fa =n(n-l)(«-2)(n-3)(n-.4)= 10.9.8.7.6 = 30240 F« = «(n - l)(n - 2)(n - 3)(n - 4)(n - 6)(n - 6)(/i - 7) = 10.9.8.7.6.5.4.3=1814400 r,o = 1.2.3.4....n= 1.2.3.4.5.6.7.8.9.10 = 3628800. * ■ Ex. 2. How many different words can be made with all the letters in the expression a^bc'de'^. OPBRATION. V{e are to find the permutation of 13 letters, of which 4 are a% 2 are c's, and 5 are e's. h' _ 1.2.3.4.5 .6.7.8.9.10.11.12.13 ^~ \£\g.\L~ 1-2.3.4 X 1.2 X 1.2.3.4.5 = 7x9x10x11x12x13 = 1081080. Ex.3. The number of variations of n-2 things 3 together: number of variations of n things 3 together :: 5 : 12. Find the value of n. OPERATION. (n - 2)(n - 3)(n - 4) : n(n - l)(n - 2) : : 5 : 12 12(» - 2)(« - 3)(n - 4) = 5n(n - l)(n - 2) 12(n - 8)(n - 4) = 5n(n - l) 12(n» - 7» + 12) = 5n=* - 5« 12n* - 84» + 144 = Sn^ - 5n or W - 79n = - 144 196n» - 2212» + 6241 = - 4032 + 6241 = 2209 14« - 79 = + 47 .-. 14n = 126, orn = 9. Ex. 4. The variations of a certain number of things taken 3 « together is 20 times as great as the number of variations of half as many things taken 3 together. Find the number of things. OPERATION. l)(n - 2) = 20 X in(i»- l)(Jn - 2) At- 2^ n(n-l)(n-2) = lOnf-y ■)(H^) n(n - 1)(» - 2) = i n(n - 2)(n - 4) and dividing both by n(n - 2) we have n - 1 = f (n - 4) whence n = 6. iAT. aw.] OOAIBINAZIOITS. 217 ExvBOUMi LXII. 1. In bow many differont wajs can six di£fereat cqimters be arranged 7 2. How niAQy YaciatioiiBaiui be formed oat pfB tbings taken (i) 4.togetber, (n) 6 togetber, and (in) all together. 3. How many dififevent words can be formed ontof the expres- sion a"6*c'd 7 4. Asauming tbat sixteen changes can be rang per ,minnte, and that the bells are rang 10 hours each day, bow loni; wqnld it require to ring all the changes that can be rang on 12 belli 7 6. If the nnmber of permutations of n things 6 together is six times as great as the number 3 together, find n. 6. A landlord agrees to board a company of 10 persons as many days as they can sit in different posttions a( table, for $5000. Assuming that the board of each is worth $5 per week, how mach. does he lose by the transaeUon? What is his loss if the |500Q is paid at once and placed at simple interest at 6 per cent, per ancum till the close of the term of agreement 7 7. Ibe.nnmber of vvu^iations of 15 things, taken n together is ten times as great as the nnmber taken (n - 1) together. Find tbevalaeofn. •8* How many diflbrent words may be made of all the letters inthe woi^ds ConttantimpUt divkibUUyf octwoorif. commemoration. 9. How many diflbrant permutations can.be formed with tbe letters in the words a/ge6ra, demotutraiionf Toronto. 10. The variations of f n things taken 3 together : variations of §n things taken 3 together :: 146 : 2. Find n 369. IThiouii hi. — J%e numher of combinations of n things _ , ,. . n(n - l)(n - 2)(n - 3). . .^n -. p + 1) iBk^np together u 1.2.3.4.. ..p DwoNSTBATiON. The number of combinations of n things two imdt^o together is evidently only half as great as the nnmber jif fM^atibns of n things two together. Since each combination ^gt^ two variations, ab, 6a, hence the combinations of n .. n^U" 1) tlimgi two together is — -r — . 218 COMBINATIONS. [SXOT. XI. Again, since there are ft(h - l)(n - 2) yariations of n things taken three together^ and each combination of three things admits of ^ 1 . 2 . 3 variations, it is evident that there are 1.2.3 times as many variations of n things taken three together as of combinations taken three together, and consequently the n(n - l)(n - 2) nnmber of combinations is 1.2.3 Similarly, the variations of n things taken p together is n(n - l)(n - 2). ... (ft - p + 1), and every combination ofp things win make 1.2.3..../) variations. Hence there are 1.2.3....}) times as many variations as combinations of n things taken p together, and consequently the nnmber of combinations is n(n- l)(n-2)....(n-y + 1) 1.2.3. • . .p 270. Thborih IV.— 7%e number of combinatioru of n things taken n-p at a time is equal to the nunUter of them taken jf ata time. Dbuonstbation. It has been shown by last theorem that the nnmber of combinations of n things taken p together is n(n-l)(n-2).... (n~p + l) -^^ , q a , and mnltiplying both numera^ Ji.A.o. . . .p tor and denominator of this expression by 1 . 2 . 3 . . . . (n - p) ve n(n ~l)(n-2) (n^p+l)x(n-p) 3.2.1 1.2.3 px 1.2,3,... ..(n-p) n(.fe - l)(n »- 2) 3.2.1 _. h Le i»-y \i l**-p Now putting n - p for p in this result, as may evidently be done, since the expression holds for all values of p which are less thpu n, we have n^-p^n-n + p^p and consequently \n \n_ ^P = \p |n-p " |n-p \p " " ^»-* that is, the Cp of n things = Cn .p of the same n things. Hence ifp>in, the number of combinations is more easily found by the supplemental formula, i. e., taken C,|.p instead of Cp. find that it ^ [SsoT. II. ■ Int. TftO.] dOiiBt^A'ttO^B, 219 I numera* Nolv.— The troth of this prinolple is alio evident from the Ihet that If, from N things p be taken, (n ~p) things will alwa3rs remain, and hence for every diflbrent set containing^ things there will be a different set left containing n - p tilings, and consequently the number of the former equals the number of the latter. Cor. 1. Hence representing combinations of n things, li 2, 3^ &c., p together, by C^j Cg, (7,, &c., Cp we have n n(n-l) ^ n(n-l)(n-2) «-i =p t.a = J 2 » ^3 = 1.2.3 .se- ctor. 2. To find the Sum of all the combinations that can be made of n things taken 1, 2, 3, &c., n together, we proceed as follows :— n n(n-l) n(n-l)(n-2) It will be shown hereafter that :j-, — pg — > T~2~% &c., are the coefficients in the expansion of the binomial (1+a;)**, 80 that (1 + x)* = 1 + 6iX + CgX^ + C3X* + &c. + Cnx\ Now writing 1 for x we hare (1 + 1)* = 2« = 1 + Cj + (?8 + C3 + kc. + C„. Hence 2" - 1 = C^ + C, + C3 + &c. + Cn, or the sum of all tlie combinations which can be made of n things taken 1, 2, 3, &c., n together = 2"- 1. Ex. 1. Required the number of combinations of 22 things taken 5 together. OPBBATION. Here n = 22 and p = 5 n(n - 1)(« - 2)(» - 3)(n - 4) 22.21.20.19.18 ^»" 1.2.3.4.5 22.21.19.3=: 26334. 1.2.3.4.6 Ex. 2. How many combinations can4)e made out of 23 things taken 19 together? OPBBATIONi Here n = 28 and p - 19, and consequdntly n -• p ^ 4 23.22.21.20 C, - Cn.p or C,9 = C, = jT^.-gX" " ^^^^' -JK--'. 220 OOMBINATIQKS. ttaoT. XI Ex. 3. What is the sum of all the «ombinationf which can be made out of 10 things taken 1, 2, 3, ko.^ 10 at a time. OPIKATION. Ci + Ca + C3 + C4 + &c. + C,o - 2»" - 1 = 1024 - 1 =* 1023. Ex. 4. Out of 10 consonants and 3 yowels how many words each containing two vowels and four consonants can be found 7 OPBRATION. I0.9.8.7 10 consonants combined together 4 and 4 will give ; 034 « 210 combinations; and similarly the combination of three 3.2 vowels two together = . , = 3. Honce the combinations of J the 10 consonants and 3 vowels = 210 x 3 = 630. But each of these combinations of 6 letters will furnish 1.2.3.4.6.6 = 720 permutations each, forming a different word. Hence the entire number of words formed will be 630 x 720 = 463600. Ex. 6. How often may a different igfuard of 4 men be posted out of 60 ? On how many occasions would a given man be selected ? OPERATION. 60.49.48.47 ^*=— 1:2:3:4- =230300 Taking away one man there remains 49, and the question nov becomes, how many combinations may be formed of 49 men taken three together. 49.48.47 C3 = — ' 2 g — = 18424, to each of which the reserved man may be attached. ExBROiSB LXni. 1. How many combinations may be made of 10 thingrs taken 3 together ? How many 6 together ? How many 8 together ? 2. How many combinations can be formed out of 15 things 5 together ? How many 7 together ? How many 11 together^ 3. How many different classes of 5 children caor he formed oat of a school containing 12 children ? '■ ' ^ ''■."'■ tSacT. XI MiT. S71.] COMBINATIONS. 221 loh can b« I ^ 1023. anjr words be found? LO.S.S.t ' 1.2.3.4 n of three inations of I ill furnish srent word. ) 630 X 720 be posted en man be estion nov of 49 men erved man 4. The whole number of combinations of 2n things is 513 limes the whole number of combinations of n things ; find n. 6. From a company of 36 policemen 6 are taken erery night 1 for special d&ty. On how many different nights may a. different geleotion be made ; and in how many of these will any particular I man be engaged ? 6. How many words of 7 letters can be made out of the 26 I letters of the alphabet, with three out of the fire vowels in every 1 word ? ^ 7. In how many ways can 10 persons be seated at a round^Jy> I table so that all shall not have the same neighbours in any two [^^ I arrangements 7 / 8. If the permutations of n things 3 together ; combinations |ofn things 4 together :: 6 : 1. Findn. 9. The number of permutations of n things p together is 10 . I times as great as their number taken p -^l together, apd the I number of combinations j) together : nuinber p -f I together ; 5 : 3. Find n and p. 10. In how many ways may n persons be arranged in a circle ? 11. With ten flags representing the 10 numerals, how many [signals can be formed, each representing a number, and not loonsisting of mor^ than five flags? • 12. How many different sums can be formed with a guinea, a llialf guinea, a crown, a half-crown, a shilling, a sixpence, a I penny, a halfpenny, and a farthing ? SECTION XII. BINOMIAL THEOREM. 271. The Binomial Theorem is a general formula |myeated by Sir Isaao Newton, for the purpose of expedir tioofAy involving any Hn(miial to any power. The formi^la things 5 ■» IP^^reBsed 88 follows : getherl MrJ^ ^ ^« .»_«.,_. K» - 1) he formed ngs taken )gether? r i»^itt»+ — qw-ia;4. '"\' _~' o**-*a;' + »(n-l)(u-2) 1.2 1.2.3 o« "W h^e.^ the (r + \) th term being n(nr-l)(n-2)...(n-rfl)^ 1.2*3.... V a*-ri^^ 222 BINOMIAIi THEORBM. tSaoT. XII. Where (a + x) is the given binomial, n, the exponent of the required power may be any quaptity positiye or negative, integ* ral or fractional, and r any positiye integer whatever. NoTB l.-JThe (r + l)th term as above is oommonly called the genenU term of the expansion, . NoTK 2.— The ooeffloientsof a;, aJ^,x' Ac, x* in the above ezpansionare, when n Ib a positive integer, merely the general expressions for the number of oomblnationB of n things taken 1, 2, 8, &o., r together (See Art. 269), and we shall therefore use the expressions (7,, C,, C^ ftc., C, to repre. sent these coeffloients, so that the formula given above may be written . 272. Since in the formula (a + »)♦» = o" + C^ o" - » x + C" a* - V + 4c., a and x represent any quantities whatever, we may write - X in place otx and we thus obtain :— (a - x)« = o» + Ci o»-» (- x) + C, o«'» (-x)»+ Ac. = a** - Ci o"-> X + C, o*-* x" - &c. The terms being alternately plus and vninus. Cor. It a = 1, (a ± x)» = (1 i x)» = 1 i Ci X + Cj x» ± C3 x» + C4 X* ± &c. ' 273. Thiorhic I. — The Binomial Thtortm it trii« inaUcaset when n is positive and integrcU. Pbkonbtration. — By actual multiplication it appears that :-« (x + o)(x + 6) = x' + (a + 6) X + oft. (X + a)(x + 6)(x + c) = x" + (o + 6 + c) X* + (o5 + oc + 6c) X + ofcc. (X + o)(x + 6)(x + c)(x + d) = X* + (o + 6 + c + d) x^ + (06 + oc + 6c + ad + id + td) x' + (oftc + ocd + bed + aid) x + aftcd. Now it is evident that in these results the following laws hold :— I, The number of term^ in the right hand side, is one more than the number of binomial factors which are multiplied together. II. 7%e ejqtonent of x in the 1st term = ^he number of binomial factors^ and it decreases by unity in each succeeding term. III. 2%e coefs. of 1st terms s unity ; coefs. of 2nd term* = sum of 2nd terms of all the binomial factors; coefs. of 3rd term ~ the sum of all the products of the 2nd terms of the bino- piial factors taken two at a ^ime; coefs, of ^th terms ^ AliTt. 372, 378.] BINOMIAL TBEORBM. 223 1 the gentrai ears that :-* ium of all the productt of iam* ucond term* taken three at a time and to on ; the latt term it the product of all the tecond terme of the binomial factore taken all together. Let us assume then that these laws of formation in the pro- duct hold forn- I binomial factors (« + a), (x + 6), (« + c), &o. Bo that (X + a)(x 4- 6)(x + c) to (x + A;) »x'»-» + ^x»-« + Bx^-^t Cx*-* + Ac... + JiT. where jS = a + b + c + k ] B = ab + ac-¥bc + &c. C= abc + acd + kc. &c. = A;c. K = abed , ,m» k. Then introducing a new factor x + 2 we have : (x + a)(x + 6)&c (x + fc) (x + = «* + (^ + *" " * + (fl + M) x« - « + &c + jn. Wherefore -tf+Z = a + 6 + c + k + l B + lji=:ab + ac + bc + +a/ + &2 + kl. kc. - ftc. Kl^^abcd kl. That is Ji + l = sum of all the second terms of the binomial factors. B + L/i^ sum of all the products of the second terms a, b, c, I taken two at a time. And so on, and Kl = product of the second terms when taken all together. Hence if the laws indicated hold good when n - 1 factors are multiplied together, they hold good also when ^ factors are multiplied together. But we have shown that they hold good when 4 factors are multiplied together, therefore they hold when 5 factors are multiplied together, and therefore also for 6 and so on, and hence generally for any number whatever. Now leto = 6 = c = d = Ac. Then jl = a + a + a + ....,,ion terms = na. B = a^ + a^+ Ac, to a number of terms - to the No. of combini^tions of n things taken two to^ getber s — ---- o\ 224 BINOmAL TflEORBM. [SaoT. XII. C s a' 4- 0^ -f ko.f to a number of tenns » to the No. of oombinationB of n tbingi taken three together n(n-.l)(n-2) . m ■ ■ --T a". And so on. 1.2.8 Kb a.a.a.a to n faeton > tf^. Alio, (X + a)(x + b)(x -f c) iko., beoomes (« -i- a)(x -i- a) n termi s(x + a)*. « * - , n(n-l) ., . _ n(n-l)(n-3) fl^«"-»+ + a». * **^ **^*' 974. Tbiosim II.— 27l« JBtnomtoJ T^orem AoUi for all vtUuet o/n eithir potUive or negativtf integral or fractional. DiKOMTBATiov. (Edlib's.) It hag been already shewn that when n and m are positive integers, fl m ni(m-l) „ j»(m- l)(«i-2) „ (I.) (l+xr=y(m)=l+- X + -\-^ x» + 1,2.3 -V V- ^ n n(n-l) „ n(n-l)(n-2) + &> (i+»)"=/«='+ r ' * -TTT^ "^ * 1.2.3 *" where /(m) and r(n) an symbols used to denote the series l + r-x + «-l) , ^ . » n(n-l) , ^ T-r — x^«c. and 1+r- x + — r^n — x" + sc. Hence tohatever may be the values of m and », f m m(m-l) „ . 1 f n n(n-l) .. . 1 Jn._x+ 12 x» + Ao.|-|n--x+-y~xH&c|. But the product of these two series will evidently be a series of the form of 1 + ox + 6x* + ex* + Arc, ascending regularly by the integral powers* of x, the letters a, fr, c, &c., being used to represent the coelficieutSi found by addition, of x, x', x", 4^c. Amt, 274.] BRTOMIAIi THBORBM. 225 Now it if erident* that the product of theie two f eriet must be of the tamt form whether m and n are poiitire or negative, integral or fraotional . Whatever therefore be the formt aiaumed bj Of bj c, kCf when m and n are positive integers, they will remain the same when m and n become fractional or negative. Bat when m and n are positive and integral we have seen that Yy maltiplying I and II together we get /(m) ^f{f^> = (1 + at)** X (1 + »)'• 3 (1 + «)••♦«» n 1 + or + ftjc* + c«* + Ac. a I + kc. m + n (m+n)(m+n'l) (>/Hii)(ffl + n"I)(TO+n-2) , a f(m •¥ n) by the notation adopted. (III). .'. Oenerally f (»n) ^ f (n)s T(rn + n) tot all values of m and n. « And since this is true for all values df m and n, for n we may write n+Tf then /"(m + n + r) = t (n + r)x y (m)a y (»») x jT (n) xjcr). Similarly /. . KP - 9) - . Pip- S)(P - 2?) 5 1 +_ar + 1.2.g» 1.2.3.3» x' + &c. IV. (l±x)-7= 1 i ^ --~V- ^'^ ' 'l.2.3 ' ^ + &0. = ^^i*+ 1.2.g» ^0' + 9)^.:p 1.2.3.9" 3:"+ &c. And these reduced general expressions shoald be carefhlly noticed by the student, and used as formulae for the expansion of binomials according as n is positive or negative, integral or Pac- tional. NoTB.— No examplea with n integral and positive are given, as there are a number such in Exercise XXXVIII. 10 10.9 „ 10.9.8 „ 10.9.8.7 Ex.l.(l + x).» = l + -x + — «- + -— x» + --_ X* + &c. # = 1 + lOx + 46x2 ^ i20x» + 210x* + &c. 6 6.6 „ 5.6.7 . 5.6.7.8 Ex. 2. CI + xV = 1 — X + — X* x" + jux. A.^^iirx) 1*^1.2 1.2.3 1.2.3.4 X* + &c. = 1 - 6x + 15x2 . 35a;» ^. ^qx* - &c. 1 1.2 , 1.2.3 , 1.2.3.4 , Ex. 3. a - x) - 1 = 1 + — X + x* + x" + X* ^ ' 1 1.2 1.2.3 1.2.3.4 + &c. = 1 + X + ';2 + x' + X* + &c. Note.— Hence it appears that in all cases when n is integral if the sign of the exponent and that connecting the terms of the binomial are both like, t. e. either both i^^us or both minu», the signs of the. expansion are all plus, but if unlike, the signs of the expansion are plvs and tnintM alter* natdy. 3 3 3(3-5) , 3(3-5)(3-10) „ Ex.4. (l+x/=:l + -X + -e^;:/x'' + . \ ,;\,^ V + 5 1.2.25 3(3-5)(3-10)(3-16) ^1.2.3.4.625 ^ ^*°- 1.2.3.125 = 1+— x+ 3x-2 3x-2x-7 1.9.26 «=• + 1.2.3.126 x» + 3x-2x-7x-12 1.2.3.4.626 «* + &Q* 228 BINOMIAL THEOREM. [sxoT. xn. 3 3 7 21 = 1 + — '--— x' + — — 3? - :r— x* + &c. 6 25 125 625 ™ . ., S-- , 3 3(3 + 2) , 3(3+2)(3+4) . Ex.6. (l-«) .=H.-x + -i-— *^ U2Tb-*' . 3(3 + 2)(3 + 4)(3 + 6) _ . + 1.2.3.4.16 ^ + ^' 3.5.7.9 2 X* + &c. 3 3.5 .^ 3.5.7 „ *^ 2 1.2.4 1.2.3.8 1.2.3.4.16 3 15 „ 35 315 Ex. 6. (o + 2x)-» = |a M + ~ jl =a-=«(l f 2a-»x)-=» ,,2 , 2.3 2.3.4 , . 2.3.4.5 s a- 2 { 1 - 4a-i« + 12a- ^ar* - 32a- »a:» + 80a- *x* - &c. J so-*- 4a- »x + 12a- *x^ - 32a- "x^ + 80a- ^x* - Ac. Ex.7. (o3+x2)~* = {«'(l +a-2x»)}"*=o*^(l + a-='x?)"* * 4 ^ 1.2.16^ P.2.3.64 ^ ' 3.7.11.15 , „ , + 1.2.3.4.256 <«■"*'>* = a""*! 1- J a-2x2+ J^o-*x*-TVsO-6x«+ii^f o-8x8 + &o.| -&c. EiCKROIBB LXIY. Expand to five terms each of the following expressions :— 15. (a*-x*)-» 16. («*-x»)* ■.^17. (a?+x-»)'* 18. (a^-x'^)~* 10. (ahn-x^)"^ 1 20. 1. (l+x)-8 2. (l + x)-a 3. (l-2x)-i 4. (l-ix)"B 5. (1+3X)-'' 1 6. 7.7 (l-2x)» 1 ^Ira:^* 8. (1 - 4x)* 9. (1 + X) " ^ 10. (1 - Jx)^ 11. (l + !x)* 1 12. -r (l-x)l 13. (a-x2)-» 14. (a» + x»)-i 21. (a + x-")-| 1 ^, > T ^/a**bx ▲itT. 876>278.] BIKOMIAL TBEOREM. 229 %^ 276. Thbobrm III.-- In the es^antion of (1 + 1)° there are only n + 1 ^erms, tohen the exponent ia positive and integral. Dbmombtratiom. — The coefficient of the (r + l)th term is C^ = n(n - l)(n - 2)(n - 3) (n-rfl) \r Now if r be such that n - r + 1 = 0, then the (r + l)th and all following terms vanish, and the series will terminate with the rth term. But if n - r + 1 ~ 0, r = n+ 1 and the (n+ l)th term is the last term of the series. NoTB.— If n is negative or ihictionfd, the series never ends, but may be continued to an infinite number of terms, since as r is necessarily integral and positive, we can then find no value for r which will render n — r + 1 = 0. . 277. TinoiuDif lY. — In the expansion of (I ■{■ x)^ when n is J^positive and integral, the coefficients of terms equally distant from beginning and end are the same. DxMONSTRATiON. — The (r + l)th term from the end having r t^rms after it is the same as the {(n + 1) - r ) th term from the beginning, i. e., is the same as the (n^r •{■ l)th term from the beginning. And since, Art. 271, the coef. of the (r + l)th tenn is Cr writing n - r for r the coef. of the (n - r + l)th term will beC..,. But it has already been shown (Art. 270) that In n C - **(»- 0(^-2) (n-r + 1) ^ i_ , ■__ ^ c, '" . 1.2.3 r \r \n-r |n-r |r that is the coef. of the (r + 1) from the beginning = coef. of (r + 1) term from the end. 278. Toftnd the general term of the expansion of (a + x)««. yp . In writing down any * say the 5th term) so aa to exhibit the factors of the coefficient thus^ n n-1 n-2 n-3 T * o**"*x*, we observe 2 3 4 I. The numerator added to denominator of each factor - n-t- 1. 230 BIXOMtAL THEOREM. SXOT. Ztt. II. The number of such factors is one less than the number of the term. III. The exponent of x is equal to the denom. of last factor. IV. The exponent of a = n - (the exponent of x). Hence the (r + l)th or the general term of the expansion - n(n-l)(n-2)....(n-r + l) 1.2.3.. ..r o**-*"**". 279. The student must note the following points with respect to this general term : — I. The gen. term of (1 + x)^ when n is a positive integer, is as above. II.- When n is positive, the gen. term of (1 - x)" = C,(- x)** n(n-l)(n-2)...(n-r+l) where (- l)** will of course be positive or negative according as r is eyen or odd, that is, according as r + 1, the number of the term, is odd or even. « III. If n be negative, the general term of (1 + x) " * = '-n(-n-l)....-{n-(r + l } ^ n(n+l)....(n+r-l) . — ^ —fT^ — '^ ^ (r^y i- —YT '^ IV. If n is negative, the general term of (1 - x)-" = ( * l)*" x /n(n+l)(n + 2) (n + r- 1) . w(n+l)(n + 2) (n + r-1) 1 — - ■_ _ ■ ■ . ., — ^* I*- x^ Since ( ^ l)*" x ( - l)** = (-.l)*=: + l. When the exponent is fractional, the sign of the general term is subject to the same laws, and C, may be written as in III and IV on pages 226, 227. Thus the general term of V. (1 +x)^= P(P-lXP-^)-..^lP-(r-l),i ^ V„. (, ,,y-7.PimXi^i^dP±(rzML^ mber of ctor. Dsion = snect to ^r, is as r+1) „ x^) rding as r of the -D" ral term III and ■) Art. 279.] BINOMIAL THEOREM. 231 — /p(p-fl)(p-2g)....(p-(r-l)j} \ VIII. (1 -x)« = (-!)• r^^ ^'^"^ ^^f '-^ a:-j Ex. 1. Find the general terms in the expansions of (1 + x)"r (1 -«0 " * , (a» - x») " * , (1 + 3x) -». 8.T.6.... (8-r+l) ^ 8.7...(9-r) G.T.of(l + x)« = + 1.2.3.. ..r V ^^.— r^— x'' G. T.of(l-r»)-i= + 1.3.5... (2r-l) 1.3.5....{l + (r-l)2} Ij X 2'- ^^ - ^ [jr X 2'" G. T. of(a2-x»)''* = o"^ (1-0-2x2)"* = _a f3.t.ll...f3 + (r-l)4} , „ + « H Ijx^' ^ a-2xy = + " jjr X 4' a « . ^ , , , /2.3.4....(2 + r-l)\ G. T. of (1 + 3x)-2 = (- 1)'*^^ ^ 'A iZxy.{^\y /2.3.4 (»• + 1)\ ( uj— ^ jS"**" = ( - ir (r + 1)3'- x*-. Since [r In den. cancels 1 . 2 . 3 . . . . r s I r in the numerator. Ex. 2. Find the general term in the exp/Rnsion of (1 + xy 4 6.2. -1. -4.... f5-(r-l)3| G. T.of (1+x)* =. J ^ ^ y ^ — x*- i 5.2.1.4.... (3r- 8)7 ~ ^ '-' Lr X 3'' ^ KoTB.— In the above expression for the general term it will be observed that we change all the negative signs in tbe numerator, and then prefix a power of ( - 1). Now if all the fbctors in the numerator are negative, ( - 1)** is the prefix, and if any even numbers of negative fttotors are changed to positive, ( - ir is still the prefix, btit if any odd number of them is changed, the sign of the product of the whole, i. e. of the general term, is altered, and p_ becomes ( - 1)^ ^ ^. In the expansion of (I + x) « therefore the sign of the general term is ( - 1)^ or ( - 1)^ * ^, according as the number of positive fltotors is even or odd. 282 BmOMIAL THBOBBM. [SaoT. ZU. In the expansion of (1 -x)t the generalterin will of itself involve ( - 1)**, ftnd this taken in connection with the above renders the sign of the general term ( - 1)*" = 1 or ( - 1)*^ * * = - 1 according as the number of positive flM^tors is even or odd. Bbbiabk.— In the above paragraph the general term merely expresses any term after negative fkctors begin to appear in the nnmorator. Bx. 3. Find the general tarm of (1 - x)* « « - * 3--2--7.... {3-(r-l)6.j , G.T.of (!-«)» = ^ I y '-^x- 3'2'7.. ..,.(5r-8) ^ Ex. 4. Find the 8th term of the expansion of (1 ■¥x)-\ Since the general term = (r + l)th term = 8th term} r-1 , , / 4.6.6.7.8.9.10.11 \ , Formula II. 8th term = ( - 1)^ ( 1. 1^.3.4.5.6.7 ) ' a - 1320x8 • Ex. 6. Find the 6th term of the expansion of (1 - «) " ^ , «„ , 1.3.6....{l + (4-l)2| , Formula VII. 6th term = U^i* 1.3.6.7 4 J6~ " 1.2.3.4x16* ~ 128** Ex. 6. Find the 7th $erm of the expansion of (1 - \x)^ „ 11.10.9.8.7.6 . „ Formula II. 7th term = ( - 1)« 1.2.3.4.6.6 (^*>' x« 462 ■ 164 „ = + 462 X -— r T— -X6 = rrz X8 729 729 243 Ex. 7. Find the 6th term of the expansion of (1 - x)^ FnrmnUirm ^'2.-3 .. | 7- (6- 1)6} 7.2.3... {(4x6)-7| Formula Till. jjs x 5» ^* \J^^^ = + 7.2.3.8.13 x" = + 182 x" 1.2.3.4.6x6126 * - ^ 30626 Since there are two positive factors in the first expression, the sign is ( - 1) *" = + 1, see note above. Ex. 8. Find the llth term of the expansion of (a "" * + x'')^ Abt. 280-282.] BINOMIAL THEOREM. 233 (a ~ * + x» Then by formula v the 11th term. .^.^ |ll.r3-l.^^...JU-(10,l)4| | ^J, ^„ ..-v.(-i)" ("-'-^-;^'--;:f''-"> )a.»» 11. '7.3.1.5.9.13.17.21. 25 1.2.3.4.6.6:7.8.9.10.1048576*° * » ««o .11 85085 J g^, 85085 „g "" ^ " i268435456 " * " " 268435456 "^ * 280. To find the sum of all the coefficients of(l+ x)», The Theorem (1 + «)» = 1 + y * + -^^^ x' + &c., is true for all values of a:. Let x « 1. n n(n - I) n(n -Dfn - 2> Then + I)- = 2" = 1 + - + -i~^ + -Lj^^ij^ + 4c. .'. 2" = sum of all the coefllcient of (1 + x)\ r" 281. Thbobem Y.—The sum of the coefficients of the odd terms in the expansion of (1 + x)^ is equal to the sum of the coefficietUi of the even terms, Dkmonstbation. — ^Put a: = - 1 ia the expansion of (1 + x)^, n(n-l) n(n-l)(n-2) Then (l-l)» = 0~ = = l-n + -^ i,2.3 + *c^ Sum of coefficients of odd terms - sum of coef. of even terms s 0, /. Sum of coefficient of odd terms = sum of coefficients of even terms. GoR. — Since the sums are equal, each sum is evidently half 2» of 2", Art. 280, and is therefore = -— = 2" " * 2 282. To find the greatest term in the expansUm of {a + x)?. ^, n(n - l)(n - 2). . . , (n - r + 1) The (r + l)th term = -^^ -^ — j7 ^^ ^a»-' ^^ 234 BINOMIAL THEOKlSk. [Bkot. XU. The rth term - n(n-l)(n-a) .... (n»r + 2) \r-l ' .n-r+l-r-x Hence the (r •(- l)th term is obtained from the rth by multiply- , n-r + 1 X ing the latter by • — . Oonsequently the rth term will be the greatest as soon as a «-r + l — becomes <1, That is as soon as (» - r + 1) a; (n + 1) x. That is as soon as r > (n + 1) a+ X r therefore must be the first whole number > (n + 1) ^^ ^' If(n+1) is a whole nutnber, then two terms are equal, a + x and each is greater than any other term. If n is negative, r is the first whole number equal to or next greater ihan (n - 1) - — — . Q> ^ X Ex. 9. What is the sum of all the coefficients of (1 + xy. , Here Art. 280, 2» = 2^ = 612. iSx. 10, What is the sum of all the odd coef. of (1 + x)^'. riere Art. 281, 2«-i = 2i« -1 = 2»* = 16384. Ex. 11. Which is the greatest term in the expansion of (1 +x) ' 3 when X = '3. Here r is the whole number equal to or first greater than •3 3 42 (13 + 1) r-r or 14 X — • or TT- which is 4, therefore the 4th term 1*3 13 13 it the greatest. (iii ExBROiSB LXy. Find the general term and the 6th term of :— • 1. (1-x)-" 2. (1 + x)-* 3. (l-^x)-8 5. (1 + xf^ 6. (I + »)"^ 7. (a - x)-i Fmjcl the general term and the 5th term of: — i. (l-2x)-» 10.(l + |x*)-| U,, ( ^^^ '^^^'^ ^^' tegers all less than r, the radix of the scale. For let N be divided by the greatest power of r it con- tains, and let the quotient be d^, less of course than r, and let the remainder be ^1. Then JV =(/»»"" + -^i . Similarly let N^^ be divided by the greatest power of r it con- tains, and let the quotient h6 d^'^ with remainder N^. Then Similarly JVg = d„.a ^"'^ + -ZV^j , and so on, and continuing thJB process until the remainder becomes < r = say d^^ we h&vte N=dnr^ + dn-i r'^-^+ kc + d^r^i- d^r^ + do. Where any of the coefficients d„, d^.i, &c., d^, d^, d^, dg^ may vanish, i. e., become = 0, but none can be > or == r. In other words, these coefficients, or digits as they are called, may havfi any value from to r - 1 inclusive, and consequently in any scale r there occur r digits, including zero. (See National Arithmetic.) 286 PROPERTIES OF NUMBERS. [SxoT. xnt. 284. To express any number in any proposed scale :— Let N be the number and let r the radix of the proposed scale. Then by last Art., the given number may be written as s dnr^ + dn.ir^-^ + &c. + rf, r» + d^ r* + df,. Dividing this by r we get a complete quotient with remainder d^f the right digit of the number in the proposed scale. Dividing this complete quotient by r, we get another complete quotient with rem. dl,, which is the second digit of the number. And proceeding thus as long as we get a quotient divisible by r, we obtain as remainders the su jcessive digits of the number. (See Arithmetic.) >^ 286. To prove the rule for reducing a pure repetend to its equi- valent vulgar fraction. Let R = the gi^n repetend, and let it contain r digits, and let r='it8 value. Then V= 'RR&c. (i). Multiplying each by lO' we have 10" r = i2-2?iJ &c. (II). Subtracting (i) from (ii) R But since r = the number of digits in the repetend, 10** - 1 will be AS many 9's as there are digits in the repetend. Repetend V = As many 9's as there are digits in repetend ' 286. To prove the rule for reducing a mixed repetend to Us equivalent vulgar fraction. Let Fsthe value of a mixed repetend in which jP represents the finite part and R the repetend, and let F and R contain respectively /and r digits. Then V- 'FRR &c. Multiplying these by 10^*'' we have 10/*T= FR'RR &c. (I). Also multiplying them by 10^, 10 >^ r= FRR &c. (II). Subtracting ii from i, FR F (lOf*--lOf)V=FR-F That is, r= xo/(10'^-i y But 10^ is unity followed by at many ciphers as there are n})its w /, i, e., ^ manjr cipl^ers m there ft?^ 4igita m -FJ tb§ ABT. 284-289.] PROPERTIES OP NUMBERS. 237 finite part, and lO**- 1 is as many 9*8 as there are units in r, i.e., as many 9'8 as tliere are digits in J2, the repetend. Whole repetend minus the finite part. ' ' "As many 9's as figures in repetend followed by as many O's as figures in finite part. 287. Thiorkm I. — If from any number the sum of its digits be st^tractedf the remainder is divisible by the radix of the scale decreased by unity. Dbmonstration. — Let r be the radix of the scale, and let a + br + cr^ + dr^ + &c. be the number. Subtract a-\-b ■{• c +d + &c. the sum of the digits. Then the rem. = 6r-6 + cr'-TC+dr*-d+ &c. = 6(r-l)+c(r^l) + rf(r» - 1) + &c., which (Art. 80) is evidently divisible by r - 1 i.e., by the radix decreased by unity. 288. Thiorbk II.— If the sum of the digits of any number is divisible (r - 1), that is by the radix decreased by unity ^ then the number itself is divisible by one less than the radix. DiMONBTRATioN. — ^For let N = the number and S = the sum of its digits, and since S is by hypothesis divisible by (r - 1) let S = m(r - 1). Then Theorem I, N- S is also divisible by r-1, ..letN-S = p(r-l). Then by substitution we have N-m (r -1) =p (r- 1) .: N= p(r - 1) + m(r - 1) = (r - l)(p + m), and since the right-hand member is evidently divisible by r- 1 .*. also the left- hand member Nia divisible by r - 1. Cor. In any scale such that r - 1 is divisible by 3, if the sum of the digits of any number be divisible by 3, the number itself is divisible by 3. For let N and S represent the number and the sum of its digitt), and let S = 3m and r - 1 = 3q. Then N- S = p(r-1) = 3pq .-. N-Sm = 3pq .: N = 3(pq + m). That is, N is divisible by 3. * Hence in the common scale a number is divisible by 3 or by 9, according as the sum of its digits is divisible by 3 or by 9. 289. Thiorbm in..~-Iffrom any number the sum of the digits standing in the odd places be subtracted, and to it the sum of the 288 PROPBBTIBS OF NUMBBRS. [Sbot. XIU. digits standing in the even placet be added, then the reeult is divtS' ible by the radix increased fry unity. DmoKSTBATioK. — Let r be the radix and let the number be a + br + cr^ + dr* + er* + &c. Add- o + fr-c +rf -e + &c. The result is 6r + 6 + cf' - c + rfr»+ rf + «r* - e + &c., which is equal to b(r + 1) + c(f - 1) + rf(r" + 1) + e(r* - 1) + &c. But r + 1, r" - 1, r« + 1, r* - 1, &c., are all (Art. 80) divisible by r + 1, .'. fr(r + 1) + c(r* + 1) + d(r^ + I) + 4^o. is divisible by r+1. 'Cor. Hence in the common scale any number answering the conditions given above is divisible by 11. 200. Thxorih lY.'^Ifin any number the sum of the digits stand- ing in the even places be equal to the sum of the digits standing in the odd places, then the number is divisible by the radix increased by unity. Let Ns the number, iSf = the sum of digits in the even places, and 8 1 the sum of the digits in the odd places. Then Theorem III, N+ S- 8,iB divisible by r -l- 1. Bat since by hypothesis S= S^fii follows that 8- Si = .'.NIb divisi- ble by r + 1. 291. To prove the commxm rule for testing the accuracy of mul- tiplication by casting out the 9'«. DaicoNSTRATioN.—It follows from Theorem IL that any number in the common scale will leave the same remainder when divided ;[.by 9 that the sum of its digits will leave when divided by 9. Let then 9a + c be the multiplicand and 9fr + (2 be the mul- tiplier. Then Slofr + 9frc 4- 9ad + cd will be the product. Now if the sum of the digits inthe multiplicand be divided by 9, the rem. ittc, if the sum of the digits in the multiplier is divided by 9 the rem. is d, and if the sum of the digits in the product be divided by 9, the rem. is evidently the same as the rem. obtained by dividing cd by 9. 982. Thbobrm Y. — The product of any thrse consecutive numbers in the scale of 10 is divisible fry 1.2.3, i.e., fry 6. Aet. 290-9B8.] PROPBRTIBS OV 17TJMBEBS. m DiMONiTRATiON.— Every number must be of the form of 3m or 3m 4- 1, or 3m + 2, because every number when diyided by 3 must leare or 1 or 2 as remainder. .*. The product of any three consecutive numbers may be represented by 3m(3m + l)(3fii 4- 3). But 3m is a multiple of 3 and of the other factors 3m + 1 or 3m 4- 2 one must be even, and must therefore be divisible by 2, .*. 3m(3m 4- l)(3m 4- 2) must be divisible by 1 . 2 . 3, i.e., by 6. 393. Thiorbk VI. — The product of any r coruecutive number t is divisible by 1.2.3.. ..r. DiifONSTBATiON. — Let fi be the least of the numbers, and let n(n4-l)(n 4- 2). ... (n 4- r - 1) be represented by "P, for all 1.2.3.4. .. .r values of n and r. T»,n„ P n(n4-l)....(n4-r -2) n4-r-l - /n-l \ n-1 ( n- l)n(n + l)(n4- 2). . . .(n 4- r - 2) = .^r-,x ^ +.F,.,= 1.2.8....r * Now if we assume that .P^.| is an integer, or in other words that the product of any (r - 1) consecutive integers is divisible by 1.2.3....r. Then since as above shown «P, * , . ,i*r + n^r . i ^e have «P, » n-i-Pr + int., an integer for all values of n and r, and writing in succession n - 1, n - 2. . . . 3. 2 forn we obtain ..,/», = „.,Pr + int., Ac. 3 Ac. ,P, = ^,4-int. ,P, = ,P^ 4- int. Adding these equals and cancelling, we « .> -. . » . ^ 1.2.3.4. ...r have ,P, = ,P 4- sum of integers, but ,P, = r irTI Z = 1- .•. ,P, = 1 4- sum of integers = an integer. Hence if .P^.i is an integer, then also ,P^ is an integer. But it has been shown Theorem V that ^, is an integer therefore also J*^ is an integer, and therefore also JP, and so on, .*. JP^ is an integer, that is n(n 4- l)(n 4- 2). ... (n 4- r - 1) is divisible by l.Z*3*aa»f« uo INEQUALITIES. SECTION XIV. [Sect. XIV. INEQUALITIES, VANISHING JRAOTIONS, INDETER- MINATE EQUATIONS. INEQUALITIES. 294. In addition to the axioms given on pages 16, I7j the student will find it advantageous to remember the fol- lowing propositions : I. If the sarne qttantity be added to or subtracted from two ttn- eqtuUs, the sums or differences are unequal. Thus if a > 6 then a ± c > 6 ± c. II. If two unequals be both multiplied, or both divided by the same positive quantity, the products are unequal, as also are the quotients. Thus, if a > &, a - 6 is positive, and if m be positive then also m(a - 6) is positive, and .*. ma > mb; similarljr 1 - a b •zria-b) is positive, .-. — > nL m m in. Jf the terms of an inequality be multiplied or divided by any negative quantity, or if the signs of ail the terms be changed^ the iign of inequality must be reversed. Thus, if a>6then o-6>0 or -b>-a,OT^a<-b', so also if a > 6 and - m be any negative quantity, a - 6 is positive .*. m(a - 6) is negative, .•. m(b-a) is positive 1 .*. m6 > wa or ma < mb. Similarly "Z" (^ - <*)w pos. b a a b mm mm, IV. If any number of inequalities, all having the same sign of inequality, i.e. all > or all <, be atl niultiplifd together, left-hand members by left-hand members, and right by right^ then the resulting products will form an inequality with the tame sign. ABt. 2M.] IltEQUAtitTItid. 241 Thus, if a > 6, c > (i, e >/, then ace > bdf. Y. If&th arid n b*> positive quantities^ and a> b, then a'>> b" and l^a > J^b. Thus, a > 6, /. last article, a''> 6*, .*. a'> 6', and eo on? .*. a« > 6" ; similarly JlJ^a > J^b Vl. 7/* any nunU>er of inequalities having the same sign be added together, the sum is an inequality of the same kind. Thus, ifo>6, c>dand e>f then a + c+ c>6 + d + /". Note.— It does not, however, follow thtit if one inequality be sub- tracted from another, the difference is an inequality of the same kind. Thus, if a>& and c><2 it does not always follow that a - c>& - tf, since a may be nearer in magnitude to c than 6 to d; for example, although 7 > 6 and 6 > 2, 1' - 6, is not greater than 5 - 2, i. e. 1 is not greater than 8. YII. If the same quantity or two equal ^mntities be divided by each side of an inequality ^ the sign of inequality vnll be reversed. Thus 5 > 3 but "5" < "o"* ^'®* S <^ 5 ; so also if a >6 then by m a m dividing m by each we have Ei. 1. Shew that if a be pos. and b>a then jr^ > a .^ Since 2 > multiplying by ab we have 2ab > .*. also a* + 2ab + 6» > o2 + 6" and dividing each by (o* + 6^) (a + 6) which is 1 a + b positive since a and 6 are both positive, we have —-7 < rjTp and multiplying each of these by a - 6 which is negative, because > a we have, proposition ui, — rr > JjaTTp* x6 +y6 Ex. 2. Shewthatx2+y''<-^— ^-^2^»_^j^^yi. Because (Art. 134) 2xy < x' + y^, multiplying each ea6h by xy we have 2xhf^ < x'V + «!/*, And adding x*-x^- x V -xy^i-y* to each we have x* - x^y ^'»Y--xy>' + y* 3x + 13 [ From 1st inequality, 2x<10.*. x<.5. From 2nd inequality, 2t>6 .'.x> 3 .*. X is >3 and < 6, i.e. is any whole number between 3 and 5. Hence x = 4. EXRROIBB LXVI. Find the limit to the value of x in the following inequations : 1. 7a;- 13 < 22. 3. 7a:-l<3« + ll. X X X X X 4.2x + 5>Ya:- 10. i3 to find the ax a' bx b* 6. Given -=-+6a;-a6>-r- and -=--ar+a6<-ir ,. . « " 5 7 7 limits of X. 6. Prove that a' + 1 is equal to or greater than a' + a accord- ing as a = 1 or a > 1. 7. Prove that o' + 1 > o* + a when o is negative and numeri- cally < 1. a b 8. Prove that T" + "T > 2 when a and b are both positive or both negative. 9. Given i(x+ 2) + ix < i(x - A) + 3 and i(x + 2) + ix' > i(x + 1) •!• i to find the value of x in whole nun^bers. 10. Shew that a^ + b^ + (:^> ab + ac + bc unless o = 6 » c. 11. Shew that o6c > (o + 6 - cXa + c - b)(b -^-c^a) assuming that a, b and c are unequal. 12. Shew that (1 + a + o^)^ < 3(1 + o» + a*) unless a = 1. ^ 13. Shew that ab (a + b) + bc (b + c) + ca (c + a)> Qahc and 'K 2(o' + 6' + c') when o, 6 and c are positive quantities. 14. If a:* - o' + 6" and y^ = c' + cP shew that xy > ac + bd. 16. If a > 6 shew that V(a + &)(« -b) + ^b(2a -b)>a. f 16. Shew that (a + 6 + c)» > 27a6c and < 9(a» + 6' + c»). ^17, Prove that (a + 6)(& + c)(c + a)> Sabc. jp2 ^ 3^ _ Yl . 18. If a: be real prove that ~2T2x~^f ®*"* ^*^® °® value betwen 5 and 9. n' - n + 1 19. Shew that ofn. n" + n + 1 lies betwen 3 and h for all real values Sxcm ^y. ARTS. 295-297.] VANISHINO FRACTIONS. 248 ambers. klity, 2r>6 between 3 [uations : ■l2-'>9- [) find the 4- a accord- nd nnraeri- positive or • b s c. assuming = 1. > 6abc and s. + bd. > a. no value eal TaluM VANISHING FRACTIONS. 295. A vanishing fraction is one which assumes the form of -rr when some particular value is given to some particular letter in both numerator and denominator. Thus, "T^TT is a vanishing fraction when 6 = 0, because then it becomes = "q". ' ' 206. Now it will be readily seen that in the above ex- ample, and indeed in all others, the peculiarity arises from both numerator and denominator having a common factory which factor = under the assumed conditions. Thus, in the example p.}- < above we have ■ ~ ^, and striking out the common factor a-h which = when b = a the expression becqmes a + b or 2a since 6 = a. 297. In order therefore to find the value of the fraction or more properly the limit to its value, we endeavour to find out the common factor involved, and casting it out, the result required is obtained by a simple reduction. Ex. 1. Find the value of Here X -a OPBBATION. X* - o* (a: - a)(x + a)(3^ + a^) when ar = o. = (z + a)(ar2 + <,2 j X — a X — a Now making x = awe have tlgds - 2ax 2a^ - 4a'. Ex. 2. Find the value of x^ — a"* jp7» _ £!»» X - a OPBBATION. when X - a. Here = x*"-* + ox** - » + a" x™ " ' + a' x** " ♦ + &c., to m X —a ' 244 INDBTBRMINATB EQUATIONS. [8«ot. XIV. A«T. terms and when x^ a this expression becomes = a™"* + a"'"*4' a"* ^ + a^-^ + Ac to m terms = ma^-^. X — a + *J2ax — lo? Ex. 3. Find the value of when x - a. OPERATION. XX ^JlfLt ^^'^ (a: - a) V^ - <* { V^ ~ " + V2a | V(a? - a)(x + a) V2a EXBRCISK LXYII. Evaluate the following vanishing fractions : l-x» 1. -:; — - when x = I. 3. » - o'x' X - a when X = o. ^- xa^Ia" ^**®*^ * ~ "' x2+2x-35 x^ + |x - I 6. - x" + 6x - ax=* - aft x^ - ox + 6*x - a&* 2 wheax«fl. 7. ox' + oc* - 2acx bi'-2bcx + bc* when X = c. ox — X* ^- a*-2a»x + 2a^'^r^ ^^^'^ * = «• x» + 2ax'' - a^x - 2a» ^' x» - 13a2x+ I2a8 ^^®° *='<'• INDETERMIIfATE EQUATIONS. 298. It has been already stated, Art. 122, that when* there are two or more unknown quantities involved in a single equation, the number of solutions is unlimited^, and the equation ifi indeterminate. ▲9T. 298-^OC.] INDETBRMINATE EQUATIONS. 245 TbuB, 8x + 2iyz:ill is an indeterminate equation because tlie number of vfduss which may be assigned to x and p is indefinite. This number may, however, be decreased : 1st by rejecting all fractional values ; 2nd, by rejecting all negative values; 8rd, by rejecting all numbers that are squares or cubes, &c. 299. Thbobbm I. — 2%e indeterminate equation ax ± by = o admits of at least one solution when a is prime to b. Demonstration. — ax ±by = c .-.xs a ; and substituting in succession 0, 1, 2, 3.... (a -1) for y, a being prime to 6, the several remainders must necessarily be different. For if any two values ofy&av and i/ give the same remainder r, q and ^ being the quotients, then c ± bv = aq + r and c ±bv' = a^ + r. There- fore i 6» ? bv' = a(g - j'), that is 6(» - i/) s a(q - ^) or 6 (t>' - r) = a (5 - g') ; that is 6 (t> - v') and b (t/ - ») are divisible by a without a remainder. But by hypothesis b is prime to a .*. v - v' is divisible by a which is impossible, since V and v' are both by hypothesis less than a, and consequently v-i/ and v'-v are less than a. Hence the remainders are all different and their number = a and each is a positive integer less than a, consequently one of them must = 0, .*. x is an integral number for a certain integral value of y less than a, and these integral values of x and y satisfy the equation ax ± by = c. Ex. 1. Find integral values of x and y which satisfy the equation 5x + 23y = 170. SOLUTION. 170 -23y Here x = r and substituting in succession 1, 2^ 3, Ac, for y and we find that 5 will do. 170-115 56 Thus, = 11 = « ,*. a: = 11 and y = 5. 300. Thbobkm II.— The equations ax i by = c cannot be solped in positive integers if a and b have a divisor which does not tUso divide c. DKMOHSTRA.TION.— For if it be possible let a and b have a com* mon measure m which is not also a measure of c, and let a con- tMn ^1 p times, and let b contain m, ^ times. Then ax^by-cUi 24^ mbimmmsATis equations. [isBor. xiv. ^0 equivalent to pmx ± qmy s c, orpx i jjf = ~. And since both p c and q are integers, and — is a fraction, it follows that x and y m cannot both be integral. XoTB.— If a, b and o have a common measure the eqaation may be divided through by this, and thus a may be made prime to b. In the fol- lowing articles this is always assumed to be done. 801. Given one solution of the equation az f by = c in positive integers to find the general solution. Let « s /3 and y = 7 be one solution of the equation ax + bys:c. a y-y Then a$ + bys c = ax + by .'.a(fi-x) = b (y-y) .'.-^s ——. a Now since -r- is in its lowest terms, a being prime to b ; .'. whatever multiple y~y is'of a the same multiple is jS - a; of b. Let y - 7 = aif, then $-xsbt where / is an integer, since we are only to obtain integral values. Therefore y = y + at and x = fi-bt is the general solution. Similarly writing - 6 for 6 we obtain for the general solution of ax- by = CjX-fi + bt and y =: y + at. Hence if one integral solution of the equation ax ±by ^c can be detected, the others can be readily found by giving different integral values to f in the equations x = fi^bt] y = y + at. Ex. 2. Given 3x + 4y s 39 to find the positive integral values of X and y. SOLUTION. Here x = 1 and y s 9 is evidently one solution. Then x = 1 - 4/ and y = 9 + 3^. Now let if = - 1, then a; = 6, y = 6, let ^ = - 2 then x =: 9, y = 3. NOTB.— Since the values of x and y may be found by substituting for t in the general solution x= fi'^bt, y-y +at, successively the values 0, d: 1, i 2, ± 8, kc, it follows that the values of x and y taken in order constitute two arithmetical series, and consequently that as soon as two contiguom values of each are determined, the rest may be written at once. 802. Thbosbm.— TAe number of positive integral solutions is limited/or ax + by = c, but unlimited/or ax r by = c. A&T. 801-006.] INDETStlMINAtE IfiQtJAttONd. 24t DsMOMSTBATioM. — I. By Art. 301 it appears totax + hye^ the general solution is x^ fi- bt and ysy + at where x = and y = yis one solution and t is any integer positive or negative. Now since by hypothesis x and y are both to be positive, it is manifest that fi~bt must be positive, that is bt must be less than /3, that is t is limited to integral values which are less than p y. Hence the number of positive integral solutions of ax + by = c is restricted. II. Similarly in the general solution of ax - 6y - c we have x = fi + bt and ysy-i- at where x = fi,y = yia one solution and t is any integer positive or negative. Now since by hypothesis x and y are to be positive, fi + bt and y + at must be positive and since ^, b and y are positive it is manifest that t may be any negative integer such that bt ro m .(If .*. The pos. int. solutions are x s 30 or 11 and y = 3 or 8. Ex. 6. Solve in positive integers 41a; + 68y s 2789. BOLUTIOir. 27y 27y,- 1 Dividing by 41 we have x + y + -jr- = 68 + ^, .*. — ^- - ig 41 41 81y-3 81y-3 mt. ; multiplying by 3 we have .. ii .. .•. 2y rr— is int. 82y-81y + 3 y + 3 y + 3 .*. ij , that is —.y- is int. Lei -^ = t%'iMnysiiit-'3, Substitute this value of y in the given equation and 41x = 2789 - 68 (4U - 3) » 2789 + 204 - 68 x Alt. 2993-68x41/ ^ ••• * = 41 = 73 - 68^ Hence x s 73 - 68/) x = 6 ( , y = 4W-3}-*-y = 38. r*'*^'^^ It is evident that this is the only int. post solution, for 73 - 68/ must be pos. int., so also must 41/ - 3 .*. 68/ < 73 or / 3 or />ff and the only positive integer between J|, and A is 1. NoTB.—The student will not fail to observe the artifice made use of, in the 2nd line of the solution, to avoid using « large multiplier, and the trouble of searching for it, since it must be such as to render the ooeffi* cient ofy divisible by 41 with a remainder 1. v^ a r<;^/»« 3x-7y + a-16 > to find the positive integral Jix. 6. ixiven 5a; + 3y . 4a; = . 4 J values of x, y, and z. SOLUTION. Multiplying the upper equation by 4 and adding the two together we have 8y 9 l7x - 25y = 60, and dividing by 17 we get x - y - T» = 3+7^ V-. ^y±±is integral. 17 lAtix i« . . 16y + 18 So also is ^^ll_ !^, and so also is y rr-- integral. . y-18 * • 17 17 is integral = /, say, then y s 17/ + 18. Then l7x s 60 + 25y = 60 + 25 x 17/ + 460 = 510 + 25 x 17/, , X = 30 •!• 25/ and y - 17/ •!- 18. R 250 INDBTERMINATE BQX7ATI0NS. [Siot. XIV. Hence a; s 5, 30, 66, Ac, and y « 1, 18, 36, Ac. Bnt % also has to be positive and integral, and therefore the only yalues of x and y which are admissible are x » 6 and y ■ 1, and consequentlj 2 = 8. Bx. 7. What is the least number which when dirided by 4, 6* and 7 shall leave remainders 1, 3 and 6 ? SOLUTION. Let the number = 4r + l = 6y + 3-7s + 6. Then 4« - 6y » 2i .-. (i) 2x - 3y = 1 .'. X - y - ^ = 1 m Then y = 2/» - 1. y + 1 is int. ~ m, say Also (u) 6y - 7a = 2, that is 12ot- 6 - 7« = 2 /. 12wi - 7« « 8 6in 1 6m - 1 16ni - 3 /. -^ - s + ^ a I + -r ••. — = — is int. .*. r — is int. 7 7 7 7 -3 is int. = ^, say, then »i = 7< + 3. Hence y = 2m - 1 = 14^ + 6 - I = 14/ + 6. 6y + 2 3 1 42/ + 15 1 X = =-j»+^- + - = 21« + 8. 2 6y - 2 84/ + 30 ~ 2 And« = -^^=: ^ =12/ + 4. r Consequently « = 8, y = 5, and z = 4. And the required-number s 4x + 1 » 33. Ex. 8. In how many ways can £80 be paid in sovereigns and' guineas ? SOLUTION. Let X - number of sovereigns and y = number of guineas. y Theni n shillings 20 « + 2i y = 1600 •*. ar + y +— = 80. .'. y 5 20/. And 20x = 1600 - 21y - 1600 - 21 x 20/. ,'.X a 80 - 21/. I7y » 2 C. 13x + 21y - 89 7. 12»-41y = - 17 8. 37x + 43y = 357 9. 22x - 43y = 6 10. 7x + 26y =177 11. 99x - 160y = 335 12. 17x - 4y = 22. t Find a positive integral solution of the following : 13. 2x + 3y iz = 29 3x + 6y - 3z = 9 I 14. 4a:-5y-6«= 17) J 2x + y + ll« = 47$ 15. In how many ways can the sum of $697 be made up by 'bank notes of the respective value of $3 and $5 ? 16. In how many ways can $27.30 be paid in twenty-five cent and ten cent pieces ? 17. What is the simplest way for a person who has only guineas to pay £1 10s. 6d. to another who has only half crowns ? 18. Find two integral square numbers whose sum is a square. 19. Find two integral square numbers whose diflference is a square. 20. A basket of apples is known to contain between 90 and 100, and it is found that when they are counted four at a time, there are two over, and when counted six at a time there are also two over. How many are there in the basket 7 21. Find the least integer which when divided by 6, 8 and' 10 respectively shall leave remainders 1, 6 and k.y 22. How many fractions -are therewith denoafmators 10 and 15, whose sum is>^? ^^ 23. A person bought 50 barrels of fruit, consisting of apples, pears, and cranberries, for $250 ; the apples cost $2 per barrel, the pears $5 and the cranberries $4, how many barrels were there of each ? /^u^Lu ^ 24. How can a debt of £100 be paid with £5 notes, £1 note /I jU&d <»rdwn pieces ? ' 25. Divide 25 into two parts, one of which may be divisible l^y !^,and the other by 3. — 1m. Divide 24 into three such parts that if the first be multi* U2 mSOliLlANltfOUS ViStLOlM^, plied by 36, the aeoond by 24, and the third by 8, the sum of the three producti may be SI 6. 27. Find a perfect number, t. e. one which is exactly equal to the sum of all its divisors. 28. What is the least odd integer which divided 10, 12, 14 shall leare remainders 7, 9 and 11 respectively 7 29. A person buys 100 head of cattl^ of three different kinds for $500. For the first he gives $60 a head, for the second $80, . and for the third $2, how many were there of each kind ? MISCELLANEOUS EXERCISES. / 1. Simplify J [1(1 - a)\ - HHH^a - 6)}). , 2. Prove that («> + 1 -x"')* - («» - 1, - x - =0' = 4 (a:«-x-=«). • 3. Find the O. C. M. of o» + 2ab + b\ o» + 6», a* - 6» and fl» + 2a% + 2o6» + 6'. x~b x-a 6' w 4. Find the value of . where x = . . b'-a I 6. Given x + y + ««3(« + «-y) = 6(«-a:-y)=:15to find the values of x, y and a. / 6. Find the value of 6'^135 - 3.^40 + 2'^626 - 4'^320* I 7. Given x^ + 1 » to find the values of x. 8. If a : bwb'.e^ and 6:c::c:c;, show that o + 6:ft + c;:6 + c:c + rf. 9. Shew that if a: c:: 2a -> 6:26 - c, then will a, \b and \c be in harmonic progression. ' 10. In the series o + o /l — j »* + a T 1 - - J *^ + a l\^ (,.!)= + &c., the sum to infinity \Bp times, the Sum of the first n terms. s» 11. Reduce. X' -X x*-x — and -z — T-T — = to their simplest x*+o'x' + a* form. 12. Find the cube root of 343x6 . 441*»y + 777xV - B31*V + 444!B^ - 144xy» + 64y«. MISOBUiAlfEOnS BXBR0I8BS. 258 ab 13. Simplify **»-« X x^-P X x^w-"*, an4 also — x^-? be ca ^ 14. Find the product of 2«'+y+ Jx-'y»into 2x''-y + Jx-Vj of ** + dx + 6* into x" - ox + fc^", and of x"* + y* into x" + y«. 15. Simplify ?^-^:i^? + 2-Vi--?^ ' 3V6 + 2V3 3V6 - 2V3* 1 16. Find the yalue of 2x + 1* IT. Find the value of 1 1 2x4-1 4 (2x - 1) * 4 (2x + 1) "^ 2 (2x - l)(4x2 + 1). 18. Find the yalues of x in the equations a c a — c (0 x+a *" x + c " X + a~c' (n) V(x - l)(x - 2) + V(« - 3)(x - 4) = 2. ^ 1 __1 1 / (™) x»-2x-16"'"x2+2x-36" x»-13x-48 ' ^* ■ b + c md |e be I -f- 19* If n s l~3~;} and 6 be the O. mean between a and c, n'^b^ 1 then a ■ 1-, ^ill he the H. mean between n and — . a* + b^ n 20. .i and B can together perform a piece of woric in u wxys, which w4 and C can finish in 6 days, and B and C in c days. Find the time in which each can perform it separately. 21. Find the values of a* 6» . (o - 6)(o - c) ~ (c - 6)(6 - o> (6 - cXc - a) ' ?2. Shew that a^ " [ 4^ ■)•■ (o + 26 + 3c)(a + 26 - 3c)(a - 26 + 3c)(2 6 - o + 3c) 1662 . 254 MISCELLANEOUS EXEROISES. 23. Find the two factors of oS + 6^, and the two factors of 24. Simplify « + y + y a? + y + ^ a; 25. Find the I, c. »i. and also the G. C. JIf. of x^ + 3a;y - 28^^^ «' - 2a:jf - 8j/' and x' - 5a;y + 4y^ 26. Find the general expression for the sum of a geometrical series when r = + 1. 27. If by the notation a^we represent the 6' factpfi. BaSOSLLANEOUS EXEHCIS£;S. 267 66. If four quantities are in "ontin.ued proportion, the first has to the fourth the triplicate ratio which it has to the second 6*7. Find the integral values of x which satisfy the inequaUty a:2any even number of terms, the sum of the latter half is three times the sum of the former half. a b 77. If the jt. mean between two quantities be -r- + — +2, a b a b and the H. mean be t" + 2, then the G. mean will be -r- - — . a ; p. a , 78. If a, 6, c, be in H. progression, then will ae''b-ab-c' X 79. If r + « + < = V, where r is constant and » oc — and t oc xy\ and when x = y = 1, v s 0, and when x = y = 3, v = 8^ and wheQ j; s 0, D - 1, fii^4 V in terms of a? and y. 258 MISCELLANEOUS EXERCISES. ^ 80. 3olye with respect to x the equations (i) {(a + 6) X + o - h]{(a + 6) « + 6 - c} = 4a6, ax b b (ii) 1 = a; + — . ^ ^ b a ax 81. Fin^ the continued pfoduct of (aV 6)(a + &)(a' •!-&') + &c, to n 4- 1 factors. . 83. Divide x* - (a + 6 +p)x» +(ap + bp-c + q)3^ -(aq + bq^l ' cp)x -qchy x^-px + q synthetically. f 83. Find the square root of o^x^ + 2o6x* + (b^+ 2ac)x2 4- c*x-» + 26c. 84. Simplify 1 1 "[(x + a)(x - 6) + (X - a)(x + 6)j "^"[j } [(x+o)(x+6) (x - a)(x-6) .^■} 85. Find the G. C. M. of x* + pV + p* and x* + 2px» + p^x* - y*. 86. Find the /. c. m. of 2J(x2 + x - 20), 3\(x^ - x - 30) and 4i(x2_l0x + 24). 8*7. Solve with respect to x the equation ^a* - l)x2 - 2(o6 + l)x + 62 - 1 = 0. 88. Simplify the following expression x»+x-» + 2(x + x-i) /x^-i y a:»_ a: '» - 2(x - X -1) • ( x^TTy 89. Prove that if to any square number there be added the square of half the number immediately preceding it, the sum will be a complete square ; viz., the square of half the number imme- diately following it. 90. A cistern is furnished with two supply pipes A and B^ and a discharge pipe C. If A and C be left open together for three hours, and C be then closed, the cistern will be filled in \ an hour more ; if B and C be left open together for five hours and Cbe then closed, the cistern will be filled in 1) hours more; or it can be filled by leaving A open for I§ hours, and B \ hour. Jn what time can the cistern be filled or emptied by A^ Bj and C, separately. -^ 91. Find the G. C. M. of 2x5 ^ 2x* - 6x» + 4x'« - 9, and 3x^ + 3x' - lOx* - X + 3. 92. Find the /. c. m. of apx* + (o^ + bp) X + 65, and aqx"^ - (ap - 65) x - bp^ Also of (r* - xy) ; (x* ~ y^j and ^xy + 3/*), :t?::«H*::5*sa5E=rK5i-' MISCELLANEOUS EXEBCISES. 259 b. ■6«) + &c, oq •\r hq "^ I ' »)(«-6)J p^x^-p*. 30) and Ided the sum will er imme- and JB, ether for led in i )urs and 8 more; i hour. Bf and and 3x^ bp, 93. Solve the equations X -2a 2x + 6a 0) (n) 3 x-1 x + l ~3~ x+1 X + 2a 13" 2 ' x-l (ra) Va: + 4 + ^/2x + 6 = V^ia? + 34 (IV) x'(v ~ 1) + 3y(a;2 - 1) = ./x" + 3^ and a;»y = 5 94. Form the equatic whose roots are 2y 3 and - 2 ± V - 3 ^ 95. Simplifyo-(o-.ri.)-{-(-{-o-(-m-|-(m- radii 4, 6, IPand 7 respectively, shew that they can all be (C f made into a single circle of radius 12, assuming that the area of a circle varies as the square of its radius. 123. "I^iven the first term of an A. series s ll, and that the sum of the first 3 terms ~ the sum of the first 9 terms, to find the series. 124. Given any two terms of a G. series to construct it. 125. Find the G. series whose 1st term = 3, 6th term = ^f, and sum of first five terms = 2^^. 126. Prove that the latter half of 2n terms of an A. series ic one-third of the sum of 8n terms of the same series. 127. If 8^ denote the sum of n terms of the series 1+5+9 + &c. and S^ denote the sum to (n- 1) or to n terms of the series 3 + 7 + 11 + &c., prove that 5i + S^ = (S^ - S^)\ 128. Find the 7th, the 10th and the general term in the' expansion of (1 + x-")~^* 12 9. Fo rm the equation whose roots are 1, - 1, 2,-2 and/^ 130. Assuming that - 1, 1 and 1 are three roots of the equation a;5 + 2x* - 3x' - 3x2 + 2« + 1 = ^o find the other two roots. 131. Find what quantity must be added to each term of the ratio a : 6 in order to make it four times as great as the ratio c : d. /2-V3\i_ y2_ \^2 + V 3y " 1 + V3 132. Shew that 133. Given ^^ 1 ly 1 3« = T } tofiadx,y,« in positive integers. 134. Find the value of the vanishing fraction when X = y. 135. The sum of two numbers is 45, and their I. c. m. is 168) what are the numbers 7 a-Wy" (x - y) 262 laSOBLLANEOUS EXBROISES. 5 ( (a: + 2)(x-4) > 1^ (g + 3)(x - 6) 9''(«+ 4)(a:-6) 2_ ( a; + 5) (a - 7) 92 13*(x+6) (x-Sf* 685 to find X. 137. Given ^**a?- «»ts^6~ * J ^ fi"** ***® values of « and y. 138. Prove that the fraction ^ on being cojurerted into a decimal will continually produce, successively^'^ order, the digits 0, 1, 2 .... 9 inclusive with the exception of 8. 139. Prove that the roots of ax' - bx =c?x -ab are rational. 140. Solve the equation (a + x)(b + x)s nab. 141. Find the value of x in the equation 1 + V^ =^ 6x. 142. Given V* + ^/x - 1 = ^x + 1 to find x. 143. Solve with respect to x, y and z the equations a" 6« x+y + z = — = — * X y c» 144. If a number be multiplied by 4, and the same number reversed be multiplied by 5, the sum of the products is exactly divisible by 9. Prove this, and infer the general proposition of which it is a particular case. 145. Simplify (a + 6) (6 + c) - (a + 1) (c + 1) - (o + c) (6 - 1). 146. Find, without actually multiplying, the product of /xV \ /xy N (•~--xy + 9Jinto(^-3>+3J 147. Find, without actually dividing, the quotient of (ox -f b'y)'^ + (ex + dyy + (ay - bxy + (cy - dxf by x' + y*. 148. Extract the square root of a* (x2+ 4) - 2a (x + 2)+ Aa^x + 1 by inspection. ^f 149. Find the G. C. M. of a* ^b'^-c^-k- 2a6, and a" - ft* - c^ + ^ 2bc by factoring. , 150. Divide synthetically 4x* + Sx" + 1 by x' + 2x - 1 obtain- f ing the exact remainder, and also four terms of the remainder expressed in descending powers of x. * 15 15 m 151. Expand -^ — - in ascending powers of x. i ■■ * T X MISCELLANEOUS EXBROISES. .268 ; and y. ed into A rder, the (. itional. ) number 1 exactly h it is a (6-1). of ax + by)'^ iaH + 1 62 - c2 + obtain- (nainder 152. Sim. lify (jf,+ ;^j) X (^, -^^) 163. Divide (-^ ^^ by (-r^ ^—\ \a^c b + c J '^6 + a + cj 164. Bedace to a single fraction in its lowest terms 3{x - 2) 1 1 1 (x-l)(a;-3) " X- 1 " (« - 2) " x - 3 156. Prove that (gy + 1+ 2x)(a;y + 1 + 2y) + (a - y) 2 (x + l)(y + 1) xY+l-a^a.ya = (x-l)(y-l) i66. Find the conditions necessary in order that the equations lax* + 6x + c = and a^x' + 6ix + c, = may have (i) One root common. (ii) Roots equal in magnitude, but of contrary signs. ,.^' , , x+1 2x-l 3x + 4 5x-6 167. Solve the equation 2 3 3 ,i.« r., (x-l)(x + 4) (3 + x)(2-x) ^ ^ ^ 168. Given ^ — '^ .. ^ =: ^^ —^ to find the value f X + oj X "" X Dfx. 169. Find the value of x in the equation l+2x 1 + X + VI + 2X l-2x i_a.-Vr^^ 160. Find x in the equation (X - 1)» (n - ly + 4n =P. ^ (X ^ 1)* (n - 1)* + 4n iEind shew that if n be positive and x real, the value of the left hand members always lies between n and ^ 161. Find the Jl.y G. and N. means between I and |. 162. If H. be the harmonic mean between a and 6, prove that it is also the H. mean between (H- a) and (JET- 6) 163. Find the STth term of the series 6 4- ^^i^ + V + &c., and also the sum of the sums of the first 31 terms and 42 terms. 164. Find the sum of n terms of the series 3} + 2 + 1^ + &c. 165. Find the sum of n terms of the series l-0*4 + 0«16^ b*64 + &c., and also the diflference between the sum *o infinity and the snm to n terms. :264 MtSOBLLANSOUS BXlfikoiSES. 166. There are p arithmetical series, each continued to n terms ; their first terms are the natural numbers 1, 2, 3, Ac, and their common differences are the successive odd numbers 1, 3, 5, ke. Prove that the sum of all of them is the same as if there were n such series each continued to p terms. 167. Find the continued product of x - ^xy + y, ar + '^xy ■¥ y and x^ - xy 'i- y*. 168. Find the value of y (x' - Syy + x (x^ + 3y)* when x= 5 and y =: 8. 169. Extract the 4th root of 16a« - 9Q(^b + 216a>6' > 216a6'' ■<- 816*. 170. If a : 5 : : c: d shew that Jl 1. 11 L 1. f± L * _ JL' J_ J_ /"^ A £ \ 2b'3c'*'4d^ad \^ ~ 3 "T^^J 2x + 3 Ax + 5 3x + 3 171. Solve the equation ^j-^ - ~-^ + z_— l giving the rule and reason for each step of the operation. 172. Solve with respect to x the equation 1111 X X +b~ a a + b 173. When x :s a+1 and y = ab + a reduce X + y- 1 to its ^ o6+l ' ab + 1 x + y + 1 lowest terms. • 174. Shew that 2(x - y)(x - «) + 2(y - «)(y - x) + 2(« - x) (x - y) can be resolved into the sum of three squares. 175. Divide a* + 6* - c* - 2ay + Aabc^ by (a + 6)* - c^. 176. Find the G. 0. M. of x* - land x' " + x^ + x^ + 2x7 + 2x*+2x' + x2 + x+ 1. 2x + 3 x + 2 x-7 177» Reduce ■ . -.r TT." ^ . , - z — r-rrr- — rr *<> a (x + 5)(x+l) x^+1 (x + 5)(x-l) simple quantity. a + 6V"l a-b ^f - 1 178. Reduce —■ =: + ■=z to a simple quantity. a-b^ - 1 « + 6 V- 1 179. If four positive quantities be in A. Progression, the sum of the extremes is equal to the sum of the means ; but if in G. or Ji. Progression the former sum is the greater. Required proof. 18| the greal 18 is] 18 184 18{ thedi by th thedi 186 long, seooni per ho rates f 187. duct two is thd'sq 188. 189, 190 IJ? 191 '^ 2cxy ( 192 193 194 (6 + c; plest : 196 losoauuwKocs bzbboisw. m 2x7 + tity. sum in G. luired- 180. Sl^ew thAiin an asoendtng J. lertei if the le^t term b« the oommon diflbrenoe, the 8om of (2n < 1) ternii ii n times the greateat term. r- a , yy^" . i 181. Solye with veipeot to x the equation 182. Oiyen 3x +x^ * 3104 to find the yalnei of 9, 183. Find the yalue of « in the equation x+ a X" a 6 + x h -mx X - a " x-¥ a h - x " b -k-x 184. Giyen « + V {** + V^' + »6} » U to find the yalneg of «. 185. Find a number of two digits, such that when divided bj the difference of the digits, the quotient is 31 ; and when divided by the sum of the digits and the quotient increased by 17| the digits are inverted. 186. Two horses Jt and JB, trot twice round a course two miles long. £ passes the post the first time 2' before Jt^ but in the second round Ji increases and B slackens his pace by 2 miles per hour, and A does the round in 2' less than B, Find their rates and which horse wins. 187. With any five consecutive integers, the contioned pro- duct of the first, middle, and last, added to the cubes of the other two is equal to the product of the middle number by the sum of the'squares of the middle three. Required proof. 188. Prove that x* + y* + (x + y)* = 2(a^ + xy ^f)\ 189. Multiply x* + y' + xV + «y* by «* -Y ^ «^ + xy*. 190. Find the value of ax* - J x* when x s (o+6)" ± (a- 6)*. ; 191. Divide ax» + 2exyz + djf* + aa^^y + ») + bf{x + x) +1 2cxy (x + y) by a; + y + «, synthetically. I 192. What is the quotient of x"' - 1 divided by x» - 1. 193. Simplify 1 - {1 - (1 -x)} + 2x - (3 - 6x) + 2 - (- 4 + 6x). !94. Express a(h + c)» + ft(c + o)» + c(a + 6)» - f(a - 6) (a - c) (6 + c) + (6 - c)(i - a)(c + o) + (c - a){c • 6X« + *)} "» its Bim- plestform. 196. IBzpress in the simplest form the sum of (i + c - d)x ^(c^a - 6)y + (a + 6 - c)» (c + a - 6)x + (o + 6 - c)y + (6 + c - «)• (o + 6 - c)x + (6 + c - a)y + (c + - *)« 8 njt^ 2M MIIOIUJlHBOUB BXBBOItM. IM. Find th« prodnot of («* •!• 6«*y 4- 12xy» -i- Sy>) b7 («■ - eaf«y + Uary" - 9f) aIm of (o + 6 V-^l)(« - * V""!)- 197. PindtheTftlueof(a+6-»-c)(6 + c-o)(c + a-*)(a + ft-0 Alio the product of (»" + 1 + ar>) by («■- 1 + ar>). 198. DlTlde (2x*-8a^ + 4ar«y« - 6«/ + 6y*) bjCaVj Md alio («*•(■ 4x -t- 8) bj (z* •«- 2x -I- 1). 199. Find \>j inipeotion the quotient of (8« - y") f (« -^y) and of («• - op*" + a^x - o") f (« - «). aOO. Find by factoring the O. 0. H. of A^ (1) *«-3«-4,ar«-a«-8andx« + »-a0. ,<^,^/ '^ (11) 8x« + 4»» - 3* - 4 and 2x* - 7x* + 6 (ni) («• + o*)(** - o») and («• + o*)(«* - a*). 201. Find the 2. c. m. of (i) «•-(!«- 2a', *• + ox* and ax* -a* (ii) «• - x*y - a'x + a'y and x* + ax* - xy"- ay* 202. Find the ralue of (a + b -c)* - d* (& -h c - o)« ~ (f a (c + a - 6)» -

'*" (c+a)»-(6+i«)« ^ , X* + «*-«'+ 2xy 208. Reduce n — rs — ,» . «... to ite loivreit termi. x'-y*- a* + 2y2 204. Simplify the expression g* + a*b a(a'-b') a»6 - 6» 2a6 (a-f6)6 a*- 6* 206. Reduce fa+ i^) ("a- ^U (^-^ + ^ ^ a - x/ \ a+ xj \o - X o + xy to a simple quantity. X -f 2a X + 26 4aft 206. Find the value of — - + when x ■ r x-2a x-26 a + 5 20 T. Find by inspection the square roots of (I) x*-4x* + 8x + 4 (n) 4x*» - |x«» + ^x«" ^ , fl^ ft" c« a c b <«°)5r + ?-+^-2--2y + 2- 208. If oV + 6x + 6c + 6« be a perfect iqnare, shew that 1 c ^.->^^u^y^ut UlBOEUJXf^OVB BZBR0I8BS. 267 by w - 2ab a- a zl) Itbat a09. Solrt with reipeot to x the equationi (i) mtuB -f OAin B n'x + am' 8-« 2X-11 x~2 (II) -a i^T '^ "6- 310. Find the ralues of x in the equations tx+2. 80/g-i\ ^^^ ei^si "* 'B\x-iJ (II) «« - 2a* - 26x - 3tt» + 10o6 - 36» = 211. Find the values of x, y and« which satisfy the equations X - aw ax + y (I) — 1~ ■ 1 = ^ 6 c (u)*'" + xy + y* = 37 and X + y = 7. 212. Solve the simultaneous equation « (X + y) s o« + 6' ; x (y + «) s ft" + c» ; y(« + x) = c' + a' 213. The difference between the ages of «tf and B is twice as great as the difference between the ages of B and C, and the sum of the ages of Ji and B is half as much again as the age of C; six years ago it 'was only one-third more. Find their ages. 214. Sum the following series : (i) li + 3 + 4i to 12 terms, (ii) 1} + 2^ + 3^ to n terms. (Ill) V2 + SV3 + yi to infinity. 216. Ifa|.ay.a3....at = ax^ then will Oj + a^ 4- as + .... a, = a, . Ta~f • Required proof. f^: 216. Given (x + 6)(x + 1) = 4V2x + 1 (x - 1) to find x. 217. Find the value of x in the equation (3x - 4)(6x - 1)(1 - 2x») = 4, 218. Find to 4n, 4n + 1, 4n + 2 and 4n + 3 terms the sum of the following series 1+1 + 2-2 + 3 + 4 + 4-8 + 5+ 16 + &C. 219. The number of matches in the side of a certain rectangular bunch is > 10 but < 20, while the number in the end is < 10. When the digit expressing the number in the end is written to the left of the expression for the number in the side, the number 2U mBowuMSEOvs jatsiiciisBd, go formed is to the wbble number of matohes in the bnnch as a certain number a is to 2 ; but if this digit is written to the right of the expression for the number in the side, the number thus formed is the whole number of matohei as a - 10 : 4. Also a seconc;. bunoh similar in form to the first, and con- taining as many matches in its perimeter as there are matches in the first bunch, contains four times as many matches as the the first bunch. Find the whole number of matches in the bunch. 220. Shew that, in the preceding problem, if the last condi- tion had not been given, the solution found above would have been the only integral solution of the problem. 221. A person travels by railway from Stratford to Toronto and back. In coming down he finds that when he travels by express he is as many hours on the way as his fare is ^.ents per mile, but i^ben he travels by the accommodation train he is half as many hours on the way as there are units in the square of the number of pents in his fare per mile, the fare being the same by both trams. In returning, the express by which he travels goes slower than the express by which he came down by an average (including stoppages in both cases) of as many miles per hour as there are cents in his fare per mile, the fare being the same as in coming down. He now calculates that if the fare had varied as tlie speed of the trains, he would have gained a cent a mile by taking the accommodation train to Toronta*^the fare on the express to Toronto remaining the same — and in returning he would have gained as many cents as there were miles in the average speed (including stoppages) of the train. Find the distance from Toronto to Stratford, and the fare be tween them. 222. Given V«" + 25 {x\x^ + 9) (V«* + 25 - 1) -46} = 6a:» + 226 to find the values of or. . 223. Two persons engage to dig a trench 100 yds. long for $100, but one end being more difficult to dig than the other it is agreed that the one digging the harder end shall receive $1*26 per yard, while the other rei^eives but $0*75 per yard. At the termination of the job it is fbund that they' each receive $60. How many yds. did, each digf Shiew algebraically that this problenSt is impossible. MiddBLtiAiraOUS BXBROtSEEl. 260 i bnneb as a tten to the the number a - 10 : 4. t, and eon- are matches iches as the ches in the last condi- wonld bare Toronto and J express he le, but "Fben many hours I number of both trams, lower than (including as there are I in coming xied as the e bj taking express to jronld have rage speed itance from 324. A iquare and a rectangle are (i) equal in area, (li) equal in perimeter. The number of square inches in the area of the square is m times the number of linear unitd in its perimeter, and the number of squarei units in the area of the rectangle is n times the number of linear units in its perimeter. Find the length of the sides of the rectangle. 226. Two boys find upon trial that the distances to which they can respectively throw a stone are in proportion to their ages, and that the throw of the elder is 24 feet longer than that of the younger. After the lapse of a year they try again with the same stone and find that the elder can throw it but 22 feet farther than the younger, and that the gain of each is in the same ratio to the age of the other. Also the H. mean between their ages at the latter trial is equal to the quotient obtained by diriding the length of the longest throw made by the difference between the ji. mean of the 1st throws and that of the 2nd throws ; and if the antecedent of the ratio compounded of the ratio of the throw of one to his age in the first instance and the ratio of his gain to the age of the other on the second trial, be multi- plied by i of the product of their ages on the secoud trial the ratio of which the resulting ratio is the duplicate, will be the the same as the ratio compounded of the ratio of the throw of one to bis age at the first trial, and the reciprocal of the ratio of his gain to the age of the other at the second trial. Find their ages and the distance to which they throw the sione. = 6a:»+226 s. long for the other all receive per yard. cb receive ANSWERS TO EXERCtSHS. EZXROISB lY. 1. 2. 18 3. 14 4. 2 5. 3 6. 7. 48 8. 16 9. 48 10. 11. 24 12. 2700 14. 5 < 6 15. each = 16. 6>5 17. each = 10 18. each - 2 19. 2 20. 44 21. 19 22. - 112 23. -3 24. 22 25. 8 EXBRCISB y. 1. 43a. 2. -26a6''. 3. 19(a + 6 - c2). 4. 27o(x - y»)*. 5. 27a-13y + 23. 6. 16(a: + y) + 28a-20o5c. 7. 6(o + b)x - 19(c + d)y - 23(rf +f)z. 8. 15a%'x^ + 12a'»62x' - 13a'6 V - Ila'bh\ EXERCIBB YI. 1. 3a + 3c ; a + 3c ; 4a + 46 - 7c. 2. 8a& - 7fly + 13c<£. 3. - a^x^- 7(0 + 6)- 12a;^i/- 20. 4. 2a - 26. 6. 5xy + 14a6 + 17. 6. 5 + 8o-56 + 8c. 7. 6a6+6xy-5cd-m+16c. 8. 17 - 25ffi''x+20xy. 9. mV. 10. 18Va-8V3+14V4 + 6Va + 19Vc. 11. 20xy > lOay + 2^/x + 25^y. 12. 4(ax + 6y - czy + U^nTTh + 16(a; - y). EXBROISB YII. 1. o + 6 + c + m + 3p + x + y. 2.- 3x« - 5c" 3. 7c + 4x* + 2 (x-y). 4. lOx^y - a»6 + 7 6. 6a+156 + 5a6-3m='rt-l-5xf y. 6. 6'i/x - b*ja + y + 18 7. 4a:' - 2y' + Zy^ + 2y. 8. 3-^x2 + xy-yz ^am- *la*y + x'y - m ANSWEBi TO BXmiOISBS. 271 10 !h = 10 Enaciu YIII. 1. ay« - ll«y» + 1 la«» + 4ry + 20m. 2. 14a - 14e - IZ^cTHt^ + 4«y» + m*. 3. 2c(f - 3(o + 6)yx» - y. 4. 4(arV + y» - «■)* + 14a*a:* - 14Vm. 6. 16 + 7V8 - 23y - 9Va"^ 6. 67» - 2c - \U - 25x 4- 12y •^ abed. 1. U-m-Bc-e. 4. 6 + m. 7. 2 10 a+1 EXBROISB IX. 2. 2a-2b-2c. 3. x-5a-2 5. llo-3c-5/i + m. 6. 2o'-c'-m*. 8. 5a2+7a:+3m»+2x» 9. 8a»6c-2«. 11. a -86 -6c. 12. -a -5am -2c -17 x6c. I3c({. 'a+19Vc. 8 x'y- 1. 2. 3. 4. 5. 6. 7. 8. 9. JO. 11. 12. EXBRCISB X. (a- 6) + (c - rf) - (« - m) - (/+ r) - (s - 1>) + (w + x) (o - 6 + c) - ( - x) (a-6 + c-(i-e + m)-(/'+r + «-»-ti>-x) {a-^6-c)}-{rf + («-m)}-{/+(r+»)} + {y+(«)+x)| { (o - 6) + c } - {(d + «) - m } - [(/+ r) + « j + {(t> + w) r :s } }a- (6 - c+d) I - { e - (m-/-r) } - {« - (t) + ii>+ r)} {(a-6+c)-d}-{(e-m + /^) + r}-j(«-i>-«;)-x} ja-(6-c)-d}-fe-(m-/) + r|-{»-(t»+to;- xj {o-6 + c-(rf + c-m)|- {/+ r 4- « - (r + uj »■ r\| {(a - b + c ^ d) - e + m\ - {(/ + r + s - v) - w - X I \a~(b-c)'-d- («-m)}-{/+ (r +•«) -r - (t»"»-x)[ EXBROIBB XI. 1. 3a -3&; 4ax + 46'x - 4x» ; 3;>"x - 36j)'x - 3c^"x. a. om - 6% + m'p + x» - 3flx» - bx^ - Sm'x* + 6«»x> + m*x». 3. 7 + oo; +• 3ay - 46x + 4xy -ac^- 3c^ - mhf. 4. o'm - o'tt - 2acp + 20cq - c^m + c'n. .c * y m 6. «-6--= - -= + -:---=--r AmmOMi to vaaassiMi e. « + — - xyz xyz xyjs xyz 7. amx»axy»a*c-ahc+ay-r +• rr — - r^-^—t ' 2a-c 20-0 2o-c 2c 3m 4p 8. 36cd - Bahd + 36/m -». 36/n •* -=-o "" tt + -^-s* ox ox* ox* 3. i. 6fl/n + (1 + 9a)x + (3 + 15a - 2m)y. 2. (4 + m)a + (2m + 3a)x + (3x - 4 + m + Ba)y. 3. 5(2o - X -^ 6c) + 2(6 - 2c) - 3m. 4. (2a + m)x - (3am - 2c + a)xy + (3a - 2cm - 6 - /')l/''t 5. { 3(a # 6 + c) - (6m -^ c)a } y - {c + 2(1 + 3a)m | x - c(2 -o)z. 6. j 11(0 + 6)m + 3(cy + u^, y - { 3(a -fr + c)x 2(a+3)c|xy + 3(m + a)c - 2acp. ExBROi!«K XIII. 1. a*-4a»y+7aV-6a2/'+2j/*;a^-a*6-2a'6«+2a«6''+2a6*-6«4 2. 2a'»m» + lOa^m^xy - 3amx«y2 - 9xV; 9a*x* - Sa'x*- 3a«x - Sa'x" + 3ax* + 3ax2. 3. o*^ + m" I 2a* - 2aPry - 2o''x + 4,a^y^ + 2a'x=y ^ 2axy* - 2oa%' + 2y*. i 4. x* - 1x^ + 5x + 28 0^ - a*. 5. aS - 4a'62 ^. 4^253 _ 17^6* - 126«. 6. a262 - a^c^ + 2a6c2 - bh^. 1. 06 - 6a*62 - 10a'6« - 6a26* + 6". 8. 3x3 + 4a6x2 - 6a26«x ^ 4o'6' ; x* + x'' - 4x2 + 6x - 3. 9. x8 + 2x6 4. 33.4 + 2x« + 1. 10. 6y8 - 5x«y5 - 6xV + 21x'y' + x*y^ + I5x*; o'"*" + o"6** 4.a''*6*4' 6"**". 11. 30a» - 5a*- 207a' - I78a2 + 79^ 4. 72. 12. o^x" +a(6 + c)xy + 6cy2; aa'ft+i- a"'+*6«-a'"6'*-»'+rf**icP + 62«-P-6*-»cP. 13. o'**2 - a'cP + aV - o'*m' + c^'m' - mV + a"*x''- cPx''+ g***^ 14. a' - 2a*x + 3a'x* - 3a*x' + 2ax* - x''. 15. |6oc (2c -^ m) - 36c (2c - 12a + 36 - m) - 96(2a ''m)\m + |2am(c + 36) + 4ac(c- 36)+ 26c(c + 36) - 6m(c + 36^ Ja^» 2. 3. 4. Ai^fiWBRS TO Bxmtotdaii m -"Vt^ EXK&OIBB XIV. 1. hbc\ ea'y; 8tt| "xyz^ 2, - 26cm ; - or* j Dmsj/ ; 3** 3b*c llbx 3axy 6" 3. &xy Hot ^ 6a* 16a;s 3)c ]xy + Ja6*-6«, -2oa^' 4- o«6« m c CXBRCISB XVi 2lbc Sxy 2m 4x c X 11 2x ly 5a 5ax 2 ^- 5a B5xy 3a Bmxy 4. - abc 4mxy a'^c^ 4ay 5a« :. + _1 + 3»u;y 3m 2xy EXBROISB XVI. 1. a: * y ; a" + 2a6 + 62 2. m*+ 2mx + x^ . 3. 9a:* - lOa;^ + 5xt^ 30a; 4. o* + 4a6 + 6" ; x'^y^ + a;y -!- 1 5. X* + 2a;» + X* - 4x - 11 6. a^ - a*/ft - am* - m' 7. 1 - a + a" - a' + &c. I a + a" + o" + &c. ; 1 - 2m + 2m* - 2in' + &c. ; and 1 - 3a; + Ta;^ - lOr* + l7x* - Ac 8. 2a* - 6am + 4>ii'' 9. 2a» - 3a6» + 56» 10. o + 6 + c 11. 36x» - 27x'^ - 16xy« + 12y« 12. 2a« - 36 EXBROISB XV n. 1. o« - Bay + 9y« ; 9o« + 12ax+ 4x2 j g^r j >,« .- 42xy + 49 ; 4a'x* 12ox' + 9x« ; 4a« + 12a»xy« + 9a8xV 2W A178WBBS TO IfiXBRCISlSd. 2. a» - 9«« ; 4a« - 9y« ; 9a»6* - a:*y* ; Am* - 9«*y«* .^. 9a« - 4ar«y« ; 4a« - 49 ; 9 - x« ; 4 + 20ay + 25o«y« j 9ft« - 24ax«y' + 16a5*y«. 4. x« + 5x - 66; 9o« + 9a - 10 ; x* - ISjp + 86 ; x« - 4* - 21 ; «* - 3x + 2. 6. a* + a'x + o*x' + a'x^ + o'x* + ax' + x* ; a" - o*x + o'x* - a'x' + ax* - x« ; m* - m'a + m'a' - mo* + a* ; c* + x* is not dir. by c + X. (See Theorem xiii.) 6. a' "- a^xy + o'x'y''-a^xV+ a^x*y* - a'x''y'+o*x*y8 - tfix'^y'" V o'x^y' - ox^y^ + x'"y^<» ; a'm" + a^m''r + a^wi^r" +' a^m'r' + Q^wi^r* f a'wi'r** + a'm'r^ + amr'' + r" ; o' + m*«" ii not div. by a - MS (see Theorem xr) ; <^ + a^yz + ay^z* + j/"*". 7, X + 4 ; X + 8 ; 2x - 1 ; Sa'x - o*. EXBROISB XYIIL L a* - 2a6 + 6* - c« ; a* - 6* + 2ic - c« ; a« - 6« - 26c - c«. 2. 16 - 9a« + 12oc - 4c« ; 4a« - «« + 6m«x - 9m* ; 4x'^y« - 4a* + 12av-9y*. 3. 4a« - 12ac + 9c' - 4x« + 12xy - 9y^ ; o« + 6arf + 9d^ 4c* • 16COT - lewi*. 4. 9o« - 6a»i« + 7n* - 4 + 4xy - x«y* ; 4o* - 12a««* + 9«*' 1 - 2y* - y*. 6. 37a6 - lOo* - 26ft' - 36 6. 75a« - 12axy + 23x«y« 7. l-x^" 8, a* ' * •• x** "J tt'* ~ * ^-/ Rtbrciss XIX. 1. (a-7n.)(a* + am + ?/i^) 2. (a + c)(a* - o'c + «*c ' - oc* + c*) ^ 3. Not resolvable, a "'■ u li.tt*'!^ i^^^ A.rz(^(^ + b')(t^-b^}=(a-\-b)(a-'b) (a«-o6 + 6«) (a«+cft+6«) 5. (a - x) (a^ + ax + x*) (o° + a»x' + x^) 6. (a-6)(*i''+a9ft + «862+a75»^jj654 + afi6«+a*65 + o»6T + o'&« +069 + 6" 7. (fl? + j»V)(a + mx)(a ~ 7/ix) AltSMhBRS to B&EROI^liS. 2Y6 X* - a*x* ', dir. by a^iu'r' + t diT. by l6c - e*. 9^^ 1 8. (2a+x)(16a*-8a"x + 4o'x»-2ax' + x*) 9. (9 + 4c»)(3 + 2c)(3 - 2c) 10. (3TO-2c)(81m* + 54m»c + 367»V + 24»ic"+16c*) 11. (a+x)(a«-a«x + a*x'-o»x'+o»x»OT^» +»»««) 15. (a - c)(a + cXa^ + c'Xa* + c*Xa* + c") (a" + ac + c') (a" - oc + c'Xa* - a^c^ + c*)(a« -aM + c«)(oi6 -a'c* + c*«) 16. (a« + 7»")(a6« - a^m" + m«*) IT. (a + c)(a - c)(o» + c'Xo' - ac 4- c^X*^^ + ac + c') (a* - oV + c*) (o6 -a?c' + c8) (o8 + aV + c^) (a"- a^c^ + c") (a^*- o'c" + c'«) (o' « + a9c9 + ci "Xa^e - a^ »c' » +c='«) 18. (mi« + ci6)(m'»-?ft'6ci6+c83)(;a96-m*''c*'» +c96) 19. (o» + ma)(ai3-aiOTO2 + a«m*-o6^*'+o*ffi«-a'»i'<» + m*') 20. (am - py(ja?ni? + amp + ja") (o^m' + a'»i'^' + j)*) (a^'ni^" + 1. 2. 3. 4. EXBROISB XX. a - 2x 14a' - 43x* - 4ax 3V3 + 6V6 + 2V5 - 8V* - V2 - 4ax" + a'x' - 3o»« a e-t-M + a'x'*' — a**x*'"P-x"'*' 5. o»-i-a"-»x + a»-»x2-aP-*x' + ^-«x*- a' n- 8 «.8 a*"' + x' a + x 6. 7. 8. (a«- 9. 10. 11. 12. 13. 14. (X - 17)(x + 3) 1 + 1+1 + 1 + 1 + Ac, to infinity, = oc (a - x)(a + x) {a? + ox + x') (a'' - ox + x') (a* + oV + x*) ■a'x' + xS) 3^m\d?x - 2/))2 - 89J x6 - 2x» + 1 and a" - 4o'6' - Sa'i' - I7a6* * l«y x" - ax + 6 (a«+7ft'°)(a^« + nii«)(a» +m«)(o* +m*)(o'+m«)(a+m)(a - m) o** - c** 16. i 16. 2o(a« + 36*) 17. 2fl(a - iH) m AXaWIM so lUMWIWi BxiAotii XXI. 1. 6ah*m 4. x-t-a t. «-» 2. 3a«»i« ft, tt»(a - x) 8. o»(x-l) 8. «y ; , 6. m»(o»-»») 8. X - 1 fiXlROISI XXII. 1. x + 2 6. a -26 9. a- 2 2. x-i 6. o-ft 10. 4(a-6)^ 3. -X 7. 6x'»-3x + 4 11. a»+a«-5d4-3 4. x + 4 8. aft - by 12» a' + 2ab^2b* BXIROISK XXIII. 1. 12o"6''xV 2. 12a2xV«* 3. (x» - x^^ - xy + y»)* 4. x8+x''y+x*y'-x'y*^xy''-y'' 6. 4x^-4x*-4x"+4x» 6. 36a'-36o66-36fl66 + 30&' t. x» - lOx" + 21x 8. o* - a' - ox" +^ 9. a*-10o»+85a=-60a + 24 10. 60(a'" + a96 - aH^- 2o^6»- 2a66*+2a*66+ 2a^b^ + o»6» E7.BR0IBB XXIV. I. 2. x-y 2o+i»-m* 3o' + >» c 3. - n 6. a + c 9. iO. 11. 12. 13. rJ*''ab + b* a-b 0-6 a^ + ab + lt» a»-6» r a^ + m* 0? axy^ ^* Pxin+ay^y*^ ^** 11 7. 3 - 5x X 16. 7_ LI x-4 x + 3 17. 18. 19. 20. 21. 22. 23. X + 2y 4- 3y» 2x^3xy-5y« a-6 o2 + a6 + 6» a-TO c + d m. + 2p X + o • o + TO 2x + 3 x + c 2x' •<- 3x ^ 6 7x-5 a + m x'-a' + 2am -m^ o*-o*x* + x* a ' «»- o*V+ o'x*- a*x**|-x ' •* AlffiWBItS TO EXmOISBS. ^ BxiBOiu XXV. 1. . 2a'z' - axy 4- 3 - 2a 2. 3. 4: 6. ax a»+l *' a-l Sax + 9a •-' yx • 3y •> 8a' 4- 80 «T3 3ax - 3rty • 2a - y* x-y 3o»x - oy* - 2xy'' + am + mx 6. 7. 8. 9. 2xy(g + to) z + 2m 2b(3a* + 6») a + 6 2m» a* + 111^ 200? a*-¥x* a-i-x EZBSOIBC XXVI. 1. 4to - 4 + 3 5to 4. 6m'+5iB|»+6p*+— - m-p 2. a + x + 2«» a-x y + x^-xy+y 5. a--r 6 »'(i+y) 3.x + y^-x^-xy+y^~-^^-^ 6. l + 5a- ^^^ 6(4a+ 1) 4-x I A 1. 2. acim V^dm EXBBOISB XXVII. 16*111 bcd» 3. -. bcdm ' &e(jm ' bcdn ' bcdm xy am by mxy ' mxy * mxy Bbxy Za^xy eabm 4. 6. 7. I2abxy* 12abxy* IZabxy (1 +my (I -my 1 - m^ ' i - m' 5. x(3^ -f) x + y x(x^ + i/)t x(x^ + fy 6x^ + 6xy 8x + 2y 2x^ - 5xy + Sy" lea^/ft 16a '- 40^^" 6mr3mx 6a*yii(2+»)' 6a»»»(2+x)' 6a'm(2 + x) 278 ANSWERS TO BXBRCISES. 3fl«'-3a 4x'-Ax 3g« + 3 3j:* + 2g' - 3 j - 2 *• 3(*» - 1) » 3(xM)' 3(a:«-l)*°* 3(a!=' - 1) 6o' > 6a'6 2a a-6 EXIBOIBB XXVIII. 4am+Zm-2bc 26m a:*y+3xy + 2a-26 6. 11. m.^~2mp-p* 8. 4. 6. a:y«+3y' 4o6 6»-o« 332a: + 63x« 63 a'+xy'+i/* m' '-;>* 14-12a I 2x 10. y 3x« 1. -— fia 2. 2 2a -26 6. - BXKBOISB XXIX. o^+a^m+am'+m" my Aax - 4x* 3. 4. 6. 3y 3z»-3 2a 4- 26 8. 9. 10. 3 a:»-llx + 28 x» am (a - 2)' 2a 12. 13. 14. 2ac- 26e a6+6c+ac+6' 14x-20a:» 1 - 6x~« + 4** 1» aJbc 11. X + a X + rf 12. - 13. x" + 4x - 21 x» - 19x + 88 o*+o»+l a* 14. 1 i-J. 2. a + x a-x 3. EXBBOISI XXX. 0-6 a + 6 4. Sa^ - 6<^ 3a«y - 6a« X -3 6. 1 -3a:- 2 2ac- 26e o6+6c+flc+6' 14x-20x» 1 - 6x« + ix* m abc X + a ' X + d x^+ 4x- 21 x^ - 19x + 88 o*+o"+l '« -3 «- 7 e. 1 ANSWEB8 TO EXERCI8B6. 279 7. I 8. 3a> - 3a 9. 1 10. ah tt» + 6» 1. 2. 8. 4. t8o-66 10a 41 96 7a- 2x 21. ax a+2x 63 - 36a; 30X-10 6. EXBROISK XXXI. 45-18x+18a 2O0T2OX-I3 4a 9. - 1 ^' I + 4a'' 7. -a 8. a 10. xV df+e df-c 1+ I 4m - n EXIBOIBI XXXII. 1. 4,V 2. 5 3. 105 6. 19 6. 7 7. 16| 29. 31. 8. 41 9. 3 10. l7,Vr 11. 9 12. 4 13. 5, 14. 12 16. 8 16. 9 17. 120 18. - 10 19. 4 20. 15 21. 8 22. 80 23. 4 24. 26. 4 c-b 26. 27. 28. a 3b{b + e) 11a a-b» 6a* 34. Aa^-v2a-ab-b* bdf bd-\-ad'¥ be 6"+19a6-4a» 30. 86 -3a- 6 20a6 + 6*c + 6ac - 16a6c 156 + a6c - 10c 40. 2a -I- 86-2 180 + 396 - 36c 37. 4AV 72a 32. 35. 38. 41. 33. 36. 10a - 4a6> 36+ 4a a 2(26 -ly 297a 650 - 99a 3a6 -ae-' a*b* tt* + 3a6'-'A*-c-o 6c(6 - o) a6-a»-6» a6 a+6 39. i IMAGE EVALUATION TEST TARGET (MT-3) // I.I 1.0 ^lii 1^ >^ Ui 12.2 WUu Photographic Sciences CorporaJion ..»* 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) •72-4503 •sj ,\ c\ \ ^ o^ •-^^ m AKSWBES TO isasmbjgB^. 13. 14 14. 23 15. B9\^ 16. $3 ; 12, 7 19. 1803 \ 1689 EUUMB XXXIII. 1. 30; 17 5. 12; 18; 24 9. S6 2. 21 ; 42 6. $560 10. 14 3. $52-50 7. 30 11. 26 4. 64 8. 163 12. 102 17. 26| miles 18. 134|^ hours 20. A :: $2542 ; B ~ $2422 ; » $2436 71. Masic $0-55-/>f ; drawing $0'32f 22. 70 vol. Scieace; 210 vol. Travels; 210 vol. Biograiphy; 315 vol. Histocy ; 630 vol. Qeneral Literatare. 23. Niagara river, 34^ miles ; Rideau canal, 130|- mfl^s. 24. 2|^ days. n+o-c n-o+c 26. — -g and — 36. (I) 1 h. 5^rm. ; (n) 12 h. B2^tm.] (m) 12 h. 16iVm. 27. $155 and $220 28. 19U days. 29. A, $3594-50; B, $1055-574 ; 0, $1795-03; D, $743-89^ 30. 0-tVr days. 31. 68 32. $8142-85f 33. 72 lbs. abn 34. ^$11100 35. b-a feet. 36. 11 times, viz.: 1 h. 5-^^-m.; 2 h. 10^ m.; 3 h. 16i^ tn.; 4o. 37. 90^V and 5^g 38. A's- $808-42 Y^r; B's » $538-94^ '> Cf's » $1212.63iV 39.'^0 miles ; 15 m. per h. down ; 10 m. and 12 m. per h. up. 40. 5 ; $9000 41. 18 42. A'8 « $657-14^; B's $731-42f ; G'g = $7ll-42f na ma 43. 2575 44. -r-rr and r-rz" 45. 15 and 45 46. 86 weeks. anq 47. m + n na l + j» + n' l + »n + n ;aikd m.a 48 amq nq + mq + np^ nq + mq + np and anp nq + mq + np 49. 189 1. « - 2 ; y = 3 4. « = 4; yslO EziROisi J(XXiy. 2. X > 5 ; y =^' 3. X ^ ^| V y s tf| 6. X s 7 i y s 3 9* « - 24 i y s 30 14 !3 ^3 ; 12, 7 93 } 1689 Bfiograiiiihy ; 6Ti*rni. i743-89^ T*f m.; 4o. jr per h. up. ma ll-fm + n 49. 189 |y = 30 AKswniB iro fitXftOlSlS. 281 7. 10. 12. 14. X ■ X s X B mt ; *' "= 3i'/>; 8. xsl2;ysO 9. x«8;ye6 2a + 86 "IF 2ac - 6» ; y = 6a <^ 2b 19 oc -26» on -bin "•*"4Jrr36il''' a" + 6 aem(2<; - m) 15. xs 16. It. 18. 19. 20. X s XS X s X X X s 3a6 » y ~ 3a6 amc(a + c + m) me ■¥ ma - ac ■\- (^'^ ^ " cfH -f am - ac + c* m^ 4- 6n 6n + fflg a5 + 6n»l'~a6-6m 8; y = 3 8 ; y = 9 o(c^ - a' - c^ c(a^ - c* - a*) 4m- 8n 4a -36 ft-a« 3* 9; y-1 ab -J y = ab a + b C*-tt* EziROiBi ZXXV. xsll;y=2;«s3 2. C£2;y«0;sat x=l;ys2;zs..3 4. x = 4;yBl;xa-a x=l|; y = -2;2 = 2; «=-l| 6. x«2; ySj « = 4 x«li;y = 4;a=ii 6m + 16n-36 llfr + Tm*8n 236 4-411 -13m 1. 3. 6. 1. 8. X- 9. x = 76 c" - 6'c + a«6 o6» + ac"^~' y * 76 2dc - a* 6« + c» ' * = » *= 76 6»-6e« + aV a^i ' » 6» + c» ' ' a6' + ac* 10. « » 2 ; X = 6 ; y = 6 ; « = 10 11. x = 6 + c-o; y = o + c-6; «sa + &-e 12. X- op-am+an-m am-n+ap-an am-op+an-ji 2a> - a - 1 I y = 2a> - a - 1 ;«- l ' EZIROISB XLI. 1. 2a+36; a-2x; J lax- 7c 9. o'»-4ac + 4c* 2. 3am4-5«y; 4aj:' -6V 10. l-y + 3y* + 2^ 3. 2x" + 3« - 1 11. 2o' + 3a« + x» 4. aJi-y"-! 12. «» + y" 6. a-f-fr-e 13. a>.6» + c»-d" 6. 3o* + 2a + 6 14. l-Jx + a:»-Jx« ». a^-h^c^d 15. |.» + ?-^- 8. «'-3«^ + 3«y'- »" y « , EzKROisa XLII • ^ 1. 2a; + 3 4. 0"- 2a +1 7. x»-a: + l 2. a^ + 2a - 4 5. 2aa; - nhx* 8. a + 6 + c + d + « 3. 1 - 2a 6. 2x». - 3oa: + 4a* EXKBOISl XLIII. 1. a^ \ tr \ (fi \ crb*c ; o''6*c* j (^h*e^ ; a'^b^c' 2. ^a ; VP i Vc"; V^' ; V«5c ; Va^ 5 VC^^^^ i V(«^*«»'7f Ml rt •( 8. 2a6"*»i'*; 2a"*; 3oni"*; m'a"*c"'j }a4»ft-*c~'; fa'e *; Sa"*c» ; a"''6"*c'*"iii~^ or (oft'cm'j"'* ; a'm"*" or («*-*) i •8m Art. UJI. AKSWBR8 TO BX1R0I8B8. 285 4. — : -— 3 3 a»*c6-» 'o"»iii-»c*' 3o-»x>y' a-»6-V* * ao^ai^arV ' ^m^i \aK »; or Baric »ma? 2(a6)i (mn»)*«« J. 2. »(^ /«Y S^«« ^,^i^, /t.Nl ^ A» © 2oV & » ftti* mn ST'W ■-«. _'_ I — V« 6, Va»«{ a 8. o"6««c» jgmlr(i-q}f(%t^pr y4r«r« -» -M« ♦ »• t?^ 4oU* •{• Qa6 - 4a^ 6^+6 )0. a< 11. a^ + a'«'+«^ Lax*! 12. 8»'-4x'y*+ 6«'y *«"*«^+ 2i/-»»^ - »"^ - 2 »*«^-rM y 13. 2« !r*-3x -s 14. a^-oV*+a«6"*-6-^ 16. X 16. a»-2a*- xi4.a-fttt'^ -i-fl-i-2ii-i 17. o* + 1 +3a"-3a + 2o* +3-6o'* + a-» + 4a'*-a'*~2a* ftr* -i - B» - aaiS" .""I J. •.""* 18. «»-2x» + 3-2x ' + x 19. x~*y - x'y * 20. x' - 2x»y» + 3y ExntoisB XLIV. I. 4*: 343* : 16 ms>y:Hi) i f 286 .^vmm ^ vsaamm' 2. (o")" ; 9» ( 4/ (4o»)";(9a*6»)'; (16xV> i (i-r \2lJ '\I29J '\8a7 »\^27a«6«; » \, 64a»y» ^ ' (a*)* ; (81)* ; f- ~V ; (16o*)^ ; (81o«i«)^ and (266»«^)* {-i^'J s. (o* j *' ( 3 j ' (4u46« j ' ( ah^ J 5 (484J -1 (81) -J /llT649\-i " \^ 4096 ) \xY) \^' /J^.^ (<»"r; (V27)»; (8a66»)' / 262144 \i /x-»y6\J- -^(^> ^*^ '' ' \^ 126 y ' \n9) ' V40353607 / 4. V48; V126; Vi24; Vli^, (y)*} ^^4)^ 2 a 1 — 4 — 3a — »• 7^3-6} 26 V6 ; yVl4; 26^20; ^^46 a. Vi08;{/8a; ^Fs; V^;(~)^ (§1^")^ («*''*^»P'S^^ T. 3y»; »Va^ 2V5; 21-yr2j iW2i) -5-(«"m«)' ai»(»'-rf'+a^j»'')* 4 cm 6(3+10 V{6a(a+x)}; — V«; flJ^K*) ' Ug " " ^^^^ '•" *^^*^ ■*■ *^''^^ S. 3^ is the greater; 2V11 is the greatest, and 3^2^ the least.; ' ^ ' ' 10. 60Va J 4V3 + 2V1S. c*. - U. syt^jy?; (3a6=»+2a='— j-)Vac 12. (|tt«6* + to"-** - c»)V^» IS; tSViaj 60V2i 70^15; 24^12160 f*p»^i 'to '2^ the ••« :^u •/*^ 16. 4V6 + eVTi - 16Vf6-12-/36; 3V30 +V6-2HV5 17. iV6} AVl4i JViO; 4W"l30 18. f^iiMT; i»tfV2000; J-t^M j-g^Va^" 19. 10; 2(^3 - 5 '^6038848 + '^96*467; — V(a'6*^c***<«) 20.-29; l;-42;2^ihr;-r4W 21. ,V (W6 - 2V3); ?A^^jVL- >-Vi^-^W2; -. 14V6 + 8V'2l + 7712 + ^77 ' —28 \ 22. 3V 3 + 3V ar . a - 2 ^ am+m 30^2 + 24 V^? + 30V3 + Seyfo 3-:r a-m 23. «»- V«*-«*-2«-l x + l 'v.. 24. 2V3 -^ V30 - 3V2 . 26V3 - 27Vfr+ SiyS - 186 ; 12 ' 92 136 - 3V3 + 26V6 - 1 4V2 73 EzBROisi XLV. 6. V6-2 9. Va~l-1 6. 2V7 + Vn 10. ^a+b+^/d'b ^ 1. 1+V5 2. V "^ - V5 3. JV126 + yV2 1'Y (V6 + V2) 11- y(V26 +V8) '4- 4. V22 - 1 8, 5 - 3V2 12. i(h* + Va'-ft") SxiSROlSK XL VI. h yia - Va a, V20 + vs 3. V24 + 1/€ 4. ys - vWl97 Bnio ta ZLYII. 1. 8V-8i aa(yfr-l-^c)Vi'l 9.-4V-1-10V2 a. (VB> V» + VTi)v^ 10. V3 -V"^ / 8. 8-i-v-a. 4. 60 5. -29-6V6_ 6. 4(V3-Vi.6) 1 - o»V-ii+ 1 ; V- 1 ; - 1 16. o« + 6« 8. 0*- aaV- A-o 11. IVa+iy-2;iVa-iV-a 12. 7-f3V-2 13. 1 + V"^ 14. a - V- 8 1. 4 a. 6 8. 49 «. 4 9 % ExiROMi ZLVIII. 16. 81 16. Vi^ 6. 81 4A 6«-4a o'-aafr fr-c 17. !« 6. (Va-1)« 12. 18. 8a -26 an* (a -by 19. 2a a (m« + 1) 20. 2m t±iV-3 U.^^1^ 8. i8 4 ±1 ExiRoisi XLIX. 2a /- 6. i6V-l "-±56^^ 7. ia 8. ie • 18V?ii 14. i ^ V2C-1 10. iV^^^ 16. i9V2 (c-l)6 - ^v-c-i^-) a AKSWIR8 TO BZIROIBItf. 28^ BziBOiii L. 1. 6 or - 9 8. 10 o r -8 16. 16 or - 14 2. 9or-l 9. liVl-a» 17. lor-12_ 8. 10or-.2 10. 7 or- 7} 18. or i 2Vl6 - 8 4. 3 or -15 11. 4 or- H 19. lor- | 6. 5 or - 5 I 12. 4 or 6. 3 or li 7. 47ori 13. 3 or } d b 14. -or-- c « /6To / 6 + c \» 6 + 3 '''^yf—a^[vrT2i)'2r: e 2a 21. m or -o o + 6 i V2(tt« + 6») 22. V^ 16. 4or lori (-3iV-7)orO. EZKROIBI LI. 1. 6or-6| 6. lJor-31 11. f V9o» + 6»-3a 2. 16 or - 14 7. 1 or - ^ 12. 3 or - 9y^ eh 8. 6 or -41 8. — or- — a a 13. i(&iV26-4m») 4.25orl 9.H*±V61) Uy^^^'?;^ -»^ 1 2 ^ fi fot'lt lO-rrvF^' V3^ 16. i(a±Va«-4) 16. 2or-2i', EXIROIBI LII. 1. as* + 9« + 14 « 2. »* - S*" - 6«* + 8x = 8. x' - 13x' + 36x « 4. x« - 6x» - 22x* + 174x» -103x» - 600x + 700-0 6. X* - 20x1 + 164x« - 590x* + 1189x' - 1190x 4- 456 « 6. x« - 14X' + 76x* - 206x* + 283^ - 140x = 0. 7. i (3±V^^W 10- 0? 2 IV^^ 8. 3or-l li. 0or5or-2 ^f•l0i6y^6~ 12. 0or2or-; 290 18. tf s 2. AV8WXB0 TO XXIBOISIi. 14. cx* + bx+a~0 P 16. /»»-89||»>.4ji T/»(Vi'»r4j);- j; (|»"-y) Vj»"-*i BZIBOIM LIII. 1. 64 or 4 8. i 2 or i V^ 0. 10 or - 2 t. 8 or V-19 9. 2 or - 3 11. 60 or 235 13. 1, or ± V-l 16. 3 or ± V^ IT. 1, 1 or - 3 19. H* ± V^^-aaft) 2. 81 or I 4. 9 or ^ 1681 6. i 4 or i iV-W 8. 4 or - 1 10. 4 or 7^ 12. 4 or 1 14. 6or-2iV 16. 3 or 2 18. ±Y v^ -46* 20. 4 or - 5 21. iV~i;-i;iV40iV^3y; i; icifV"^); i(-iiV^) 23. 3, 2| or 1 23. or 2 i V^ 34. 4, 5 or - 1 26. 26. i;, l,.or 2 38. 2, i,orJ(9±^^^'l) a'f6-2c 2t. (-Vft-W" 29. 4,9, or J (-»33:fV-67) 30. 1 i V± V"« 31. ± -^{(vfH?«l)(Vl^+l)|* 32. 6, - 1, or i (6 i 3 V - 3) 93. i V8o«+6» - 4oc - 6 ± V -8a*:f26"-4oci:26V8«'+ft»^4 '■ ■ ■ " ■ . ' J. 4<» 84. la Vi(liV5) 35. ?,J-W2or-f IJV^^ 36. ^"'^ ▲V8WBB8 TO SZlROiaiS. 2»X y)Vp'-4i 3T. H8^V l'y)ori(5fV -T) 38. i(9 ± Y2t±2V-36) or 1K9 ± y 15 12^263) J 89. iV'n 40. 18 ± I IV"^ + Y 61 i 10 V -~3] 41. V* (T i V"^) 42. ± o-VftF, - a, or J (3 ±V 5) 43. * | or ^I—^Ll? 44. i(l iV^9)orKU V-ll) 46. i Y 3a ± a ^a* + 2a + 9 where a s ^3 -> V6 liV-3) •ft*-«r4ao KxiKoin Liy. 1. Xs T; y-a 2, X 9 13 ; y » 8 3. X s 6 or 4 ; y s 4 or 5 4. X s 8 or 7; y--7or-8 8. X s ±5or±8;y«±8ori6 6. *« ±8orT3V- 1; y = i3ori8VM 7. X s 12-^^ or 10 ; y « - ^*jj or 4 8. ;b« 7or.71;i;y«4or-6H 9. X8 11 oris J y = 134Vor-3 10. X » 3or-l;y=:lorr'3 11. X 3 2;y = 2 12. X X 266 or 1 ; y = 1 or 266 18. X = 2 or - 46 ; y s 8 or 16 14.*'a: = 6 or - 9i ; y = 3 or - 6| 15. Xa 6 or i ; y « 3 or -> i 16. Xs 2, 4, or 3TV2i ; y = 4, 2, or 3 ± VH 17. X« 5 or liV ; V = 3 or - -i^ 18. X5 i7orTVV2; y = ±4ori- I Va 19. Xa 'iC; y«i6 90. Xa 4 or 8; y«8or4 292 ANSERWS TO BXEROISBS. 21. x=3orlop2±V-33; y = lor3or2TV-33 22. a; B 2 or 5 ; y s 6 or 2. 23. X =3or-2orl(l±V-3i);y-2or-3orj(-l±V-3l) 24. X = 3 or 4 ; y = 4 or 3. 25. X = 2 or 4 or i ( - 13 4: V^^l); y « 4 or 2 or ^(-. 13 1 Vif^ 26. X = ± 6 ; y = ± 4. . , 27. X a 3 or - If ; y = 6 or 4i| 98. x.-^(l±V3)or^(l±W3);y=^(l?V3) or'^(inV3) iim.'if 29. xa±3op:F8;y = ±5 30. x«±2op±3;y = ±3ori2 31. x=±6j T4^»f; ±78V3; or ? 60V 3; y = ±3ori39V3 32. X = 6 ; y = 7, 33. X :: 8 or 152 T 64 V6 ; y = 4 or 40 T 16 V* 34. X «i^3 or ± iq_+ V23) or i i(2 + V22); y - f 8 or f |(7-V23)or±KV22 -2) ' 35. X » f (19 f V105) or |( - 13 i V - 87) ; y = K 3 i V 105) OPi(3iV-87.) 36. X s 1 or i^4 ; y > or ^-^4 37. x«iV-loriJ{V3 + V3 + VV3-l|; y = iV-Xor 1 1 {V3 + 3 V9 + V3^9 - 1} 38. X = 4, - 2, or 1 ± T*fV33 ; y= 2, -4 or -If tS V33. 39. X -9,4, or -13±V-61 } y = 4, 9, or -13 TV- 61 2 ,,-,-,-- 2 ____ ■♦ 40. X = ± J V6 (V« + 2 i Va - 2); y - ± JV* (V« + a + V*" 2)i wljere V a. 2 a>2 ^ 41. 42.1 43. Ai^dWBRS TO tilCtiBOtSBS. 29d -l±V-3l) -13lV3f7) TV3) or i 39V3 y - ± 8 or (31VT05) 33^ •2),wl)ere 41. « = aor-c"(a + l),y = -oor±o V-(«' + l) 42. xafoViT; y=±<»V(-l±V2) 43. X = ± iVV { n(a^-9 ± 3V 9 - IBa' + 2a^\] ya-^i^ {6a»-3± V9-16tt» + 2a« ) m + a m - a 44. a; = -— - ; y = —7,—, where m = ± V( ± 2 V 2o*+ 26* - 3o'') 2 2 45. *»± J (Vo«-c»±V«'+3c»); y = ±| (Va' - c« T Va' + 3 c«) where c* = o» ± V3a* - 26* 46. x = iVl4or± VK-1±V-19); y = ±Vr6oriVKi ± V^9) 47. x=lorl±V-4;y = iV6oriV2±4V-l Vi 48. «» = 1 ± V - 9^1 1 i V - 1| 62 ± V2410 or 4 ± VlO ya « -1 ± V-9V| - 1 ± V-~if - *6 T V2410 or 2 ? V^ EXIROISK LV. 1. 12 and 7 2. 10 and 7 3. 52 and 40 rods 4. l7and8or-8and~l7 6, 12 and 4 6. $90 7. 16 8. 862 9. 75 ; $3*20 10. 6 and 4 11. 10 and 14, or 84 and - 60 12. i(l i V5) and i(3 ± V5) 13. 4 yds. and 6 yds 14. I and } 15. 8 16. 3h. 23m 17. 144 miles and 180 miles 18. 16 19. 36 2fO. Coffee 12^., Sugar 26o 21. B. 30 days, C. 36 days 22. 10 x 10 x 5 23. 75 m.; ^, 15 m. per hour ; B, 10 m. per hour 24. i V6 and i (1 i V^) 25. Bacchus 6h. and Silenns 3h Bxmoisi LYI. 1. l:<£ 2. Ita 3. x + 7:«-i-l he •'Od 5. The Utter 6. y 7. oc 4. The former 8. 6 : a •!• 6 m ANClWfiBS to SXiBOtSlis. EXIRCISB LVII. be '-ad . 4. i 6 and i 4 6. 6 2p 8. 3 and - 2p + d 9. 8:7 10. $300 and $350 13. 3i ■ 14. 20n»5 : w»/» 16. (^(a - c) EXKROISK LYIII, 2. a = ly 3. a 4. X = iyVy 36 '•'= 16 + y - 6. y = 3 + 2x-x^ 7. y = 6a?=» 9945 302 ' 302x 8. y = 6 + -^ 10. 145 i EXIROISK LIX. 1. 2883 ; n (n + 62) 2. - 1628 ; n(6n > 206) 3. 238 ; i(2m + p) + i(2m + p)» 4. - 29§ 6. 60 ; 83 ; 3» - 1 6. - 77 ; - 132 ; 8 - 5n 7. 13-^f ; 211i ; -/V (6 + 2n) 8. 3 + lOJ + 18 + 25} + 33 9. 9-6-21-36-51-66 10. -l + llf + 24i + 36j + 49i + 62H'?4J +87^+100 11. 2701 12. 2n-l 14. afi 15. 39a ; a(2t - 1) 16. ± 14| ± 10, ± 6, ± 2 . 17. i 14, ± 10, i 6, i 2 18. 1, 3, 6, 7, 9, or 9, 7, 5, 3, 1 19. $l-00iVo 22. 11 .23. 2,5,8,11,14,17,20,23,26,29,32,35 24. 11, 10, 9, 8, 7, 6, 5 25. 6-c + 2rt 28. ^(27-n> 29. ± I, ± 3, i 5 30. 2, 4, 6 f^nd 8, or 8, 6, 4 and 1 1. 729; 1092 4. -6144; -4095 7. - f 8. U 12. 5 13. fH EXBROIBI LX. 2. 266; 611 3. 18^; 36f 6. -i2^V;-»7h e. -H; mi 9. 4} 10. 42f 11. ^l u.un ' 16, ics^-i) 16.1 19. 20. 21. 23. 1 24. 25. 30. 1. 2. ANSWERS TO fiXERClSti^. ^95 00 and $360 - + 9945 302x » - 206) ; 8-5n i-25i+33 P) 6, ±2 22. 11 6,5 m ) 16. -Sf^ {1 - ( - ?r> 17. 62(1+ V2) 18. ^, ^ j^ 19. 1+a + ^+^+if 20. 2 + 6 + 18 + 54+162 + 486+1468 + 4374+13122 21. 9 + 3+1 + i + i 22. 4, 24, 144 and 864 23. 5, 10, 20 and 40 or - 15, 30, - 60 and 120 24. $180, $90 and $45, or $375, - $300 and $240 25. 2, 4, 8, 12 and 16 29. 5, 10, and 20, or 46f, - 23^ and 113 30. 248 • EXBROISB LXI. (") "hi A> -hi iVi tVi tVj i> i» - J (III) - i, - i, 00 i, i, ^, ^, ^ff, T»y (IV) - iJ, - i^ - 2, 14, If, i?, i*, i^, and it (V) TTf , i, fft A, H, - li, - J, - ^n - i( (VI) - i, - i, - i, - ii « , i, i, i and i a. (I) 2 + 2f I- + 21 + 2i + 3 (II) 6 + 5-i^ + 5H 6i^ + 7 (in) 11 +6? + 4^ + 31 + 3 (IV) 2i + 2-i\,% + 2^Vr + 2m + 3f (v)6-2-^-A-§ 8. A> ^"ifiand 13 3n-l ab 4. If; liiyand--^ 6. 2 and 1^ 6. -^ and ^^ , _ab_ 7a-66'6(2-n)+o(n-l) 9. Ci; 6; 6^V 10. S^V; 5; 4gf 14. 18 and 2 15. 14 or f 17. 201 and 4 8. 13. Halfof the middle term 16. 20 and 10 EXEBOISB LXII. 1. 720 2. (i) 1680"; (ii) 2^160; (ni) 40320 8. 360360 4. 136 jrs. 222 days 6. n = 6 ' 6. Loss = $2S4ft5(5bO when the money is not paid till the end of the period. -^ immm. m 4. n^9 A, B are ours 12. 611 + 129 9009 Insout LX7. , 3,4.6....(94-r) *• IT" *' and 21*' ^- ^> V il^' J ^^ /7.9.ll....(5 f 2r)\ 8. (- ly (^ jr.-Tr— > ' •"d-^^ooo-o- ^ 9. (r+1) 2^3!' and 160«5 ift / ,,. / 6.7... .(3 + 2r)^ .385 \ \Z^ / 210 2.T....(6r-8) ^^4 ,. , 119 tJ _a 1±! -L 12. (r + 1) a » « "" » ; and 5a^ ar". 13. 1024 14. 128 15. 16. 4096 17. T)i9 4th t^rm - 32 18. The 4th a the 5th s 4}. 1970268S 19. 13th term 20. 9th » lOkh « "sgoeiT 4. s > - 10 6. « > a and < 6 IF 9.|P»f M iirownto ito izteottid* 1. n 8a Bniom LZVIL 8.1 4. li 5.-ai 6. • •fb* 7. a T 8. QC 9. 8tt a. 6 8 • (X s i.«»a,y.i a. jy, ix « 36, 19, la or 6 * jy=1,3, SorY (a;s4, 31, 38, 56,ftc '* (y»3, 11, 30, 39, ftc (x 3 3, 43, 84, 135, &o ^' (y»l, 13, 36, a (xal3, 65, 98, •• |ya6, 38, 60, ^ (x s 5, 166, 326, &c "• ly = i, SnRoiBB LXVIII. 10, 33, 36, 49, dec 3, 8, 13, 18, &e 4. X a 3 and y = 1 1, 13, 36, 37, Ao ftc 100, 199, ftc 13. X » 3, y s 3, « s 4 16. 46 6. X z 3 and y » 3 8. X s 6 and y s 4 10. X « 11 and y s 4 ^ (X = 2, 6, 10, 14, "• Jy = 3,20, 37, 64, fto 14. X s 11, y s 3, « as 3 16. 64 17. He pays 8 guineas and receives back 7 half-crowns 18. X s 3n and y » n^-> 1 where n maybe assumed at pleasure s any integral number ; and it will be found that x* 4* y* is a square 1^+1 , ^^* * ~ — oi — 'V vlio'o ^ <^&d y may be assumed at pleasure and it will be found that x* - y' is a square ao. 98. 31. 109. ^. Ifo two fractions with denominators 10 and 16 added to* gether will make ^. Prote this. 33. The problem is impossible Prove this. 34. 3, 6, 9, 13 or 15 £6 notes ; 81, 63, 43, 34 or 5 £1 notes ] 16, 33, 48, 64 or 80 orown-pieees. 36. 83 and 3; 16 and 9 ; 10 and 16 ; or 4 and 31 36. 3, 16 and 6 ; 7, 8 and 9 ; or 11, 1 and 13 38. a** X (3*^ ^-^ 1) where n may be assumed ^ to any integral numbtr. a9. ilt 80* Iat|60i9at980,a&d90at|a. 5. -ai c ;. 54 iB lleosare » Ipleaanre Ided to* notes} itegMl AKSWBBS TO BXlllOttii. 17 -21a 36 6. 3.^3 MiBoiLtAnoat Buboibbs. a 3. a + 6 4. — 8. X s 1, y s 6, s 3 9 6 *• ^^^^"^ 11. x" + l + x-«; x^-axW V2 12. 7a:» - 3ary + 4y» 13. a^ ♦ *» ♦ ' ; a6c 14. 4x* + y' + i*-V; «*-l'6*f26'x'-oV; x'*** + x^y^+ar'y' + yP ♦ « 16. Vj (es-itviB) 12x»+l 4a;» + 2«+l 16. 12*' + 6ar n. 16x*-l Bible rootB (in) x = 12 f V269. 2ahc 2abc 20. X = bc-ab'^ ^" Ac-, 18. X 3 -ia ; (ii) X has no poB- 2abc 2 = ac + bc-ab^' bc-ac + aby^ W+HF^Te 21. 1. 23. (a^ + ab^/2+b^Xa' - oft V2 + &') i («' + o6 V3 + 6') (o» - a6V3 + 6') 24. — . 25. G. C. J»f. = X - 4y ; /. e,m. = x* + 4xV - 27x»y« -34xy» + 56y* 26. 5, s na or iSf, - or a according as r = + 1 or - 1, and n an even or odd number 27. 4'9s per day 29. (i) 2059fi^ i (n) 5H+4H +4 + &e (in) 9, 6, 4, 2J, &c 30. 110 X 50 31. (i) X* + 2x^ + 3x»y» + 2xy" + y* ; . • , 59x* - lOOx + 23 (n)7x»-14x»+7x» + 33x-32 ^^^^^^ (ni) «" - * + x"* ■ • +x'» • » + X •» -'f + &c., rth term = x •»-«•• + -«8 -t« 32. (I) 6 + 2V3 ; (ii) V2x + 1 + Va« -1 33. 15a x 34. 1184040 35. x" - 2 + 3x - » 36. 1 or J (- 1 i V- 3) a' 3^-<^>^ = ^rr65y = . P 6 - a' (n) X a 0, 10, 4 or - 2 ; y = 0, 10 - 2 or 4 38. 3 and 3^ 39. i + -fr + i3f + | + W + ifr + ♦ 40. An identity 41. or i (1 i 3V -^) 42. ab + 6c + ae 48. ■lt»!+^*y-iSy'+ (P-m) X + (n-)« + (« + «)y 800 AirSWBBB TO B3UBaOI6]l6« 44. X - 6 45. (i) t s a or 6 ; (n) x s: } 46. x s f 3 of + 4V3; y«±4or±J-3'iV3; «=i2orTf V3 47. a« (6 + 1)« 6561x1* - 256x-»y« 48. x'P - 2x»' + 3x-«P 49. 107?; 81x« + 54y ? ^^^ ;p* ^. j» 4. 1 50. Any series having r = 2 51.1 52. —5 — —-z 53. xs 3a-6 or 36 --a 54. x-15;ya20 55. X* + 4x + 3 ; x» - 4xy^ + Sy" 66. x= J(o» + 3&»)* fl i I^LZ_1'^ V V«^+36V 57. 7 58. 4a 6* 59. (4a*-'/ o») ; (12a" -a) (a + 5)» 30x - 23 x (x» + !)■ ^^' * = ~2(ir::F)5 y = * (« + *) 61. 13a; -10 5x«-2x*+2x^i 62. x» - 9x + 24 X \ y 63. - - -r- + - ; x»-x+i y 2 X ' »2 x'+5x»-29x-105 64. (i) X*" - 3X™ y" + 2y2" (11) x^* - o" x^ + 2afix" + 6" 65. 4x» + 8x» + 16x + 32 ; 6a* 6» - 3a6* 67. 3, 4, 5, 6, or 7 4507a -3166 68. 30 71. (I) o* - 5a« + 25o» - 138a + 790 - ^a ^. 5^ , 4 (II) X* + 2x" + 3 + 2x-' + X -*. 36x» + 18x + 29 72. x-a T3. — 16x^.81 74. ^\jx* + ^-y* ; 64x^- 16xy* + 36x^y- 729y"2 75. x = 3V3 ; y = 2V2 13y 13 * -^ ' 2a (a - 6) 81. o'"- 6*" 82. x» - (a + 6) X - c 83. ax» + 6x + cx'^ x*-a& 84. ^^^^ 85. x« +px -Kp" 86. ^t («• - 6x» - 26x + 120) 87. 6-^1 88. x^ - 1 90. By w4 in 2) fi in 3, and C in a + 1 ""• x« + 1 4houfS 91. x'+x-S ^9,il,p^^',(a^ + %- + D* - ; 60J I I g" + 1)« «*+2x^l 5, or T -3166 ia- 4 f " . a*-4ab) 6) r-i c + 120) nd Cin ^ *■ ^9 AirSWlKS TO UtB&0I818. 801 9i. (I) lla ; (u) f V7 ; (m) B or - 12 ; (iv) » = 1 or 1 VTSJ y = 5 or i 94. **-x»-7x"-llx + 42 = 95. m 96. (o* + a6V2 + 6) (o* - a» 6» V2 +6*) a(ed - e - 6c) 6 (cd - « - ad) 98. lA 6Am 99. x = jj^:^;^ ; V - — J^j^s: i Problem indeterminate. 61«-Y0 100. J (a + 6) 101. 6x« + 10*» + 5x - 23 - ^ ^ ^^ ^ 3 or 6x» + 10«» + 6x - 23 - 61x-» - 63x-» + 79x-» + ke 102. X - y ; if y = 1 the G. C. J»f. is x» + 4x - 6 104. (a' + amV2 + m») (a»-om V2 + «") (o* + a* m? *JZ + m*) (m* - u» m» V3 + m*) 105. 1 106. 3 114. 116. 7x»+7xy + 7y» 116. 2x» + x-l 117. (2x-l) (x + l)(3x+2)(3x-2)and(2x-l)(x + l) (2x + I) (2x - 1) 1 ■(■ X + x' 119. An indeterminate equation ; an identity .118. 1 - X - X* + x" 128. 11, 9, 7, 6, &C 126. 3-2 + |-f + i?-&c 2618 -'* 391391 '" '^°* 6561 ' 1694323 129. x6 - 6x» + 6x* + 30x» - 51x» - 24x + 44 = 4frc '-ad -'» ^ 2.6.8...(3r-l) * i (-1) >* — pin? — ^ d-4c n+ 1 130. i (- 3 i V5) 131. 133* x = 2, y = 3, « = 4 134. 135. 21 and 24 136. 1 i VT9 137. x = 10, y « 8 140. I {i V4no6 + (a-6)» - (o + 6)} 141. i or ^ 1 42. | V3 a» 6» «■ 143. x = i . „ ... ', y = i-7=====; z-± 145. 6»-l. 146. "27+27. Va« + 6»+c' 147. o* + 6* + c* + <*»• 148. i|a(x + 2)^1}. 149. a + 6- c. 3x« - 4x - 1 160. 4x «. __ -— . 4x-8x->4-4r'4-7x'-.ll(r«**« '+2X-1 ' 802 AN0WBR8 TO BXlfillOISBd. 161.' 1+ «-«•-** + *• +d:T .,» .jpio + 4Q a* + 2fl"6" + 6* l.'S2. 1S3. 1 164. 1 a*-2oV + 6* '"■ " ""' (x-l)(x-.2)(x-3) 166. I. They must have a cuinmon measure; ii. The coeffi- cients of X must be= but of opposite 8igDS,and the coefficients of x* must be =, and also those of x" must be = 157. 2^ 168. 1 i 2V3 169. i i V^i or ^ 161. -*. JIf. = 1,V; G. J»f. ^ 1; /f. JIf. s Jj 163. 0; 217 26 3" • 1 166. f{i-(-D"}; f (-!)". *^*- "3 " 5^^- 167. x*+x»y* + »* 168. 43 b(a + b) 172. o or- 176. x+1 173. - 169. 2a - 3b. a 171. 6. 175. (rt-&)» + c* 2fl + 6 a +06 + 1 14x - 4x» + 14 ^ 2 (a» - b^) (X + 6) (X* - 1) ^ "*• «» + 6» 177. 181. i Vft(2 a 311. X s 1 + o« C'-ab 1-f a< ti » = 4 or 3, y a 3 or 4 6c ae ab . « ^ 213. xa±—;y = i-^, »'±- 213. J^ 16 } J, 31 J C, 34 314. 117; i {n(n+ l)H-4 -(S)"}; 8V2 + 3V3 216. 6i3Vt 317. 0, 1, V)f (4 i 1 V764). 318. 2n(4m-l) +J(1-16'»); (2n + 1)(4» + 1) + J{l-(-:2)*'***}; (4n + 3)(2n+l) + i(l-4"»**); 2 (n +,l)(4n + 3) + i{l-(-2)*'»*»| 219. 72 321. 90 miles; $2-70 222. xa0,or ylyiOVS- 70 ) or-2( V 10V29-46)or±3V^ 4m 4»i 224. I — (ml 'Jm^-n*) and— (m T V'»' - *»") n n n 4(m i V»»' - mn) and 4 (m :f Vm* - mn) 226. Ages at first trial = 11 and 16 Throws at first trial = 66 and 90 feet, And at second trial = 74 and 96 feet. ■by' THB IITD. ix i8n