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S3xaminatioii Primer in 
 
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 Presented to 
 
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 The University of Toronto 
 1^ 
 
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 ELEI^ 
 
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 OF GO 
 
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 8 th 
 
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 ^. K. (Bm & (to'» ^mmatUal »txits. 
 
 ELEMENTARY ALGEBRA, 
 
 —BY— 
 
 J. HAMBLIN SMITH, M.A., 
 
 OF GONVrLIiB AND OAIUS OOIiLEGE, AND LATE LECTUBEB 
 ▲T ST. FBTEB'S OOIiLEaB. CAMBItlDOE. 
 
 WITH APPENDIX BY 
 
 ALFKED BAKER, B.A., 
 
 MATH. TtJTOB UNIV. OOL. TOBONTO. 
 
 8 th CANADIAN COPYRIGHT EDITION. 
 NEW REVISED EDITION. 
 
 Authorized by the Education Department, Ontario. 
 Authorized by tJie Council of Public Instruction, Quebec, 
 Recommended by the Senate of the Univ. of Hali/a v. 
 
 
 
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 Entered according to ike Act of the Parliament of the Dominion of Canada, 
 in the year one thousand eight hundred and seventy-seven^ by Adab 
 Miller & Co. , in the Office of the Minister of Agriculture, 
 
 - f\'~h ^^S\ 
 
PREFACE 
 
 The design of this Treatise is to explain all that is 
 commonly included in a First Part of Algebrac In the 
 arrangement of the Chapters I have followed the advice 
 of experienced Teachers. I have carefully abstained from 
 making extracts from books in common use. The only 
 work to which I am indebted for any material assistance 
 is the Algebra of the late Dean Peacock, which 1 took as 
 the model for the commencement of my Treatise. The 
 Examples, progressive and easy, have been selected from 
 University and College Examination Papers and from 
 old English, French, and German works. !Much care has 
 been taken to secure accuracy in the Answers, but in a 
 collection of more than 2300 Examples it is to bo feared 
 that some errors have yet to be detected. I shall be 
 grr.teful for having my attention called to them. 
 
 I have published a book of Miscellaneous Exercises 
 adapted to this work and arranged in a progressive order 
 so as to supply constant practice' for the student. 
 
 I have to express my thanks for the encouragement 
 and advice received by me from many correspondents; 
 and a special acknowledgment is due from me to Mr. E. 
 J. Gross of Gonville and Caius College, to whom I am 
 \idebted for assistance in many parts of this work. 
 
 The Treatise on Algebra by Mr. E. J. Gross is a 
 oontinuation of this work, and is in some important 
 points supplementary to it. 
 
 J. HAMBLIN SMI i'H. 
 
 Cambridge, ISVi 
 
1^ 
 
 National Library 
 of Canada 
 
 Bibliotheque nationale 
 du Canada 
 
 CH. 
 
 I 
 I 
 
 \ 
 VI 
 
 VII 
 
 i: 
 
 X 
 
 XI 
 XII 
 
 xr 
 x^ 
 
 XV 
 
 XVI 
 
 XVII 
 
 xc 
 x: 
 
 XX 
 
 XXI 
 
 XXII 
 
 xxr 
 
 XX 
 
CONTENTS. 
 
 CHAP. PAGB 
 
 I. Addition and Subtraction i 
 
 II. Multiplication 17 
 
 III. Involution 29 
 
 IV. Division 33 
 
 V. On the Resolution of Expressions into Factors . 43 
 
 VI. On Simple Equations 57 
 
 VII. Problems leading to Simple Equations . . 61 
 
 VIII. On the Method of finding the Highest Common 
 
 Factor 67 
 
 IX. Fractions 76 
 
 X. The Lowest Common Multiple . . . . 88 
 
 XI. On Addition and Subtraction of Fractions . 94 
 
 XII. On Fractional Equations 105 
 
 XIII. Problems in Fractional Equations . . .114 
 
 XIV. On Miscellaneous Fractions 126 
 
 XV. Simultaneous Equations of the First Degree . 142 
 
 XVI. Problems resulting in Simultaneous Equations 154 . 
 
 XVII. On Square Root 163 
 
 XVIII. On Cube Root 169 
 
 XIX. Quadratic Equations 174 
 
 XX. On Simultaneous Equations involving Quadratics 186 
 
 XXI. On Problems resulting in Quadratic Equations . 19a 
 
 XXII. Indeterminate Equations 196 
 
 XXIII. The Theory of Indices 201 
 
 XXIV. On Surds 213 
 
 XXV. Ov Equations involving Surds .... 299 
 
viil 
 
 CONTENTS, 
 
 CHAP. PAGK 
 
 XXVI. On tttf- Roots op Equations .... 234 
 
 XXVII. On Ratio .243 
 
 XXVIII. On Pkoportion 248 
 
 XXIX. On Variation , . 258 
 
 XXX. On Akitiimetical Progression . . . .264 
 
 XXXI. On Geometrical Progression .... 273 
 XXXII. On ilARMONicAL Progression .... 282 
 
 XXXIII. Permutations , 287 
 
 XXXIV. Combinations 291 
 
 XXXV. The Binomial Theorem. Positive Integrai, 
 
 Index 296 
 
 XXXVI. The Binomial Theorem. Fractional and 
 
 Negative Indices 307 
 
 XXXVII. Scales of Notation 316 
 
 XXXVIII. On I^garithms ....... 328 
 
 Appendix 344 
 
 Answers • , 345 
 
ELEMENTARY ALGEBRA. 
 
 I. ADDITION AND SUBTRACTION. 
 
 1. Algebra is the science which teaclies the use of sym- 
 bols to denote numbers and the operations to which numbers 
 may be subjected. l» 
 
 2. The symbols employed in Algebra to denote numbera 
 are, in addition to those of Arithmetic, the letters of some 
 alphabet. * 
 
 Thus a, h, c x, y, z : a, /S, y : a',b\c' read 
 
 a dashf b dashj c dash : a^ ft^, Cj read a one, 
 
 b owe, c one are used as symbols to denote numbers. 
 
 3. The number one, or unity, is taken as the foundation of 
 all nimibers, and all other numbers are derived from it by the 
 process of addition. 
 
 Thus two is defined to be the number that results from 
 adding one to one ; 
 
 three is defined to be the number that results from 
 adding one to tivo ; 
 
 four is defined to be the number that results from 
 adding one to three ; 
 and so on. 
 
 4. The symbol +, read plus, is used to denote the opera- 
 tion of Addition. 
 
 Thus 1 + 1 symbolizes that which is denoted by 2, 
 
 2 + 1 3, 
 
 and a + b stands for the result obtained by adding 6 to a. 
 
 5. The symbol = stands for the words " is equal to," or 
 " the result is." 
 
 [S.A.] A 
 
ADDITION AND SUBTIL ACTION. 
 
 Thus the definitions given in Art. 3 may be presented in an 
 
 algebraical form thus : 
 
 1 + 1 = 2, 
 
 . 2 + 1 = 3, 
 
 3 + 1 = 4. 
 
 6. Since 
 
 2 = 1 + 1, where unity is written twicef 
 
 3 = 2 + 1 = 1 + 1 + 1, where unity is written three times, 
 
 4 = 3 + 1 = 1 + 1 + 1 + 1 , four times, 
 
 it follows that 
 
 rt = 1 + 1 + 1 +1 + 1 with unity written a times, 
 
 6 = 1 + 1 + 1 +1 + 1 with unity written 6 times. 
 
 7. The process of addition in Arithmetic can be presented 
 in a shorter form by the use of the sign + . Tlius if we have 
 to add 14, 17, and 23 together we can represent the process 
 thus : 
 
 14+17 + 23 = 54. 
 
 8. When several numbers are added together, it is indiffe- 
 rent in what order the numbers are taken. Thus if 14, 17, and 
 23 be added together, their sum will be the same in whatever 
 order they be set down in the common arithmetical process : 
 
 ^ M 14 17 17 23 23 
 
 ' 17 23 14 23 14 17 
 
 23 17 23 14 17 14 
 
 54 54 54 54 54 54 
 
 So also in Algebra, when any number of symbols are added 
 together, the result will be the same in whatever order the 
 symbols succeed each other. Thus if we have to add together 
 the numbers symbolized by a and h, tlie result is represented 
 by a + 6, and this result is the same number as that which is 
 represented by 6 + a. 
 
 Similarly the result obtained by adding together a, b, o 
 might be expressed algebraically by 
 
 a + 6 + c, or a + c + b, or h + a + c, or 6 + c + a, or c + a + 6, 
 
 or c + h + a. 
 
 9. When a number denoted by a is added to itself the 
 result is represented algebraically by a + a. This result is for 
 
ADDITION AND SUBTRACTION. 
 
 / 
 
 the sake of brevity represented by 2a, the figure prefixed to 
 tlie symbol expressing the number of times the number 
 denoted by a is repeated. 
 
 Similarly a + a + a is represented by 3a. • 
 
 Hence it follows that 
 
 2a + a will be represented by 3a, 
 3a + a by 4a. 
 
 10. The symbol — , read minus, is used to denote the ope- 
 ration of Subtraction. 
 
 Thu!» the operation of subtracting 15 from 26 and its con- 
 nection with the result may be briefly expressed thus ; 
 
 26-15-.11. 
 
 11. The result of subtracting the number b from the num- 
 ber a is represented by 
 
 a-h. 
 
 Again a-b-c stands for the number obtained by taking c 
 from a-b. 
 
 Also a — b-c-d stands for the number obtained by taking 
 d from a — b — c. . x 
 
 Since we caiinot take away a greater number from a smaller, 
 the expression a — b, where a and b represent numbers, can 
 denote a possible result only when a is not less than b. 
 
 So also the expression a — b~c can denote a possible result 
 only when the number obtained by taking b from a is not 
 less than c. 
 
 12. A combination of symbols is termed an algebraical 
 
 expression. . 
 
 The parts of an expression which are connected by the 
 symbols of operation + and - are called Terms. 
 
 Compound expressions are those which have more than one 
 term. 
 
 Thus a-b + c-die & compound expression made up of four 
 terms. 
 
 When a compound expression contains 
 
 two terms it is called a Binomial, 
 
 three Trinomial, 
 
 four or more Multinomial. 
 
ADDITWN- AND SUBTRACTION. 
 
 Terms which are preceded by the symbol + are called 'pon- 
 Uve terms. Terms which are preceded by the symbol - are 
 called negative terms. When no symbol precedes a term the 
 symbol 4- is understood. . 
 
 Thus in the expression a-b + c — d + e -f 
 
 a, c, e are called positive terms, 
 
 h,d,f negative 
 
 The symbols of operation + and - are usually called posi- 
 tive and negative Signs. 
 
 13. If the number 6 be added to the number 13, and if 6 
 be taken from the result, the final result will plainly be ] 3. 
 
 So also if a number h be added to a number a, and if 6 be 
 taken from the result, the final result will be a ; that is, 
 
 a-\-h-h = a. 
 
 Since the operations of addition and subtraction when per- 
 formed by the same number neutralize each otlier, we conclude 
 that we may obliterate the same symbol when it presents itself 
 as a positive term and also as a negative term in the same ex- 
 pression. 
 
 Thus a-a=0, 
 
 and a-a + h = h. 
 
 14. If we have to add the numbers 54, 17, and 23, we may 
 first add 17 and 23, and add their sum 40 to the number 54, 
 thus obtaining the final result 94. This process may be repre- 
 sented algebraically by enclosing 17 and 23 in a 6F..i(jKET 
 
 ( ), thus : 
 
 54-H(l7 + 23) = 54 + 40 = 94. 
 
 15. If we have to subtract from 54 the sum of 17 and 23, 
 the process may be represented algebraically thus : 
 
 64-(l7 + 23) = 54-40 = 14. 
 
 16. If we have to add to 54 the difference between 23 ami 
 17, the process may be represented algebraically thu« : 
 
 54 + (23-17) = 54 + 6 = 60. 
 
 17. If we have to subtract from 54 the difference between 
 23 and 17, the process may be represented algebraically thus ; 
 
 54-(2b-l7) = 54-6 = 48. 
 
ADDITION AND SUBTRACTION. 
 
 18. The use of limckets is so frequent in Algelira, that 
 the rules for their removal and introduction must be carefully 
 considered. 
 
 We shall first treat of the removal of brackets in cases 
 where symbols supply the places of numbers corresponding to 
 the arithmetical examples considered U: Jjts. 14, 15, 16, 17. 
 
 Case I. To add to a the sum of h and c. 
 ^ is is expressed thus : a + (6 + c). 
 i^'irst add b to a, the result will be 
 
 a + b. 
 
 This result is too small, for we have to add to a a numfc-jr 
 j/teater than 6, and greater by c. Hence our final result will 
 be obtained by adding c to a + b, and it will be 
 
 a + 6 + c. 
 
 Case II. To take from a the sum of b and c. 
 
 This is expressed thus : a-ib + c). 
 First take b from a, the result will be 
 
 a 
 
 -6. 
 
 'i'his result is too large, for we have to take from a a number 
 greater than b, and greater by c. Hence our final result will 
 be obtained by takiug c from a — b, and it will be 
 
 a~b — c. 
 
 Case III. To add to a the difference between b and c. 
 This is expressed thus : a + (6 - c). 
 First add b to a, the result will be 
 
 a + b. 
 
 This result is too large, for we have to add to a a number 
 less than b, and less by c. Hence our final result will be ob- 
 tained by taking c from a + b, and it will be 
 
 a + 6 - c. 
 
 Case IV. To take from a the difference between b and c. 
 This is expressed thus : re-(/)-c). 
 First take b from a, the result will be 
 
 a-b. 
 
 This result is too small, for we have to take from a a num- 
 ber less than b, and less by c. Hence our final result will be 
 obtained by adding c to a - 6, and it will be 
 
 a - 6 + c. 
 
ADDITION AND SUBTRACTION, 
 
 Note. We assume that a, h, c represent such numbers that 
 in Case II. a is not less than the sum of b and c,in Case III. 
 h is not less than c, and in Case IV. b is not less than c, and a 
 is not less than b. 
 
 19. Collecting the results obtained in Art. 18, we have 
 
 a + {b + c) = a + b + c, 
 
 a — {b + c) = a-b-c, 
 
 '- a + {b — c) = a + b — c, 
 
 a — {b — c) = a — b + c. 
 
 From which we obtain the following rules for the removal of 
 a bracket. 
 
 Rule I. When a bracket is preceded by the sign +, 
 remove the bracket and leave the signs of the terms in it 
 unchanged. 
 
 Rule II. When a bracket is preceded by the sign -, 
 remove the bracket and change the sign of each term in it. 
 
 These rules apply to cases in which any number of terms 
 are included in the bracket. 
 
 Thus 
 and 
 
 a + b + {c-d + e-f)=a + b + c-d + e-f, 
 a+h-'{c-d-\-e-f)=a + b-c + d-e+f. 
 
 20. The rules given in the preceding Article for the re- 
 moval of brackets furnish corresponding rules for the intro- 
 duction of brackets. 
 
 Thus if we enclose two or more terms of an expression in a 
 bracket, * 
 
 J. The sign of each term remains the same if + pro- 
 cedes the bracket : 
 
 II. The sign of each term is changed if - precedes the 
 bracket. 
 
 Ex. a-b + c-d + e-f=a-h + (c-d) + (e-f), 
 a-b+c-d + e-f=a-{b-c)-{d<-e+f). 
 
ADDITION AND SUBTRACTION. 
 
 ibers that 
 ^ASE III. 
 I c, and a, 
 
 have 
 
 moval of 
 
 sign +, 
 ras in it 
 
 sign -, 
 it. 
 
 3f terms 
 
 the re- 
 e intro- 
 
 ion in a 
 
 + pro- 
 des the 
 
 21. We may now proceed to give rules for the Addition 
 and Subtraction of algebraical expressions. 
 
 Suppose we have to add to the expression a + 6 - c the ex- 
 pression c? — e +/. 
 
 . The Sum =a + 5-c + ((^-e+/) 
 
 = a + 6 - c + rf - e+/ (by Art. 19, Rule I.). 
 
 Also, if we have to subtract from the expression a + b — c the 
 expression c? — 6 4-/. 
 
 The Difference = a + 6 - c - (c? - e +/) 
 
 = a + & - c - cZ + e -/ (by Art. 19, Rule II.). 
 
 We might arrange the expressions in each case under each 
 other as in Arithmetic : thus 
 
 Toa + 6-c * From a + 6 — c 
 
 AMd-e+f Take d-e+f 
 
 Sum a + b-c + d-e+f Difference a + 6 - c - rf + e -/ 
 
 and then the rules may be thus stated. 
 
 I. In Addition attach the lower line to the upper with the 
 signs of both lines unchanged. 
 
 II. In Subtraction attach the lower line to the upper with 
 the signs of the lower line changed, the signs of the upper line 
 being unchanged. 
 
 The following are examples. 
 
 (1) Toa + & + 9 
 
 Adda-6-6 
 
 Sum a + b + 9 + a-b-6 
 and this sum =a + a + b-b + 9-6 
 
 For it has been shown, Art. 9, that a + a=2a^ 
 and, Art. 13, that b-b = 0. 
 
 (2) 
 
 From a + b + 9 
 Take a-b-6 
 
 Remainder a + b + 9~a + b + 6 
 and this remainder =26 + 15. 
 
8 
 
 A DDlTlOJSr A AT? SUB TRA C TION. 
 
 22. We liave Avorked out the examples in Art. 21 at full 
 leiigth, but in practice they may be abbreviated, by combining 
 the symbols or digits by a mental process, thus 
 
 Toc + <i + 10 • Fromc + ci + 10 
 
 Addc-d-7 . Takec-(Z-7 
 
 Sum 2c 
 
 + 3 
 
 Remainder 2c2 + 17 
 
 23. We have said that 
 
 instead of a + a we write 2a, 
 
 a + a-J a 3a, 
 
 and so on. 
 
 The digit thus prefixed to a symbol is called the coefficient 
 of the term in which it appears. 
 
 24. Since 3a = a + a + a, 
 
 and 5rt = a + a + a + a + a, 
 3a + 5a = a + a-f(i-f-a + a + a + a + a 
 = 8a. 
 
 Terms which have the same symbol, whatever their coeffi- 
 cients may be, are called like terms : those which have diffe- 
 rent symbols are called unlike terms. 
 
 Like terms, when positive, may be combined into one by 
 adding their coefficients together and subjoining the common 
 symbol : thus 
 
 2x + 5x = 7x, 
 
 dy + bij + 8y=l6y. 
 
 25. If a term appears without a coefficient, unity is to be 
 taken as its coefficient. 
 
 Thus 
 
 x + bx=Qx. 
 
 26. Negative terms, when like, may be combined into one 
 term with a negative sign prefixed to it by adding the coeffi- 
 cients and subjoining to the result the common symbol. 
 
 Thus 2x-3i/-5?/ = 2x-8?/, 
 
 fpr 2x-3y-by = 2x- (3?/ + by) 
 = 2x-&y. 
 
 So again dx-y-4y-Qy = dx-l ly^ 
 
ADDITION AND SUBTRACTION. 
 
 27. If an expression contain two or more like terms, some 
 being positive and others negative, we muat first collect all the 
 positive terms into one positive term, then all the negative 
 teims into oae negative term, and finally combine the two 
 remaining terms into one by the following process. Subtract 
 the smaller coefficient from the greater, and set down the 
 remainder with the sign of the greater prefixed and the com- 
 mon symbol attached to it. 
 
 Ex. &x — 3x = 6x, 
 
 *7x-4x + 6x — 3x = l2x-7x = bx, 
 a-2h + 5h-4b = a + 5b-6h = a-h. 
 
 28. The rules for the combination of any number of like 
 terms into one single term enable us to extend the application 
 of the rules for Addition and Subtraction in Algebra, and we 
 proceed to give some Examples. 
 
 (1) 
 
 ADDITION. 
 
 a-2h + 3c (2) 5a + 76-3c-4i 
 
 3a-46-5c 6a-7h + 9c + 4d 
 
 4a-6h-2c 11a +6c 
 
 The terms containing 6 aif d d in Ex. (2) destroying one another. 
 
 (3) 7x-5y+ 4z (4) 6m-l2n + 5p 
 
 x + 2y-llz 8m+ n — 9p 
 
 2x- y+ 6z I m— n~ p 
 
 bx-3y— z m+ 2n + 5p 
 
 (1) 
 
 (3) 
 
 (6) 
 
 16x-7y- 3z I67n-nn 
 
 SUBTRACTION. 
 
 5a-36+ 6c (2) 3a + 76- 8c 
 2^ + 56- 4c 3a-764- 4c 
 
 3a -86 + 10c 146~l^/j 
 5a - 66 + 2c (4) x-y + z 
 2a-66 + 2c . x — y—z 
 
 da 
 
 Zx + 7y-¥\2z ' 
 by- 2z • 
 
 dx + 2y + l4z 
 
 2« 
 
 (6) 7x-l9y-l4z 
 iix-24y+ 9z 
 
 x+ 5y-2dz 
 
10 
 
 ADDITION AND SUBTRACTION. 
 
 ^ 
 
 29. We have placed the expressions in the examples given 
 in the preceding Article under each other, as in Aritlimetic, 
 for the sake of clearness, but the same operations might be ex- 
 hibited by means of signs and brackets, thus Examples (2) of 
 each rule might have been worked thus, in Addition, 
 
 5a + 7ft - 3c - 4d + (6a - 7& + 9c + irf) 
 = 5a + 7ft-3c-4tZ+ 6a-76 + 9c + 4i 
 = lla + 6c; 
 and, in Subtraction, 
 
 3a + 76-8c-(3a-76 + 4c) 
 = 3a + 76-8c-3a + 76-4c . 
 
 = 146-12c. 
 
 Examples.— L 
 
 Simplify the following expressions, by combining like eym- 
 bol« in each. 
 
 I. 3a + 46 + r'C + 2a + 36 + 7c. 2. 4a + 56 + 6c -3a -26 -4c. 
 
 3. 6a -36 -4c -4a + 56 + 6c. , 
 
 4. 8a-56 + 3c-7a-26 + 6c-3a + 96-7c+10a. 
 
 5. 5a;-3a + 6 + 7 + 26-3a;-4a-9. 
 
 6. a — 6-i; + 6 + c — d + c?-a. 
 
 7. 5a + 106-3c + 26-3a + 2c-2a + 4c. 
 
 Examples,— ii. 
 
 Add together 
 
 ADDITION. 
 
 2. 
 
 4. 
 
 a + a; and a — x. 
 
 a — 2x and 2a — x. 
 
 a + 36 + 5c and 3a-26-3c. 
 
 a - 26 + 3c and a + 26 - 3c. 
 
 2x-3y + 4z, 5x-7y-2z, and 6x + 9y-8z. 
 
 2a + 6 - 3ic, 3a - 26 + a;, a + 6 - 5a;, and 4a - 76 + 6a;. 
 
 a + 2a; and a + 3a;. 
 3a; + 7y and 5a;-— 2y. 
 
 7. l+a;-7/ and 3-a; + i/. 
 
 EXAMPLES,— iii. SUBTRACTIOK 
 I. Froma + 6 take a -6. 
 
 2 ^x + y 2.x -V. 
 
 3 2a + 3c + 4i a-2c + 3i. 
 
 4. x + y + z x-y~z. 
 
ADDITIOl^ AND SUBTRACTION. 
 
 IT 
 
 5. Fromm-n + r take m-n-r. 
 
 6 a + 6-fc a-b-c. 
 
 7 3a + 46 + 5c 2a + 76 + 6c. 
 
 8 3x + 6?/~4a 3x + 2y-5z. 
 
 30. We have givep examples of the use of a bracket. The 
 methods of denoting a bracket are various ; thus, besides the 
 marks ( ), the marks [ ], or j [, are often employed. Some- 
 times a mark called "The Vinculum" is drawn over the symbols 
 which are to be connected, thus a — h + ch used to represent the 
 same expression as that repi rented by a — {h + c). 
 Often the brackets are made to enclose one another, thus 
 
 a-[h+\c-{d-e^)\l 
 In removing the brackets from an expression of this kind it 
 is best to commence with the innermost, and to remove the 
 brackets one by one, the outermost last of all. 
 
 Thus 
 
 a-[6+|c-(fZ-e-/){] 
 = a-[6+jc-(rf-6+/)}] 
 = a-[6+ jc-rf + e-/}] 
 
 =a-[6 + c-d + e-/] 
 
 = a — h-c + d -e+f. 
 Again 
 
 6x-(2x-7)-\4-2x-(6x-Z)\ 
 
 = 5x-3x + 7-{4-2x-6x + 3[ 
 
 = 5cc-3a; + 7-4 + 2x + 6x-3 
 
 = 10x. 
 
 EXAMPLES.~iv. BRACKETS. 
 
 Simplify the following expressions, combining all like quan- 
 tities in each. 
 
 1. a + b + (da-^h). 
 
 2. a + b-{a-Zb). 
 
 3. 3a + 56 -^c- (2(1 + 46 -2c). 
 
 4. a + 6 - c - (a - 6 - c). 
 
 5. Ux-{bx-9)~\4:-3x-(2x-3)l, 
 
 6. 4x- \3x-(2x-x-a)\. 
 
 7. 15.T; -\7x + (3x + o^) ( . 
 
12 
 
 ADDITION AND SUBTI^ACTIOA. 
 
 8. a-[h+\a-{h-^a)\]. 
 
 ' 9. Crt + [4a-{86-(2rt + 4&)-226|-76]-[76l-j8a 
 
 -(36 + 4ft) + 86|+6a]. 
 
 10. h-[h-{a + h)-\h-{h--T-h)\]. 
 
 11. 2c-(6rt-?))- jc-(5a + 2ft) ' ;6)}. 
 
 12. 2a;-ja-(2a-[3a-(4a-[5fc-(6a-;r)])])}. 
 
 13. 25a-196-[36-}4a-(56-6c)}]. 
 
 31. We have hitherto supposed the symhols in every ex- 
 pression used for illustration to represent such numbers that 
 ^'.e expressions symbolize results which would be arithmetic- 
 ally possible. ■ * 
 
 Thus a-h symbolizes a possible result, so long as a is not 
 less than h. 
 
 If, for instance, a stands for 10 and h for 6, 
 
 a — h will stand for 4. 
 
 But if a stands for 6 and h for 10, 
 
 a-h denotes no possible result, because we cannot 
 take the number 10 from the number 6. 
 
 But though there can be no suQh a thing as a negative 
 number, we can conceive the real existence of a negative 
 quantity. 
 
 To explain this we must consider 
 
 I. What we mean l}y Quantity. 
 II. How Quantities are measured. 
 
 32. A Quantity is anything which may be regarded as 
 being made up of parts like the whole. 
 
 Thus a distance is a quantity, because we may regard it as 
 made up of parts each of themselves a distance. 
 
 Again a sum of money is a quantity, because we mav regard 
 it as made up of parts like the whole. 
 
 33. To measure any quantity we fix upon some known 
 quantity of the same kind for our standard, or unit, and then 
 any quantity of that kind is measured by saying how many 
 times it contains this unit, and this number of times is called 
 the measure of the quantity. 
 
ADDITION AND SUBTRACTION, 
 
 n 
 
 Rre cannot 
 
 For example, to measure any distance alonj^ a road we fix 
 upon a known distance, such as a mile, and exjtress all distances 
 l)y saying how many times they contain this unit. Thus 1(5 is 
 tlie measure of a distance containing IG miles. 
 
 Again, to measure a man's income we take one pound as our 
 imit, and thus if we said (as we often do say) that a man's in- 
 come is 500 a year, we should mean 500 times the unit, that is, 
 £500. Unless we knew what the unit was, to say that a man's 
 inc«mie was 500 would convey no definite meaning : all wo 
 should know would be that, whatever our unit was, a pound, a 
 dollar, or a franc, the man's income would be 500 times that 
 unit, that is, £500, 500 dollars, or 500 francs. 
 
 N.B. Since the unit contains itsell" oncQ^ its measure is 
 unU]}^ and hence its name. 
 
 34. Now we can conceive a quantity to be such that when, 
 put to another quantity of the same kiiid it will entirely or in. 
 part neutralize its effect. 
 
 Thus, if I walk 4 miles towards a certain object and then 
 return along the same road 2 miles, I may say that the latter 
 distance is such a quantity that it neutralizes part of my first 
 journey, so far as regards my position with resjject to the point 
 from which I started. 
 
 Again, if I gain £500 in trade and then lose £400, I may 
 say that the latter sum is such a quantity that it neutralizes 
 part of my first gain. 
 
 If I gain £500 and then lose £700, 1 may say that the latter 
 sum is such a quantity that it neutralizes all my first gain, and 
 not only that, but also a quantity of which the absolute value 
 is £200 remains in readiness to neutralize some future gain. 
 Uegarding this £200 by itself we call it a quantity which will 
 have a suhtractive effect on subsequent profits. 
 
 Now, since Algebra is intended to deol with such questions 
 in a general way, and to teach us liow to put quantities, alike 
 or opposite in their effect, together, a convention is adopted, 
 lounded on the additive or suhtractive effect of the quantities 
 m question, and stated thus : 
 
 " To the quantities to be added prefix the sign + , and to 
 the quantities to be subtracted prefix the sign -, and then 
 \vrite down all the quantities involved in such a question con- 
 nected with these signs." 
 
14 
 
 ADDITION AND SUBTRACTION 
 
 \ 
 
 I! 
 
 ir 
 
 ThiiH, suppose a man io trade for 4 years, ami to gain a 
 pounds the first year, to lose 6 pounds the second year, to gain 
 c pounds the third year, and to lose t/ pounds the fourth year. 
 
 The additive quantities are here a and c, which we are to 
 write 4- a and + c, 
 
 The subtradive quantities are here 6 and d, which we are to 
 write - b and - d, 
 
 :. Result of trading = +a~b + c-d. 
 
 35. Let us next take the case in which the gain for the 
 first year is a pounds, and the loss for each of three subsequent 
 years is a pounds. 
 
 Result of trading 
 
 = +a — a — a — a 
 = -2a. 
 
 Thus we arrive at an isolated quantity of a subtradive 
 nature. 
 
 Arithmetically we interpret this result as a loss of £2a. 
 
 Algebraically we call the result a negative quantity. 
 
 When once w have admitted the possibility of the inde- 
 pendent existence of such quantities as this we may extend the 
 application of the rules for Addition and Subtraction, for 
 
 I. A negative quantity may stand by itself, and we may 
 then add it to or take it from some other quantity or expres- 
 sion. ^ 
 
 II. A negative quantity may stand first in an expression 
 which we may have to add to or subtract from any other 
 expression. 
 
 The Rules for Addition and Subtraction given in Art. 21 
 will be applicable to these expressions, as in the following 
 Examples. » 
 
 
 ADDITION, 
 
 
 (1) 
 
 5a - 7a = - 2a. 
 
 
 (2) 
 
 4a-3b-ea + 1b= -2a + 4b. 
 
 
 (3) 
 
 To 4a To 
 
 5a -35 
 
 
 Add -3a Add 
 
 -2a-26 
 
 
 Sum a ' Sum 
 
 3a -56 
 
ind to gain a 
 I year, to gain 
 e fourth year. 
 
 ich we are to 
 lich we are to 
 
 gain for the 
 :e subsequent 
 
 a subtractive 
 
 n expression 
 II any other 
 
 ADDITIO.Y AND SUBTRACTrON. 
 
 (4) 
 
 5a + 76-12c-17 
 a-864l9c+ 4 
 
 -66+ 3c- 7 
 
 (5) 7a; - 5?/ + Os 
 
 - dx-8y+ z 
 - 1.4^ -4y + bz 
 
 SUBTRACTION. 
 
 (1) 
 
 From 
 Take - 
 
 X 
 
 y 
 
 Eemainder X + 1/ 
 or we might represent the operation 
 
 thus. 
 
 x-{-y) = x + ij* 
 
 (2) a + b-{-a + b) = a + b + a-b = 2a. ; 
 
 (3) -a 'b-(a-b)= ~a-b-a + b= -2a. 
 
 (4) 
 
 (8) 
 
 X 
 
 -3a+ 4b- 7c+10 
 5a- 9b+ 8c +19 
 
 -8a+136-15c- 9 
 -ij-[3x-\ -5x-(-4y + 7x)\] 
 
 (6) 
 
 = x-y-[3x— I —5x + 4y — 7x\] 
 = x — y- [2x -\-5x — 4y + 7a;] 
 =x — y — dx — 5x + 4y-'7x 
 = - 14a; + 2y. 
 
 7a+ 56+ 9c-l2d 
 -36 -12c- 8d+ ee 
 
 7a+ 86 + 21C- 4d-6e 
 
 ti 
 
 In this example we have deviated from our previous prac- 
 tice of placing like terms under each other. This arrange- 
 ment is useful to facilitate the calculation, but is not absolutely 
 necessary ; for the terms which are alike can be combined 
 independently of it. 
 
 * Note.— The meaning of Subtraction is \ere extended so that 
 the result in Art. 18, Case iv. may he true when 6 ia less than 0. 
 
I6 
 
 ADDITION AND SUBTRACTION. 
 
 2. 
 
 3- 
 
 4. 
 
 5- 
 6. 
 
 7. 
 8. 
 
 Examples. — v. 
 
 (I.) ADDITION. 
 Add together 
 1 . 6</, + 7?>, - 2a 4&, and Za -bh. 
 
 -5« + 66-7c, - 2a + 13b + 9c, and 7a -2% + 4c. 
 
 2x — 3i/ + Az, -bx + 4y — 7z, and - 8a; - 9y - 3z. 
 
 -a + h-c + d, a~2h-3c + d, -5h + 4c, and -5c + d. 
 
 a + h-c + 7, - 2a - 36 - 4c + 9, and 3a + 26 + 5c - 1ft 
 
 IJx — 3a — 46, 6y - 2a, 3a — 2y, and 56 - 7a;. 
 
 a + 6 — c, c — a + 6, 26 - c + 3a, and 4a — 3c. 
 
 7a - 36 - 5c + 9d, 26 - 3c - 5c?, and - 4(i + 15c. 
 
 - 12a: - 5?/ + 4;^, '3x + 2y- 3a, and 9x - 3^/ + a. 
 
 ' (2.) SUBTRACTION. 
 
 a \- 6 take —a — b. 
 
 a — h take — 6 + c. 
 
 a - 6 + c take - a + 6 — c. 
 
 6a; - 82/ + 3 take - 2a; + 9?/ - 2. 
 
 5a -126 + 17c take -2a + 46 -3c. 
 
 2a + 6 - 3a; take 46 -- 3a + bx. 
 
 a + h-c tike 3c- - 26 + 4a. 
 
 a + 6 + c - 7 take 8 c - 6 + a. 
 
 12a; ~'3y-z take 4y-5zi-x. 
 
 8a - 56 + 7c take 2c - 46 + 2a. 
 
 9jp - 4g + 3r take 52 - 3^ + r. 
 
 I. 
 
 From 
 
 2. 
 
 From 
 
 3- 
 
 From 
 
 4- 
 
 From 
 
 5. 
 
 From 
 
 6. 
 
 From 
 
 7- 
 
 From 
 
 8. 
 
 From 
 
 9- 
 
 From 
 
 10. 
 
 From 
 
 II. 
 
 From 
 
II. MULTIPLICATION. 
 
 36. The operation of finclinpr tlie sum of a numbers each 
 equal to h is called Multiplication. 
 
 The number a is called the Multiplier. 
 b Multiplicand. 
 
 This Sum is called the Product of the multiplication of b 
 by a. 
 
 This Product is represented in Algebra by three distinct 
 symbols : ,- 
 
 I. By writing the symbols side by side, with no sign 
 between them, thus, a6 ; 
 
 II. By placing a small dot between the symbols, thus, a.b; 
 
 III. By placing the sign x between the symbols, thus, 
 axb ; and all these are read thus, " a into 6," or " a times 6." 
 
 In Arithmetic we chiefly use the third way of expressing a 
 Product, for we cannot symbolize the product of 5 into 7 by 
 57, which means the sum of fifty and seven, nor can we well 
 represent it by 5.7, because it might be confounded with tlie 
 notation used for decimal fractions, as 5 '7. 
 
 37. In Arithmetic 
 
 2x7 stands for the same as 7 + 7. 
 3x4 4 + 4 + 4. 
 
 In Algebra 
 
 ab stands for the same as 6 + 6 + &+ ... with 6 written 
 a times. 
 
 (a + 6) c stands for the same as c + c + c. . , with c written 
 a + b times. 
 
iS 
 
 MUL TIILICA TION. 
 
 38. To shew that 3 times 4 = 4 times 3. 
 
 3 times 4= 4 + 4 + 4 
 
 = 1+1+1+1 
 
 + 1 + 1 + 1 + 1 [ I. 
 
 -1-14-1 + 1 + 1 
 
 4 times 3= 3 + 3 + 3 + 3 
 
 =1+1+1 
 
 + 1+1 + 1 . yx 
 
 + 1 + 1 + 1 ^ 
 
 +1+1+1 
 
 Now the results obtained from I. and II. must be the same, 
 for the horizontal columns of one are identical with the verti- 
 cal columns of the other, 
 
 39. To prove that ah = ba. 
 
 ab means that the sum of a numbers each equal to b is to 
 be taken. 
 
 :. ab= b + b+ with b written a times 
 
 = b 
 + 6 
 + 
 
 to a lines 
 
 = 1 + 1 + 1 + to & terms ^ 
 
 + 1 + 1 + 1 + to6terms( ^ 
 
 + ( 
 
 to a lines. ) 
 
 Again, 
 
 ha= a + a+ with a written 6 times 
 
 = a . 
 
 + a 
 
 to 6 lines 
 
 = 1 + 1 + 1 + to a terms ^ 
 
 + 1 + 1 + 1 + to a terms r jj 
 
 ^. ( 
 
 to 6 lines # 
 
 Noi 
 for the 
 vertic£ 
 
 40. 
 ing, wfl 
 the pr(^ 
 
MUL TIPLICA TIOiV. 
 
 19 
 
 Now the results obtained from I. and II. must be the same, 
 for the horizontal columns of one are clearly the same as the 
 vertical columns of the other. 
 
 40. Since the expressions ah and ha are the same in mean- 
 ing, we may regard either a or h as the multiplier in forming 
 the product of a and 6, and so we may read ah in two ways ; 
 
 (1) a into h, 
 
 (2) a multiplied by 6. 
 
 41. The expressions ahc, acb, hac, hca, cah, cha are all the 
 same in meaning, denoting that the three numbers symbolized 
 by a, h, and c are to be multiplied together. It is, however, 
 generally desirable that the alphabetical order of the letters 
 representing a product should be observed. 
 
 42. Each of the numbers a, 6, c is called a Factor of the 
 product ahc. 
 
 43. "When a number expressed in figures is one of the 
 factors of a product it always stands first in the product. 
 
 Thus the product of the factors x, y, z and 9 is represented 
 by ^xyz. 
 
 44. Any one or more of the factors that make up a product 
 is called the Coefficient of the other factors. 
 
 Thus in the expression 2a.T, 2a is called the coefficient of x. 
 
 45. When a factor a is repeated twice the product would 
 be represented, in accordance with Art. 36, by aa ; when three 
 times, by aaa. In such cases these products are, for the sake 
 of brevity, expressed by writing the symbol with a number 
 •placed ahove it on the right, expressing the number of times the 
 symbol is repeated ; thus 
 
 instead of aa we write a^ 
 
 aaa 
 
 a^ 
 
 aaaa a^ 
 
 These expressions a^, a^, a* are called the second, third, 
 
 fourth Po WEits of a. 
 
 The number placed over a symbol to express the power of 
 the symbol is called the Index or Exponent. 
 
 a^ is generally called the square of a. 
 
 a^ the cuhe oi a. % 
 
20 
 
 MUL T I PLICA TION. 
 
 46. The product of a^ and ci^=a'^ x a^ 
 
 — aa X aaa = aaaaa = aP, 
 
 Thus the index of the resulting power is the 5wm of the 
 indices of the two factors. 
 
 Similarly 
 
 a* -^ a^ — aaaa x aaaaaa 
 
 = aaaaaaaaaa = a^^ = a^"^. 
 
 If one of the factors be a symbol without an index, we may 
 assume it to have an indox^, that is 
 
 Examples in multiplying powers of the same symbol are 
 
 (1) axa2 = fti-^2^^3 
 
 (2) 7a3 X 5a7 = 7 X 5 X a^ X a7 = 35a3+7 = 35ai« 
 
 (3) a3xa6xrt9 = a3+6+3 = ai8. 
 
 (4) xhj X xy^ = x'^.y.x.y^ = x^.x.y.y^= x^+i, 1/I+2 _ 3.3^^ 
 
 (5) a26xa63xa66r=a2+i+5.z,i+3+7=^8.5u 
 
 Examples.— vi. 
 
 Multiply 
 
 I. X into 3i/. 
 4. 3ahc into ac. 
 7. 3a26 into 4a%'\ 
 10. 7a^c7 by 4a^bc^. 
 
 2. dx into 4i/. 
 
 5. a^ into a*. 
 
 8. 7a4c into 5a26c3. 
 
 3. 3aji/ into 4a;t/. 
 6. a^ into a. 
 9. 15ahh^\)yl2a%c. 
 12. 4a36a; by Safety. 
 
 II. a^ by 3a^ 
 
 13. 19x^1/3 by 4 (KT/V-^^ 14. 17a6%by 36c2y. 15. Qs^y^z^ hy 8x^y^z\ 
 16. 3tt6c by 4aa;t/. 17. a^&^g i^y sa^J^c. 18. 9m.2?ip by w^n^^^^ 
 ig. mfz hy hx'^z^. 20. lla^ftx by Sa^^ft^^'m^. 
 
 47. The rules for the addition and subtraction of powers 
 are similar to those laid down in Chap. I. for simple quantities. 
 
 Thus the sum of the second and third powers of x is repre- 
 sented by 
 
 x^ + x^, 
 
 and the remainder after taking the fourth power of y from the 
 fifth power of y is represented by 
 
 and these expressions cannot be abridged. 
 
MULT T PLICA TION, 
 
 21 
 
 ex, we may 
 
 But when we have to add or subtract the same powers of 
 the same quantities the terms may be combined into one : 
 thus 
 
 3?/3 + 5?/3 + 7^/^ = 15?/^, 
 8x4-5a;4 = 3x'», 
 92/'''-3?/-2?/ = 4i/5. 
 
 Again, whenever two or more terms are entirely the same 
 with respect to the symbols they contain, their sum may be 
 abridged. 
 
 Thus acZ + rt(^ = 2a(i, 
 
 5a363 + 6a363 - 9a363 ^ 2a363, 
 7a% — 1 Oa^a? - 1 '^(.(r'X = - 1 ba^x. 
 
 48. From the multiplication of simple expressions we pass 
 on to the case in which oiie of the quantities whose product is 
 to be found is a compoimd expression. 
 
 To shew that (a + h) c = ac + bc. 
 
 (a + h) c=c + c + c+ ... with c written a + h times, 
 
 = (c + c + c+ ... with c written a times) 
 + {c + c + c ... with c written 6 times), 
 
 = ac + hc. 
 
 49. 
 
 {a 
 
 Note. 
 50. 
 
 To shew that (a — h) c = ac — he. 
 
 -b)c = c + c + c+ ... with c written a — b times, 
 
 = {c + c + c+ ... with c written a times) 
 — (c + c + c... with c written b times), 
 
 = ac — bc. 
 
 We assume that a is greater than b. 
 
 Similarly it may be shewn that 
 ^ (a + b + c) d — ad + hd + cd, 
 
 (a — b — c) d = ad, — bd — cd, 
 
 and hence we obtain the following general rule for finding the 
 product of a swgle symbol and an expression consisting of two 
 or more terms. 
 
 "Multiply each of the terms by the single symbol, and con- 
 nect the terms of the result by the signs of the several terms 
 of the compound expression." 
 
22 
 
 MUL TI PLICA TION. 
 
 Examples.— vii. 
 
 Multiply 
 
 1. a + 6 - c by a. 7. Sm^ + 9m)i + lOn^ by mit. 
 
 2. « + 3& - 4c by 2a. • 8. 9a« + 4a46 _ 3^352 + 4(^2^3 i^y 2a6. 
 
 3. a? + 8a2 + 4a by a. 9. a;^?/' - ^if + x7j — 7 by ccy. 
 
 4. 2a^ - ^ 5^2 - 6a + 7 by Sa^. 10. m^ - Sm^?! + 3mn^ - 71^ by w. 
 
 5. a2 - 2a6 + 6^ by aft. 11. 12a36 - Ga^"^ + 5ah^ by 12a263. 
 
 6. a3 - 3a262 + ^3 ^y 3a%. 12. ISa;^ - 1 7xhj + bxi/ - y^ by 8xij. 
 
 51. We next proceed to the case in which both multiplier 
 and multiplicand are compound expressions. 
 
 First to multiply a + 6 into c + d. 
 
 Represent c + dhy x. 
 
 Then (a + h){c + d) = {a + h)x 
 
 = ax + hx,hj Alt 48, 
 
 = a{c + d) + b{c + d) 
 
 = ac + ad + hc + hd, by Art. 48. 
 
 The same result is obtained by the following process : 
 
 c + d 
 a + b 
 
 ac + ad 
 + bc + bd 
 
 ac + ad + bc + bd 
 
 which may be thus described : 
 
 Write a + b considered as the multiplier under c + d con- 
 sidered as the multiplicand, as in common Arithmetic. Then 
 multiply each term of the multiplicand by a, and set down the 
 result. Next multiply each term of the multiplicand by b, and 
 set down the result under the result obtained before. The 
 sum of the two results will be the product required. 
 
 Note. The second result is shifted one place to the right. 
 The object of this will be seen in Art. 56. 
 
MUL TI PLICA TION. 
 
 23 
 
 rin. 
 
 byw. 
 12a2&3. 
 
 ' l>y 8x?/. 
 iiiltiplier 
 
 38 : 
 
 + d 
 
 con- 
 
 1 
 
 Then 
 
 lown the 
 
 by 
 
 &, and 
 
 re. 
 
 The 
 
 he 
 
 right. 
 
 52. Next, to multiply a + 6 into c — d. 
 
 Kepresent c - rf by a;. 
 
 Then (a + 6)(c-d) = (a + 6)aj 
 
 = ax + 6a3 
 =a(c-cQ + &(c-(f) 
 
 = ac 
 
 ad + bc- hd, by Art. 49. 
 
 From a comparison of this result with the factors from 
 which it is produced it appears that if we regard the terms of 
 the multiplicand c — c? as independent quantities, and call them 
 + c and — d, the effect of multiplying the positive terms + a 
 and + b into the positive term + c is to produce two positive 
 terms +ac and +bc, whereas the effect of multiplying the 
 positive terms +a and +b into the negative term —d is to 
 "fTO^WQQ two negative terms — aciand —bd. 
 
 The same result is obtained by the following process : 
 
 c — d 
 
 a + b 
 
 ac — ad 
 
 + bc-bd • 
 
 ac — ad + bc — bd 
 
 This process may be described in a similar manner to that 
 in Art. 51, it being assumed that a positive term multiplied 
 into a negative term gives a negative result. 
 
 Similarly we may shew that a-b into c + d gives 
 
 ac + ad — bc-bd. 
 
 53. Next to multiply a-b into c — d. 
 
 Represent c-dhy x. 
 
 Then {a-b){c-d) = {a-b)x 
 
 — ax — bx 
 = a{c-d)-b{c — cC) 
 
 = {ac - ad)- {be -bd),h J Art 49, 
 = ac — ad-bc + bd, 
 
 "When we compare this result with the factors from which 
 it is produced, we see that 
 
 The product of tlie positive term a into the positive 
 term c is the positive term ac. 
 

 H 
 
 MUL TIPLICA TTOl^. 
 
 The product of the positive term a into the negative 
 term — d is tlie negative term - ml. 
 
 The product of the negative term - h into the positive 
 t» rm c is the negative term —he. 
 
 The product of the negati\e term - h into the negative 
 ten 1 - d is the positive term M. 
 
 The multiplitjtion of c - (i by a - 6 may he written thus : 
 
 a-h 
 
 ac — ad 
 
 - be + hd 
 
 ac~ad~hc + hd 
 
 54. The results obtained in the preceding Article enable ua 
 to state what ig called the Rule of Signs in Multiplication, 
 which is 
 
 "T/ie 'product of two positive terms or of two negative terms 
 is positive : the product of two terms, one of which is positive and 
 the other negative, is negative." 
 
 55. The following more concise proof may now be given, of 
 the Rule of Signs. 
 
 To shew that (a - h){c - d) = ac — ad — he ■{- hd. 
 First, {a- h)M= M+M+M+ ... with M written a-h times, 
 ' ' =(M + Af + iV/+ ...with M" written a times) 
 
 - (M + ilf + Af + . . . with ili written h times), 
 = aM-hM. 
 
 Next, let M= c — d. 
 
 Then aM= a (c-d) 
 = {c — d) a 
 = ca-da. 
 
 Alt. 39. 
 Art. 49. 
 
 Similarly, 6il!f=c6-c?6. 
 
 .*. {a-h){c-d) = {ca — da) — {ch — dh). 
 Now to subtract (ch - dh) from (ca — da), if we take away ch 
 we take away db too much, and we must therefore add dh to 
 the result, 
 
 .*. we get ca-da — ch + dh, 
 which is the same as ac-ad-hc + hd. Art. 39. 
 
 mu 
 un( 
 
 unl 
 
 pri 
 
 T 
 
 rigi 
 eacl 
 
 ( 
 
 (5)] 
 
MUL TIPLICA T/GN: 
 
 25 
 
 le negative 
 
 he positive 
 
 lie negative 
 
 3n thus ; * 
 
 So it appears that in multiplying {a — h){c — d) we must 
 multiply each term in one factor by each term in the other 
 and prefix the sign according to this law : — 
 
 TVhen the factors multiplied have like signs prefix + , when 
 unlike — to the product. 
 
 This is the Rule of Signs. 
 
 8 enable us 
 tiplication, 
 
 ative terms 
 positive and 
 
 3e given of 
 
 i — h times, 
 a times) 
 n h times), 
 
 39. 
 
 49. 
 
 :e away cb 
 add dh to 
 
 39. 
 
 56. "We shall now give some examples in illnstration of the 
 principles laid down in the last five Articles. 
 
 Examples in Multiplication woi'ked out. 
 
 (1) Multiply a; + 5 by a; + 7. (2) Multiply a; - 5 by a; + 7. 
 
 X'+ 
 
 x + 
 
 5 
 
 7 
 
 a;2 + 
 + 
 
 5a; 
 
 7a; + 35 
 
 a;2 + 12a; + 35 
 
 x— o 
 
 
 a; + 7 
 
 
 x^ — 5a; 
 
 
 + 7a;- 
 
 -35 
 
 a;- + 2x- 
 
 -35 
 
 The reason for shifting the second result one place to the 
 right is that it enables us generally to place like terms under 
 each other. 
 
 (3) Multiply a; + 5 by X - 7. 
 
 x + 5 
 
 x-7 
 
 (4) Multiply a; - 5 by X - 7. 
 
 X- 5 
 
 a;- 7 
 
 x2 + 5x 
 -7x-35 
 
 x^- 5x 
 
 - 7x + 35 
 
 x2-2x-35 
 
 x^-12x + 35 
 
 (5) Multiply x2 + 2/2 by x^ - if. (6) Multiply 3ax - r^hy by 7ax - 2hy. 
 
 3ax— 5by 
 lax— 2hy 
 
 2ld^x^-36ahxy 
 ^ , - 6ahxy+l0hY 
 21aV- 4^.a6x2r+ lOl^y^ 
 
 x^ + y^ 
 x^ — y^ 
 
 X* + x^y^ 
 — xhj'^ — y^ 
 
 x*-2/* 
 
.A^N 
 
 36 
 
 MUL TIPUCA TIOX. 
 
 57. The process in the muUij)lication of factors, one or 
 both of whicli contains more than two terms, is similar to the 
 ^processes which we have been describing, as may be seen from 
 the following examples : 
 
 Multiply 
 
 (1) x'^ + xy + ifhyx — y. 
 
 x^ + xy + y^ 
 ^-y ^ 
 
 x^ + xhf + xy"^ 
 
 — x^y - xy^ — y^ 
 
 (2) a2 + 6a + 9 by a^-6a + 9. 
 
 a^ + ea +9 
 a^-6a +9 
 
 X^-y9 
 
 -6a3-36a2-54n 
 
 + 9a2 + 54a + 81 
 
 a4-18a2 + 81 
 (3) Multiply 3x^ + 4xy - y^ by Z^ - 4xi/ + y\ 
 3x2+ 4;jj^ - y^ 
 
 3x2 
 
 4a;?/ + y^ 
 
 9a;* + 12x3|/ - Zx^y^ 
 
 - 12x^2/ ~ 1 6x2^/2 + 4x?/3 
 
 + 3x2?/2 + 4x1/3 - 2/* 
 
 9x4-16x22/2+ 8X1/3 _^ 
 
 (4) To find the continued product of x + 3, x + 4, and 
 x + 6. 
 
 To effect this we must multiply x + 3 by x + 4, and then 
 multiply the result by x + 6. . . 
 
 x+ 3 ... 
 
 x+ 4 
 
 X2 + 
 + 
 
 3x 
 
 4x +12 
 
 X2 + 
 X + 
 
 7x + 12 
 6 
 
 X3 + 
 
 + 
 
 7x2 + i2x 
 6x2 + 42x + 72 
 
 x3 + 13x2 + 54x + 72 
 
 NoU. The numbers 13 and 54 are called the coefficients of 
 a;2 and x in the expression x^-i 13x2 + 54x + 72, in accordance 
 with Art. 44. 
 
MULTI PLICA T/ON. 
 
 27 
 
 (f)) Find the coiitimu'd product of./; Fa, x-\-h, and x-^ c. 
 
 x + a 
 x + b ' 
 
 x^ + ax 
 + bx + ah 
 
 x^ + ax + bx + ah 
 x + c 
 
 x^ + ax^ + bx^ + abx 
 + cx^ + acx + bcx + ahc 
 
 05^ + (a + 6 + c)x^ + (a6 + ac-\- bc)x + aha 
 
 Note. The coefficients of x"^ and x in the expression jusi 
 obtained are a f 6 + c and a& + ac + he respectively. 
 
 When a coefficient is expressed in letters, as in this example, 
 it is called a literal coefficient. 
 
 Examples.— viii. 
 
 Multiply 
 
 I. a; + 3 by cc + Q. 2. a; + 15 by a; -7. 3. cc- 1£ l)y x + 10. 
 4. ac-S by 50-7. 5. a-3bya-5. 6. y-Ghy y + Vi. 
 7. x2-4bya;2+5. 8. x^-Qx + ^hy x^-iSx + b. 
 
 3a + 2 by a'^-3a2 + 2. 
 
 9. a;2 + 5x — 3 by a;2 — 5a; — 3. 10. a^ 
 11, x'^ — x + lhy x^ + x—\. 12. a;2 + a-// + 7/2 by :c- - a:?/ + 7/2. 
 
 13. a;2 + iCT/ + 7/2 by X - y. 14. a'- - a;-' by a* + aV + a;*. 
 
 15. x^- 3a;2 + 3x - 1 by a;- + 3a; + 1. 
 
 16. a'3 + 3x^7/ + 9x?/2 + 277/3 by a; - 3?/. 
 
 1 7. ft^ + 2a% + 4rt//- -f 86^ by a - 26. 
 
 18. 8a^ + 4a26 + 2a62 + feM)y 2a-6. ■ 
 
 19. a'> - 2a26 + 3a62 + 4?>-^ by a^ - 2a& - 362. 
 
 20. a3 + 3a26 - 2a62 + 36'' by a2 + 2a6 - 362. 
 
 21. a2 - 2ax + 4x2 ]jy ^^2 ._ ^ax + 4x2. 
 
 22. 9a2 + 3ax + x2 l)y 9a--3ax + x2. 
 
 23. X'* - 2ax2 + 4a- by x^ + 2{/x2 + 4a2. 
 
 24. a'^-\-W + c'^-ah — ac-hc\)y a-^h-vc. 
 
 25. X2 + 4X7/ + 57/2 Ijy 3.3 _ 3^,2^ _ 2 j;,^2 + 3^3_ 
 
 26. a6 + cc? + ac + 6rf by a6 + cd-ac- hd. 
 Find the continued product of the following expression : 
 27. x-a,x + a, x2 + a2, x* + a*. 28. x - a, x f 6, x - c. 
 
a8 
 
 MUL TIPLTCA TTON, 
 
 29. 1 - a;, I + a, 1 + a;2, 1 4 a:*. 
 - 30. x — yfX + y,x^ — xy + y^,x'^ + xy + y'^. 
 31. a — x,a + X, a'^ + re-, a** + x*, a^ + x^. 
 
 Find the coefficient of x in the following expansions : • 
 
 32. {x-5){x-())(x-\-7). 33. (.'B + 8)(a; + 3)(.r-2). 
 
 34. (x-2)(x-^){x-\4). 35. (a; - a) (x - 1) (x - c). 
 
 36. (x2 + 3a; - 2) (x2 _ 3a; + 2) (x* - 5). 
 
 37. (a;2 - a: + 1) {x^ + x-l) {x^ - a;^ + 1). 
 
 38. (aj^ - mx + 1) (a;2 — mx — 1) (x'* - m^x — 1). 
 
 58. Our proof of tlie Ru'e of Sif,Mis in Art. 55 is founded 
 on the sup[)Ositiou that a isjf^'reater than h and c is greater 
 than cL 
 
 To iiichide cases in which the midfiplkr is an isohited nega- 
 tive <[uantity we must extend our definition of Multiplication. 
 For the definition given in Art. 36 does not cover this case, 
 since we cannot say that c shall be takiiU — d times. 
 
 We give then the following definition. " The operation of 
 Multiplication is such that the product of the factors a — h and 
 c — d will he equivalent to ac — ad — hc + bd, ichatever may be the 
 values of a, b, c, d.^^ 
 
 Now since 
 
 (a -b) (c-d) = ac- ad — bc + bd, 
 make a = and (? = (). 
 
 Then (0-6) (c-0) = 0xc-0x 0-.. 6x0. 
 
 or — 6 X c = — 6c. 
 
 Similarly it may he shewn that 
 
 -6 X -d= +bd. 
 
 I 
 
 Examples.— ix. 
 
 Multiply 
 I. a^ "by _5, 2. a^hy —a\ 3. ^26 by —ah". 
 
 4a% hy —3ab^. 5. bx^y hy -Gxy"^. 6. a^-ah + b'^hy —a. 
 
 4- 
 7. 
 
 2a^ + 4a^-5ahy -2a- 
 
 8. —a^ — rt.2 — rt, by —a— 1. 
 
 9- 
 10. 
 
 II. 
 
 12. 
 
 13- 
 14. 
 
 3xhj — 5xy^ + 4y^ by — 2a; — '3y. 
 — 5m2 — 6m7i + 7n^ by — m + n. 
 13r2-17r-45 by -r-3. 
 Ba;^,':; — Qz^ by - a; — s. 
 x° + x^y - a;^j/- by —y — x. 
 
 x^hy —X- y. 
 
 7.^3 
 
 -y^-xy- 
 
 x^y- 
 
 -} 
 
18 : • 
 c-2). 
 
 III. XNVOLUTION. 
 
 18 founded 
 is greater 
 
 ited nega- 
 i})licatioii. 
 this ease, 
 
 )crat{on of 
 s a—h and 
 may he the 
 
 -ah\ 
 
 6- by - a. 
 a—\. 
 
 50. To this part of Algebra ])elongs the process called 
 Involution. This is the operation of multiplying a quan- 
 tity hy itself any number of times. 
 
 The i)Ower to which the quantity is raised is ex])res>ed by 
 the number of times the quantity has been employed as a 
 factor in the operation. 
 
 Tlius, as has been already stated in Art. 4;'), 
 d^ is called the second power of a, 
 a? is called the third power of a. 
 
 (10. When we have to raise negative quantities to certain 
 powers we symbolize the operation l)y putting the quantity in 
 a bracket with the number denoting the inc?ex (Art. 45) ]»laced 
 over the bracket on the riglit hand. 
 
 Thus (-«)•'' denotes the third power of —a, 
 ( - 2,^;)'* denotes the fourth jDower of — 2.«. 
 
 Gl. The signs of all even powers of a negative quantity 
 will be ^positive, and the signs of the odd powers will be 
 negative. 
 
 Thus {-af^{-a)x{-a) = d\ 
 
 {-af = {-a).{-a){-a) = a:^.{-a)=^-a}\. 
 
 (52, To raise a simple quantity to any power we multiply 
 the index of the (piantity by the number denoting the power 
 to which it is to be raised, and prefix the proper sign. 
 
 Thus the square of a^ is a^, 
 
 the cube of a^ is a'*, ■ 
 
 the cube of - x^yz^ is - 3[pyh\ 
 
30 
 
 INVOLUTION/. 
 
 63. We form the second, third and fourth pcsvers of a + 6 
 in the following manner : 
 
 a + 6 
 
 a + 6 
 
 + ab +b'^ 
 a +b 
 
 + a% + 2ab^ + b^ 
 
 (a + bf = a^+'^a^ + 2a^l)^ 
 a +b 
 
 a* + ^cv^b-\-3a%'^ + ab^ 
 + a% + 3a^b'^ + 2ab^ + ¥ 
 
 (a + by = a* + '4a?bTia%'^ + 4ab^ + ¥. 
 Here observe the following hiws : 
 
 I. The indices of a decrease l»y unity in each term. 
 11. The indices of b increase by unity in each term. 
 III. The numerical coefficient of the second term is always 
 the same as the index of the power to which the 
 binomial is raised. 
 
 64. We form the second , third and fourth powers of a -6 
 in the following manner : 
 
 a-b 
 a-b 
 
 ab 
 
 a 
 
 ab +/>* 
 
 'a^-2a^ll-ab'^ 
 - a'^b + 2ab'^-¥ 
 
 (a - by = a'' - da^b + 3dl^^ 
 a -b 
 
 a'^-:Wb + 2a:^b'^-ab^ 
 - a^b + 'Sa%-'-'daP + b^ 
 
 {a - by - (I* - 4a% + GaP¥~'-4ab^ + b\ 
 
 ^ 
 
INVOLUTION. 
 
 3* 
 
 jrs of a + 6 
 
 ;erm. 
 ^erm. 
 
 n is always 
 \vhiuh the 
 
 ers of a - 6 
 
 Now observe that the ji^wcrs of a- 6 do not differ from the 
 powers of a + 6 except that the terms, in which the oM powers 
 of h, as 6\ 5^, occur have the sign - prefixed. 
 
 Hence if any power of « + & be given we can write the 
 corresponding power of a - h : thus 
 
 since {a + bf = a^ + rm^h + lOaW + lOaV)^ + 5a¥ + W, 
 {a - hf = f/' - ba*h + lOcrb'^ - lOa'-^ft^ + ba¥ - h\ 
 
 65. Since {a + hy = a^ + ¥ + 2ab and (a - bf = a"^ + b'^ - 2ab, 
 it appears that the square of a binomial is formed by the 
 following process : 
 
 "To the sum of the sc[uares of each term add twice the 
 product of the terms." 
 Thus {x + y)~ = x'^ + i/ + 2xy, 
 
 {x + 3y = x- + 9 + 6x, 
 {x-bf = x^ + 25~l0x, 
 ' {2x - 7yf = 4x2 + 49i/2 - 28xy. 
 
 66. To form the square of a trinomial : 
 
 a + b + c 
 a + b + c 
 
 a^ + ab + ac 
 + ab + b'^ + be 
 + ac + bc + c^ 
 
 a2 + 2ab + 62 + 2ac + 26c + c". 
 
 Arranging this result thus a2 + 62 + c^ + 2ab + 2ac + 26c, we see 
 that it is composed of two sets of quantities : 
 
 I. Tlie squares of the quantities a, 6, c. 
 
 II. The double products of a, 6, c taken two and two. 
 
 Now, if we form the square of a - 6 - o, we get 
 
 a - 6 - c 
 
 a — b - c 
 
 a? - ab - ac 
 -ab + ¥ + bc 
 
 -ac + be + c2 
 
 (i^-2ab + b-~2ac + 2bc\-ci 
 
 The law of formation is tlie same as before, for we have 
 
!l! 
 
 1[ 
 
 1 
 
 3« 
 
 INVOLUTION. 
 
 I. The scpares of the quantities. 
 
 II. The double products of rr, - ?>, - c taken two by two : 
 the sign of each result being + or - , according as 
 the signs of the algebraical quantities composing it 
 are like or unlike. 
 
 67. The same law holds good for expressions containing 
 more than three terms, thus 
 
 (a + 6 + c + d)2 = a2 + 62 + c2 + d2 
 
 + 2a6 + 2ac + 2acZ + 26c + 2M + 2cd, 
 (a-6 + c-(Z)2=a2 + &2 + c2 + rf2 
 
 - 2rt6 + 2ac - 2af? - 26c + 26(^ - 2cd 
 
 And generally, the square of an expression containing 2, 3, 
 4 or more terms will be formed by the following process : 
 
 " To the sum of the squares of each term add twice the 
 product of each term into each of the terms that follow it." 
 
 EXAMPLKS.— X. 
 
 Form the square of each of the following expressions : 
 I. 05 + a. 2. x-a. 3. x + 2. 4. .r- 3. 5. .7;2 + ^2^ 
 
 6. x^-y^. 7. «^ + />^ 8. a^-6"\ 9. x-^ij-\-z. 10. x-y + ^. 
 
 II. m + n-p-r. 
 14. 2x2-7ic + 9. 
 17. a^ + P + c^. 
 
 12. a;- + 2a;-3. 
 
 15. x^- + if-z\ 
 
 18. x^-y^-z\ 
 
 20. x^ - 2^/2 + 5^2. 
 
 13. a;2_6a; + 7^ 
 16. x'^-4xY^ + y*. 
 19. x + 2y-3z. 
 
 Expand the following expressions : 
 21. (x + rt)^ 22. {x-af. 23. (x+lf. 24. {x-iy. 
 25. {x + 2f. 26. (a2-62)3. 27. {a + h + c)l 28. (a-6-c)3. 
 29. (m + n)2.(m - ?i)2. 30. (m + 7i)2.(7??,2 _ ^j,2^^ 
 
 68. An algebraical product is said to be of 2, 3 dimen- 
 sions, when the sum of the indices of the quantities composing 
 the product is 2, 3 
 
 Thus ah is an expression of 2 dimensions, 
 
 a^bh is an expression of 5 dimensions. 
 
DivisroM. 
 
 ,vo by two : 
 icorcling as 
 mposing it 
 
 containing 
 
 hd - 2cd 
 
 ining 2, 3, 
 cess : 
 
 twice the 
 ow it." 
 
 69. An algebraical expression is called homogeneous when 
 each of its terms is of the same dimensions. 
 
 Thus x^ + xy + If- is homogeneous, for each term is of 2 dimen- 
 sions. 
 
 Also 3./;^ + 4;«-// + 5'//^ is homogeneous, for each term is of 3 
 dimensions, the numerical coelticients not affecting the dimen- 
 sions of each term. 
 
 70. An expression is said to be arranged according to 
 powers of some letter, when the indices of that letter occur in 
 the order of their magnitudes, either increasing or decreasing. 
 
 Tlius ihe expression a^ 4- a~x + ax^ + x^ is arranged according 
 to descending i)Owers of a, and ascending powers of x, 
 
 71. One expression is said to be of a higher order than 
 another when the former contains a higher power of some dis- 
 tinguishing letter than the other. 
 
 Thus a^ + a-x + ax^ + x? is said to be of a higher order than 
 fl2 + rtx + .r^, with reference to the index of a. 
 
 )ns : 
 
 . x^-^y\ 
 
 x-y + %. 
 Qx + 1. 
 4x^y'^ + y\ 
 y-3z. 
 
 x-iy. 
 
 a-b- cy. 
 
 ....dimen- 
 composing 
 
 IV. DIVISION. 
 
 72. Division is the process by which, when a product is 
 given and we know one of the factors, the oilier factor is deter- 
 mined. 
 
 The product is, with reference to this process, called tho 
 
 DlVlDKND. 
 
 The given factor is called the Divisor. 
 
 The factor wliich has to be found is called the Quotient. 
 
 73. The operation of Division is denoted by the sign -r-. 
 Thus ah-~a signilies that ab is to be divided by a. 
 
 The same operation is denoted by writing the dividend 
 
 over the divisor with a line drawn between them, thus — . 
 
 a 
 
 In this chapter we shall treat only of cases in which tho 
 dividend contains the divisor an exact number of times. 
 [8.A.] g 
 
r 
 
 U 
 
 Divisioisr. 
 
 Case I. 
 
 74. "Wlien the dividend and divisor aro earli included in 
 a single term, we can nsnally tell l)y inspection the factors of 
 which each is composed. Tiie quotient will in this case be 
 represented by the factors which remain in tlie dividend, when 
 those factors which are common to the dividend and the di- 
 visor have been removed from the dividend. 
 
 Thus 
 
 aft 
 
 a 
 
 a 
 aaaaco 
 
 = 3a, 
 
 add, 
 
 =zaa = a~ 
 
 Thus, when one power of a number is divided by a smaller 
 power of the same number, the quotient is that power of the 
 number whose index is the difference hehveen the indices of the 
 dividend and the divisor. 
 
 Thus 
 
 a^ 
 Sab 
 
 -rtl2-o, 
 
 = 5a2/;. 
 
 ■ a' 
 
 75. The quotient is unity when the dividend and the 
 divisor are equal. 
 
 Thus 
 
 a 
 
 xhf_ , 
 
 and this will hold true wnen the dividend and the divisor are 
 compound quantities. 
 
 Thus 
 
 a + b x^'-y^ 
 
 Examples. — xi. 
 
 Divide 
 I. a;° by x\ 2. x^^ by x^. 3. r/''?/^ by xy. 
 
 4. a^y^r.(^ hy x^f-z. 5. 24ah^-chy 4ah. 6. 'J'la'-bhH^yQa^'^c. 
 
 7. 256a367(;'J by Wahc^ 8. 13:31?/i'0h'V^^ '\y Umhiy. 
 
 9. 60a V?/^ by 5a'//. 10. 9G??-*ftV by Ubc, 
 
 /b. 
 
 contii 
 
 divi( 
 
 and 
 
DIVISIOA'. 
 
 35 
 
 I iiichuled in 
 lie factors of 
 this case be 
 ndend, when 
 [ and the di- 
 
 by a smidler 
 power of the 
 indices of the 
 
 nd and the 
 
 J divisor are 
 
 )y xy. 
 
 y llm-ii^p^. 
 
 Case II. 
 
 76. If the divisor be a single term, while the dividend 
 contains two or more terms, the quotient will be found by 
 dividing each term of the dividend separately by the divisor 
 and connecting the results with their proper signs. 
 
 ax + hx , 
 
 Thus 
 
 X 
 
 a^x^ + a^x^ + ax „ „ 
 
 ,-=a^x' + ax + 1, 
 
 ax. 
 
 12xV 4- 16,-022,3 _8a:7/2 _ „ , , 
 
 4xy^ 
 
 -xii. 
 
 EXAMPLES.- 
 
 Divide 
 
 1 . x"' + 2x- + X b}"- X. 4. w^xx* + m-p-x"^ + vi/^p^ by mp. 
 
 2. If -y* + y^- y^ by ?/2. 5.1 6a"lx?/ - 28aV'^ + Ad^x^ by Aiv^x. 
 
 3. 8fi3 + \Qa% + 24rt?j2 by 8rt. 6. l^x^' - ^^^Y - \Qxhf by ^xhj, 
 
 7. Sb/i^/i" - 547w''>i'' + 27«i.^H22? by ^mhi\ 
 
 8. 12j:Y-8/jV-4a;Vby 4x3. 
 
 9. 169rt46 - 1 1 7a3/,2 ^ 9lf<2J l^y 13^2, 
 
 10. 3(; //V + 228//c4 - 13363c5 by 1962c. 
 
 77. Admitting the possibility of the independent existence 
 of a term affected with the sign -, we can extend the Exam- 
 plea in Arts. 74 — 76, by taking the first term of the dividend 
 or the divisor, or both, negative. In such cases we apply the 
 Rule of Signs in Multiplication to form a Rule of Signs in 
 Division. '^ 
 
 ■ah 
 
 Th 
 
 lis smce 
 
 axh= -ah, we conclude that 
 h= —ah, 
 
 a X 
 
 b 
 — ah 
 -b 
 ah 
 
 '~h '' 
 
 = -a. 
 
 ■ a. 
 
 a: 
 
 —ax -b = ahf 
 
 and hence the rules 
 
 I. When tlie dividend and the divisor have the .same 
 sign the quotient is positive. 
 
 II. When the dividend and the divisor have different 
 * signs the (quotient is negative. 
 
i 
 
 I 
 
 DIVISION. 
 
 78. Thf3 following Examples illustrate the conclusions just 
 obtained : 
 
 (1) = —mx. 
 
 (3) - 2^ = to,=. 
 
 (5) T 1.— - — -fr + a¥- a^o + a-\ 
 
 — ah 
 ■16 
 
 4x1/ 
 
 
 Examples.— xiii. 
 
 Divide 
 I. V2rt6by -9a&. 
 
 - 60a8 by - 4^3. 
 
 - 84x^1/^ l)y 4r''?/^. 
 
 - 18TO3n2 by 2mn. 
 
 -128a362cby _86c. 
 
 6. — a^x^ — a%2 _ ffr,. i^y _ ^3,^ 
 
 7. - 34rt3 + 51 a2 - 17aa;2 by 1 7«. 
 
 8. _ HaW - 2U^IP + 32aV)^ by - 4a^b'^. 
 
 9. - 144«3 + 10£.c2|/ _ c)Qxif by 12cc. 
 
 10. 62a;3«2 _ Jj5^7^i _ Py4^2 j^y _ J2g2, 
 
 Case III. 
 
 V9. The third case of the operation of Division is that in 
 which the divisor and the dividend contain more terms than 
 one. The operation is conducted in the following way : 
 
 Arrange the divisor and dividend according to the 
 powers of some one symbol, and pUice them in the 
 same line as in the process of Long Division in 
 Arithmetic. - ' "" 
 
 Divide the first term of the dividend by the first term 
 of the divisor. 
 
 Set down the result as the first term of the quotient. 
 
 Multiply all the terms of the divisor by the first term 
 of the quotient. 
 
 ' Subtract the resulting product from 'he dividend. If 
 there be a remainder, considv^x' it as a new dividend, 
 and proceed as before. ^ 
 
-DIVISION-. 
 
 37 
 
 iclusioiis just 
 
 -a + h. 
 
 77 + 2. 
 
 - ax. 
 )yl7a. 
 6« l)y - 4a%\ 
 U' W 12a;. 
 
 ti is that in 
 teiins than 
 way: 
 
 :ns to the 
 lein ill tlie 
 division in 
 
 ! first term 
 
 uotient. 
 i first term 
 
 ^idend. If 
 
 w dividend. 
 
 The process will best be understood by a careful study of 
 the following Examples : 
 
 (1) Divide a^ + 2ab + ¥hya + b. (2) Divide «= _ 2ab + b'^hya~b. 
 a + b)a^-h2ab + b'^[a + b a-b)a^-2ab + b'^i^a-b 
 
 a- + ab 
 
 ar — ab 
 
 
 ab + b"^ -ab + b'^ 
 ab + b^ -ab + b^ 
 
 (3) 
 
 Divide x^ - if by x"^ - y"-. 
 
 ■ 
 
 X- - f') x^ - f (x* + x2i/2 + \f 
 x^-xhf 
 
 (4) 
 
 xh/ - if 
 xY-xY 
 
 xY - )f 
 xY - y^ 
 
 Divide x^ - 4a^x* + Aa^x"- - a« by x^ - aK 
 
 a;2- 
 
 ■ a^) xf^ - Aa^x^ + 4a^;/j2 - a^ (x^ - Za^x^ + a^ 
 x^-a^x^ 
 
 -3a2x4 + 4rt4j;2_et6 
 
 -3a2.x4 + 3aV 
 
 \ 
 
 a%2_ct8 
 
 a V - a« 
 
 (5) Divide ^xy + x^ + y^ -\hj y + x-\. 
 
 Arranging the divisor and dividend by descending powers 
 of a, 
 
 x + y - 1) x^ + 3xy + y^ - I {x'^ - xy + x + y^ + y + I 
 ■« x^ + x^y~x^ . 
 
 -xhj 'rx^+^xy + y^ -1 
 -xhj-xy^ + xif 
 
 a;2 + xy'^ + 2xy + y^-l 
 x^ + xy-x 
 
 xif + xy-\-x-^y^ 
 
 Xf + y^-y-i 
 
 xy + x + y^-l 
 
 xy + y^-y 
 
 x + y-l 
 ' x-^-y-X 
 
38 
 
 DIVISION. 
 
 n'H 
 
 %^ 
 
 80. We nmst now direct the attention of the student to 
 two points of great importance in Division. 
 
 I. The dividend and divisor must be arranged accord- 
 ing to tlie order of the powers of one of the symbols 
 involved in them. This order may be ascending or 
 descending. In the Examples given above we have 
 taken the descending order, and in the Examples 
 worked out in the next Article we shall take an 
 ascending order of arrangement. 
 In each remainder the terms must be arranged in 
 the same order, ascending or descending, as that in 
 which the dividend is arrancjed at first. 
 
 II. 
 
 O" 
 
 81. To divide (1) 1 -a;* by ^3 + ^2 + ^^ + i^ 
 
 arrange the dividend and divisor by ascending powers of Xy 
 thus : 
 
 l+x + x^ + x^ 
 
 -x-x^-x^-x^ 
 -x-x'^-x^-x^ 
 
 (2) 48a;2 + 6 - 35x-^ + bSx'^ - l^x^ - 23a; by Gx^ - 5a; + 2 - 7^3^ 
 arrange the dividend and divisor by ascending powers oi a;, 
 thus : 
 
 2 -5x + 6a;2 - ^x^) 6 - 23a; + 48a;2 - 70a;3 + SSa;* - 35a;^ (3 - 4a; + 5a;2 
 6-15a; + 18a;2-21x3 
 
 _ -8a;-|-30.«2-49x3 + 58:c* 
 
 -8a; + 20a;2-24a;3 + 28a;* , • 
 
 10a;2-25a;3-[-30a;-*-35a;5 
 10x2 -25x3 + 30a;* -35a;5 
 
 EXAMPLES.— XiV. 
 
 Divide 
 
 1. x2+15a; + 50by x + 10. 5. x3+13x2 + 54x + 72 by x + 6. 
 
 2. x2-17x+70 by x-7. 6. x^ + x^-a;- 1 by x+ 1. 
 
 3. x2 + X - 12 by X - 3. 7. x^ + 2x2 + 2x + 1 by x + 1. 
 
 4. x2 + 13x + 12 by X + 1. 8. x^ - 5x3 4. 7^^2 + g^; + 1 by x2 + 3x + 1. 
 
 9. X* - 4x3 + 2x2 4. 4;^ + 1 l,y a;2 - 2:): - 1. 
 10. x4-4x3 + 6x2-4x+lbyx2~2x+l. 
 
DIVISIOX. 
 
 39 
 
 II. x* - x2 + 2x - 1 by V? + a: - 1. 12. y> - 4^2 + 8:c + 16 by x + 2. 
 • 13. .(;'^ + 4c27/ + 3x?/2 + 12//3by a; + 4?/. 
 
 1 4. ft^ + 4a36 + 6a2/)2 + /ia\? + 7j^ by a + h. 
 
 15. ^'^ - 5a46 -f iOrt'562 _ lOttS^s ^ 5^jj4 _ 55 i^y ^ _ 5^ 
 
 16. a;< - i 2i^' + U ).>;'- - 84.>: + 45 by x? - 6.c + 9. 
 
 17. rt'^ - 4a46 + 4rt3/;-' + 4rt-'63 _ Yiah^ _ i2&-'5 by a2 - 2a& - 3?;-. 
 
 1 8. 4rt2ic^ - VLaH^ + 13a^a;2 - 6a-^x + a« by 2a£- - Zo?-x + a^ 
 
 19. a;* - a;2 + iix - 1 by x2 + a; - 1. 
 
 20. a* 4- a^y? - 2a'* by cc^ 4. 2rt2, 23. a;^ - 1/*' by a; - 1/. 
 
 21. :<;2 - 13x2/ - 30i/ by x - 15?/. 24. d?-lr^ 2hc - c-hya -h + c. 
 
 22. x-' + y^ hy x + y. 
 
 25. &-3i2 + 3?>3_&4i,y 5_i^ 
 
 26. a^-b^-c^ + cl^-2{a(l-hc)hya + h-c-d. 
 2y, x^ + y^ + z^- Zxyz hy x + y + z. 28. a;^^ + 2/^*^ by a;^ + v"^. 
 
 29. p'^+p(i + 2pr - 2g2 + 72?' - 3)-2 by 2^ - <Z + 3?-. 
 
 30. a8 + a%^ + a464 + ^2^0 + 58 }^y ^4 + ^35 + ^2^2 4. ^^:: + ^1^ 
 
 31. JC^ + x^y^ + x^y^ + xhf + y^ by a;'* - x^y + a;^?/^ - xy"^ + 7/^ 
 32. 4x5 -a;3 + 4a: by 2x2 + 3a; + 2. 33, a5-243bya-3. 
 
 34. JfciO-^byF-1. 
 
 35. x^ - 5x2 - 4Gx - 40 by x + 4. 
 
 39- 
 
 36. 48x3 _ 76^a;2 _ 64c^2_^ 4. iQSa^ by 2x - 3a. 
 
 37. 18x'' - 45x3 + 82x2 _ 67^ + 40 by 3x2 _ 4-^. ^ 5^ 
 
 38. 16x* - 72a2x2 + Sla^ by 2x - 3a. 
 
 81x4 _ 256a* by 3x + 4a. 41. x^ + 2ar^ -a'^x-'z ' by a;2 
 
 a- 
 
 40. 2a3 + 3a26 - 2a62 - 36''bya2 - W. 42. a* - a262 - I26*bya2 + 362. 
 
 43. X* - 9x2 _ Q.j^y -y'i\,y x^-\-^X->ry. 
 
 44. X* - Qx^y + 9x2|/2 - 4?/* by x2 - 3xi/ + 2^/2. 
 
 45. X* - 81?/* by X - 3?/. 
 
 46. a* -166* by a -26. 
 
 47. 81a* -166* by 3a + 26. 
 
 48. 16x* - 81?/* by 2x + 3?/. 
 
 49. 3a2 + 8a6 + 462 ^ loac + 86c + 3c2 by a + 26 + 3c. 
 
 50. a* + 4a2x2 + 1 6x* by a2 + 2ax + 4x2. 
 
 5 1. X* + x2?/2 + ^ by x2 - x?/ + 2/2. 
 
 52. 256x* + 16x2j/2 + y^ by 16x2 + Axy + ?/. 
 
 53. 7^-\-xhj-y?y'^-\-^-2xif + y^hy 7?-^x-'ij, 
 
I I 
 
 li 
 
 40 
 
 DIVISION. 
 
 54. ax^ + Sa^rcS _ 2a3.c - 2«^ by a; - a. 
 
 55. ii^-x^ bya; + 
 
 a. 
 
 56. 2a;2 + r (/ - 3//^ - 47/« - ics; - is^ i)y 2a; + 3y 4- %, 
 
 57. 9u; + 3x-^ + 1 4.o3 4. 2 1 )y 1 + 5a; 4- aA 
 
 58. 12 - 38.C + 82a;2 - \\±,9' + 106^:* - TOr' by "Jx^ - 5a; + 3. 
 
 59. ic'' + if by X* - ^y + a;''^//''^ - rc?/^ + ?/'*. 
 
 60. (rt^x'2 + ])^\f) - {ar\i' + a;-7/-) by aa; + 6?/ + at + a?/. 
 
 61 . a& (a;- + 2/-) + x\j{o?' + ft'-^) by aa; + hj, 
 
 62. a;4 + (27;-^ - ((-')a;2 + 64 i^y _,.2 + ^a- + /yj. 
 
 82. The process may in some cases 1)6 shortened by the use 
 of brackets, as in the following Example. 
 
 a; + 6^ a;^ + (» + 6 + c) a;- + (a6 + ac + 6c) a; + ahc ( a;^ + (a + c) a; + ac 
 
 X' 
 
 ■ + 6a;2 
 
 {a + c) A'- + (ah + ac + ho) x 
 (a + c) aj- + (ah 4- be) x 
 
 acx + ahc 
 
 acx 
 
 ■\-ahc 
 
 X 
 
 \) ufi ~ mx'^ + nx^ - nx^ + nix - 1 (.r* - {m, - 1 ) x 
 
 X 
 
 ;6 _ .^4 
 
 - (w - n - 1) a;- - (m - 1) a; + 1. 
 
 nx"" 
 
 -(m-1) x^ + 
 
 - (m - 1) a;* + (m - 1) x^ 
 
 a;-' 
 
 nx'' 
 
 -(m-?i- 1) 
 
 - (wi - -^i - 1) x^ +{m-n-\) x^ 
 
 - (m - 1) a;2 4- ma; 
 -(m-1) a;2 4-(m- 1) 
 
 a;-l 
 
 X 
 
 x~\ 
 
 -XV. 
 
 EXAMPLES.- 
 Divide 
 
 1. a^ - (a^ - 6 - c) a-2 - (6 - c) aa; + he by a;^ - «a; + c. 
 
 2. 2/^ - (^ + w + w) 1/ + (/7?i + Zri-t- mn) y - linn hy y-n. 
 
 3. xP -(7n-c) x^ + (n-cm + d)x^ + 
 
 (r + c/i - dra) x'^ + (cr 4- dn) x + dr by x^ - mx~ + nx + r. 
 
 4. a;4 4- (5 + a) x3 - (4 - 5a + 6) a;^ - {Aa 4- 56) a; + 46 by a;^ 4- 5a; - 4. 
 
 S' x^-{a + h + c + d)x^-\-{ah-\-ac-k-ad^hc-\-hd + cd)x'^ 
 
 - {ahc 4- abd + acti + hcd) x 4- a6cd by a;^ - (a + c) x + ac. 
 
DIVISIOI^. 
 
 41 
 
 ^ by a; + a. 
 
 j:^ ~ 5.f + 3. 
 xy, 
 
 by the use 
 "+ c) a; -f ac 
 
 -l)a;+l. 
 
 -\)x 
 
 h 5a; - 4. 
 c) a; + ac. 
 
 83. The Ibllowiiig Examples in Division are of great 
 iiiiportance. . 
 
 Divisor. 
 
 Dividend. 
 
 Q 
 
 COTIENT. 
 
 iC + l/ 
 
 x'^ - y^ 
 
 
 X-y 
 
 x-y 
 
 x^-y'^ 
 
 
 x + y 
 
 x + y 
 
 ^ + y"^ 
 
 
 x^ -xy + y^ 
 
 x-y 
 
 x^ - y^ 
 
 
 x^ + xy + ^- 
 
 84. Again, if we arrange two series of binomials consisting 
 respectively of the sum and the diilerence of ascending powers 
 of x and y, thus 
 
 x-\-y, x'^ + y'^, x^ + y^, ^ + y^, r'' + y'', r''' + ?/", and so on, 
 x-y, x^ - 7/-, x^ - y^, «■* - y*, x" - y-', x'' - )/\ and so on, 
 
 x + y will divide the odd terms in the upper line, 
 and the even in the lower 
 
 x-y will divide all the terms in the lower, 
 but none in the upper. 
 
 Or we may ]3ut it thus : 
 
 If w stand for any whole number, " 
 
 X" + y" is divisible by x + y when n is odd, 
 
 hj x-y never ; 
 
 a;" - ?/*• is divisible hy x + y when n is even, 
 
 hy x-y always. - 
 
 Also, it is to be observed that when the divisor is ^ - y all 
 the terms of the quotient are p sitive, and when the divisor is 
 x + y, the terms of the quotient are alternately positive and 
 negative. 
 
 Thus —--"- = a;3 + a;-?/ 4- .r?/^ + 9A 
 x-y J J J 7 
 
 ^ = x^'-:ihj + o:hf-o?y^ + xh/-xy^ + if, 
 
 x + y 
 
 — 7™ = jfi - xhi + xh^ - x^ifi + a;?/* -y^. 
 
 x + y 
 
Ill 
 
 "T' 
 
 42 DIVISIOiV. 
 
 85. TliL'si! ]H'operti('s may Le easily reincniltcred by takinji; 
 the four aimplest cases, thus, x + y, x~y, x^ + i/, x^-y^j of 
 which 
 
 the first is divisible by cc + y, 
 
 second x-y, 
 
 third neither, 
 
 fourth both. ^■ 
 
 Again, since these properties are true for all values of x and 
 y, suppose y^'i, then we shall have 
 
 --;-, =«-l, - , =.r+l, 
 
 x+l X ~ i 
 
 , —X'~X+1, =x-' + x+ I. 
 
 X+i ' X - I 
 
 Also 
 
 cc'' + 1 
 
 ■ , = x' - a;^ + .t;^ - .r + 1 , 
 
 £C+ 1 
 
 a"- I 
 
 :r = X^ + X^ + y? + X^ + ,r + 1. 
 
 %- 1 
 
 Examples.— xvi. 
 
 Without going through the process of Division write down 
 the quotients in the following cases : 
 
 1. When the divisor is m + 92, and the dividends are 
 respectively 
 
 m^ - v?^ ipp? + ?i^, m^ + n-^, 'in^ - n^^ 771" + 71^. 
 
 2. When the divisor is m-n, and. the dividends are 
 respectively ^ 
 
 3. AVhen the -divisor is a + 1, and the dividends are 
 respectively 
 
 4. When the divisor is y -I, and the dividends are 
 respectively 
 
 f--l,y^-l,y^-l,y7-l,,f-l. 
 
taking 
 
 F X and 
 
 clown 
 Ids are 
 
 V. ON THE RESOLUTION OF EXPRES- 
 SIONS INTO FACTORS. 
 
 8f). We shall discus?? in this Chapter an operation which 
 is the opposite of that M'hich we call Multiplication. In Mul- 
 ti[)lication we determine the product ot" two given factors : in 
 the ()j)eration of which we have now to treat the i^roduct is 
 given and the factors havn to he found. 
 
 ^7. For the resolution, as it is called, of a product into ita 
 component factors no rule can be given which shall he applic- 
 able to all cases, but it is not difficult to explain the process 
 in certain simple cases. We shall take these cases separately. 
 
 88. Case I. The simplest case for resolution is that in 
 which all the terms of an expression have one common factor. 
 This factor can be seen by inspection in most cases, and there- 
 fore the other factor may be at once determined. 
 
 Thus a'^ + ah=a{a + h\ 
 
 2rt" + 4a2 + 8rf = 2n (ti2 + 2(* + 4), 
 ^x?y - 1 Sxhj^ + 54,r?/ = 9x?/ (a;- - 2xy + 6). 
 
 Examples.— xvii. 
 
 Resolve into factors : 
 
 1. 5x'"- 15a;. 
 
 2. 3r'' + 18x2-6.>;. 
 
 3. 49r-14?/ + 7. 
 
 4. 4^^(/-12a;2v2 + 8a;y^. 
 
 5. x!^ - ax^ -^hx^ + ex. 
 
 6. ZxSf - 2\xY + ^7xY- 
 
 7. 54a^l/^ + 108a%^ - 243(^^6''. 
 
 8. 4rixh/'^ - 90x^1/7 _ 360a;*j/8. 
 
i 
 
 •P"P 
 
 RESOLUTION- INTO FACTORS, 
 
 89. Case II. The next case in point of .-simplicity is thiit 
 in which four terms can be so arranged, that the first two have 
 a common factor and the last two have a common factor. 
 
 Thus 
 
 x^-\-ax-\-hx-\-ah={x^-irax)-'r(hx-{-ah) 
 
 = a; (a: + a) + & (ic + a) 
 = (x + h) {x-\- a). 
 Again 
 
 ac - ml - he +i)d — (ac ~ («I) - (he - hi) 
 = a{c-d)-h(c-d) 
 
 = {a~h) {c-d). 
 
 Examples. 
 
 Resolve into factors : 
 
 1. x'^-ax-hx + ah. 
 
 2. ah + ax - })x - x^. 
 
 3. hc + hy - ey - y'^. 
 
 4. hm + mil + ah + on. 
 
 -XVUl. 
 
 5 . a hx^ - cixy + hxy - y\ 
 
 6. ahx - ahy + cdx - cdy. 
 
 7. cdx'^ + dmxy - cnxy - mny^. 
 
 8. ahcx - h'^dx - acdy + hd^y. 
 
 90. Before reading the Articles that follow the student is 
 advised to turn hack to Art. 56, and to observe tlie manner in 
 which the operation of multiplying a binomial by a binomial 
 produces a trinomial in the Examples tlicre given. He will 
 then be ])repared to expect that in certain cases a trinomial 
 can he resolved into two binomial factors, exumplus of which we 
 shall now give. 
 
 91. Case III. To find the factors of 
 
 Our object is to find two numbers whose product is 12, 
 
 and whose sum is 7. 
 These will evidently be 4 and 3, 
 
 .-. x^ + Ix +12 = {x + 4) (a; 4- 3). 
 
 Again, to find the fiictors of 
 
 jk2 + 57).r + 6/A 
 Our object is to find two numbers whose product is ^h'^, 
 
 and ^^•hose sum is bh. 
 These will clearly be 3?) and 2A, 
 
 .-. X- 4- 5/At; + G//-2 = {x-\- 36) {x + 2/)) . 
 
RESOLUTION INTO FACTORS, 
 
 45 
 
 iity is that 
 , two have 
 :tor. 
 
 e student is 
 manner in 
 a l)inomial 
 
 n. Hg will 
 a trinomial 
 
 of which we 
 
 •t is 12, 
 u is T. 
 
 ■ct is 6/)2, 
 II is 5&. 
 
 Examples.— xix. 
 
 Resolve into factors : 
 
 x^+Ux + 30. 
 X'^+llx + QO. 
 7/2 + 13?/ +12. 
 7/2 + 21?/ + 110. 
 ?7i2 + 35m + 300. 
 w2 + 23m + 102. 
 a'^ + 9ab + 8lA 
 .'c2 + l3ma; + 36m2. 
 
 9- 
 
 lO. 
 
 II. 
 
 12. 
 14. 
 
 15- 
 
 16. 
 
 7/2+i9n7/ + 48?i2. 
 z- + 29j>.-;; + 100^2^ 
 :>j* + 5:c2 + 6. 
 
 xh/+l8xij + :i2. 
 x^i/ + 7x-^y-+ 12. 
 m^o+lOm^+16. 
 
 •);,- 
 
 ■27712 + 14022. 
 
 93. Case IV. To find the factoib of 
 
 !b2-9.c + 20. 
 
 Our object is to find two negative terms whose pro(hiot is 20, 
 
 and whose sum is -9. 
 These will clearly be - 5 and - 4, 
 
 .'. a;2 - 9x + 20 = (x - 5) (:/; - 4). 
 
 Ex AM PLES.— XX. 
 Resolve into factors : 
 
 I. 
 
 x2-7a:+10. 
 
 6. 
 
 7t2_57ri + 56. 
 
 2. 
 
 a;2-29a; + 190. 
 
 7- 
 
 x'^ -7x^ + 12. 
 
 3- 
 
 i/2-237/+132. 
 
 8. 
 
 a26^-27o6 + 2(). 
 
 4. 
 
 7/2 _ 30?/ f 200. 
 
 9- 
 
 Mc«-lli2c3 + 3o. 
 
 5. 
 
 7i2-43u + 460. 
 
 10. 
 
 rA/';;2-13a;7/;3 + 22. 
 
 92. Case V. To find the factors of 
 
 x^ + bx- 84. 
 
 Our object is to find two terms, one positive and one ncf'alive, 
 whose product is - 84, and whose sum is 5. 
 
 These are clearly 12 and - 7, 
 
 .-. x^ + 5.C - H4= (x + 12) {x - 7). 
 
46 
 
 RESOLUTION' INTO FACTORS. 
 
 EXAMPLES.—XXi. 
 
 Eesolve into factors : 
 I. a;2 + 7a;-60. 
 
 a-2+12a;-45. 
 a2+iia-12. 
 a2+l3a-140. 
 
 5. &2 + i35_3oo. 
 
 6. 62 + 256-150. 
 
 7. a;8 + 3a;*-4. 
 
 9. mi<^ + 15m^- 100. 
 
 10. ?i2+i7^j_39o. 
 
 94. Case VI. To find the factors of 
 
 a;2-3x-28. 
 
 Our object is to find two terms, one positive and one negative, 
 whose product is - 28, and v/hose sum is - 3. 
 
 These will clearly be 4 and - 7, 
 
 .-. «;2-3x-28 = (a; + 4)(aj-7). 
 
 I. 
 
 x^-bx- 66. 
 
 2. 
 
 a;2 - 7x - 18. 
 
 3- 
 
 ?>i2 - 9m - 36. 
 
 4- 
 
 n'^-Un-GO. 
 
 
 ^/- 131/ -14. 
 
 Examples.— xxii. 
 
 Resolve into factors ; 
 
 6. s;2 _ 15^ _ 100. 
 
 7. .T^o - 9x5 - 10. 
 
 8. cH^-24cd-im. 
 
 9. 'nl^?^2 _ ^,|,3jj _ 2. 
 
 10. /24-5j9Y'-84. 
 
 95. The results of the four preceding articles may be thus 
 stated in general terms : a trinomial of one of the forms 
 
 x^-\-ax + 6, aj2 - ax 4- 6, x^ + ax- 6, 7? -ax- 6, 
 
 may be resolved into two simple factors, when 6 can be re- 
 solved into two factors, such that their aum, in the first two 
 forms, or their difference, in the last two forms, is ec[ual to a. 
 
 96. We shall now give a set of Miscellaneous Examples on 
 the resolution into factors of expressions which come under 
 one or other of the cases already explained. 
 
RESOLUTION INTO FACTORS. 
 
 47 
 
 Examples. — xxiii. 
 
 Resolve into factors: 
 
 1. :c'^- 15:'J + 3G. 
 
 2. a;'^ + 4a:-45. 
 
 4. x^ - Zmj} - lOyjt-. 
 • 5- ?y^ + 7/3-00. 
 
 6. x^ 
 
 cc^ 
 
 110. 
 
 8. 
 
 .''J- 4- mx ■\-nx-\- mn. 
 
 9- 
 
 if-4i/ + ^. 
 
 10. 
 
 x'y - abx - cxij + ahc. 
 
 II. 
 
 a;2 + (rt - h) X - (th. 
 
 12. 
 
 x^-{c- d) X - cd. 
 
 13- 
 
 ah^ - hd + cd - ahc. 
 
 14. 
 
 4.<;2-28a;?/ + 48/. 
 
 7. x' + Sarc^ + 4a2x. 
 
 97. We liaA'e said, Art. 45, that when a nnniber is multi- 
 plied by itself the result is called the Square of the numbi-r, 
 and that the figure 2 placed over a number on the right hand 
 indicates that the number is multiplied by itself. 
 
 Thus «2 is called the square of a, 
 (c - ?/)- is called the square of x-y. 
 
 The Square Root of a given number is that number 
 Avhose square is equal to the given number. 
 
 Thus the square root of 49 is 7, because the square of 7 
 is 49. 
 
 rt' 
 
 So also the square root of a^ is a, because the square of a is 
 ' : and the square root of {x - ijY i** ^ ~ 2/; l>ecanse the s([uaro 
 of oj - 2/ is (x - yf. 
 
 The symbol sj placed before a number denotes that the 
 .s(|uare root of that nn mber is to be taken : thus ^J2h is read 
 ^Uhe sqiuire root of 2b." 
 
 Note. The square root of a positive quant" may be either 
 
 positive or negative. For 
 
 « 
 
 since a nmlti plied by a gives as a result a'^, 
 and - a multiplied by - a gives as a result a% 
 
 it follows, from our definition of a Squaie Hoot, that either a 
 or - a may be regarded as the square root of a'^. 
 
 But throughout this chapter we shall take only the 'positive 
 value of the square root. 
 
48 
 
 RESOLUTION INTO FACTORS. 
 
 I j 
 
 98. We may now take the case of Trinomials which are 
 'perfect squares, which are really included in the cases dis- 
 cussed in Arts. 91, 92, but which, from the importance they 
 assume in a later part of our subject, demand a separate con- 
 sideration. 
 
 99. Case VII. To find the factors of 
 
 a;2+ 12^ + 36. 
 
 Seeking for the factors according to the hints given in Art. 
 91, we find them to be x + 6 and a; + 6. 
 
 That is x^ + l'2x-\-ZQ = (x + 6)2. 
 
 EXAMPLES.— XXiV. 
 
 Resolve into factors • 
 
 1. (iy^ + 18x + 8l. 
 
 2. a;2 + 26x + I69. 
 
 3. a;2 + 34a; + 289. 
 
 4. r f 2i/+l. 
 
 5. «2 + 200;3 + 10000. 
 
 6. .^4^.14,^2^.49^ 
 
 7. .^•2 + lOxy + 25yl 
 
 8. m4+16m%2 + 64?i4 
 
 9. xf^ + 24x3 ^ 144^ 
 
 10. a;Y + 162x1/ -K 6561. 
 
 100. Case YI II. To find the factors of 
 
 x--l2x + 36. 
 
 Seeking for the factors according to the hints given in Art. 
 92, we find them to be x - 6 and x - 6. 
 
 That is, x2 _ i2x + 36 -- (x - 6)2. 
 
 if! 
 
 Examples.— XXV. 
 
 Resolve into factors : 
 I. X--8X+16. 2. X--2SX + 19C). 3. rr2_36x + 324. 
 
 4. i/2 - 40?/ + 400. 5. ^2 - 100^ + 2500. 6. ^r*- 22x2+ 121. 
 
 7. x2 - 30xj/ + 226?/. 8. m^~'S2mht^ + 2mn\ 
 
 9. x«- 38x3 + 361. 
 
 I : 
 
RESOLUTION INTO FACTORS'. 
 
 49 
 
 101. Case IX, "We now proceed to the movst important 
 case of Resolution into Factors, namely, that in which the ex- 
 pression to be resolved can be put in the form of two squares 
 with a negative sign between them. , 
 
 Since 
 
 m 
 
 ,2 — '17,2 __ 
 
 n^=(m + n) (m - n), 
 
 we can express the difference between the squares of two 
 (piantities by the product of two factors, determined by the 
 following method : 
 
 Take the square root of the first quantity, and the square 
 root of the second quantity. 
 
 The sum of the results will form the first factor. 
 
 The difference of the results will form the second factor. 
 
 For example, let a- - Ir be the given expression. 
 The square root of a? is a,. 
 The square root of W' is h. 
 The sum of the results is a + 6. 
 The difference of the results is a - 6. 
 
 The factors will therefore be a + 6 and a - 6, 
 that is, a^ - Ir = {a + b){a- b). 
 
 102. The same method holds good with respect to com- 
 pound quantities. 
 
 Thus, let a^ - {b - c)- be the given expression. 
 The square root of the first term is a. 
 The square root of the second term is & - c. 
 The sum of the results is a + b-c. 
 The difference of the results is a — b + c. 
 
 a' 
 
 (b - c)- = {a + b-c){a-b + c). 
 
 Again, let (a - b)'^ -(c-d)'^ be the given expression. 
 The square root of the first term is a-b. 
 The square root of the second term is c-d. 
 The sum of the results is a-b + c~d. 
 The difference of the results is a-b-c + cf. 
 :. {a-by-{c-df = {a-b + c-d){a-b-c + d). 
 
50 
 
 RESOLUTION INTO FACTORS. 
 
 103. The terms of an expression may often be arranged 
 30 as to form two squares witli the negative sign between 
 them, and then uie cvpiessluu can be resolved into factors. 
 
 Thus a^ + V^-c^-d^ + ^ah + ^cd 
 
 = {a^ + 2ah + h'')-((-'--2cd + d'-) 
 
 = {a + hf-{c-dy 
 
 = {a + h + c-d){a + h- c + d). 
 
 Examples. — xxvi. 
 
 Resolve into two or ni(jre factor 
 
 I. x^-y^. 2. X 
 
 4. a'^-x'^. 5. 
 
 7. x^-1. ■ 8. 
 
 x^ 
 
 m: 
 
 9. 
 
 16. 
 
 3. 4/;2-2l 
 6. a"-l. 
 
 ,A)t 
 
 49,- 
 
 10. ^\xhf--l2\a%'K)^i. (a-hy 
 
 13. {a + hy-{c + dy. 
 
 14. {x + yy-{x-yy. 
 
 15. x'^ - 2xy + y^ - z^. 
 
 16. {a - hy^ - {m + nyK 
 
 12. x'^ -(iii-ny 
 
 24. 2xy-x^-y-+l. 
 
 _ 1/2 _ /■/2 
 
 25. x^~2yz-y 
 
 26. a2-4^^^-9(,'-^+126(;. 
 
 27. «*- IQlfi. 
 17. «2 _ 2(/c + c2 - //^ _ 2Z»t? - d-. 28.1- 40^2. 
 
 18. 2hc-~V^-L^ + dK 
 
 19. 2a;i/ + a;2 + 7/^ - Ti'^ 
 
 29. (r + Ifi -c'^-d^- 2ah- 2cd 
 ^^o. a^ -¥ + c' - d^ - 2ac + 2bd. 
 
 20. ^mn-ini^ -'n?-\-a?-\-h'^-2ah. 31. 3a'^.c^-27 
 
 aa;. 
 
 21. (ax + />7/)2-l. 
 
 22. {ax + hy)'^-{ax-hyy 
 2-v l-a2_^j2 + 2((6. 
 
 •^2. rt 
 
 ■•//- 
 
 33. (5x- 2)'^ -(..;- 4)2. 
 
 34. {7x + 4yy-(2x + 3yy 
 
 35. (753)2 -(247)- 
 
 104. Case X. Since 
 
 x-^ + a 
 
 ■'-a3 
 
 ^ = x^ -ax + a^, an<l — — - — x^ + ax + d^ (Art. 83) 
 
 a' + (t 
 
 a; -a 
 
 we know the following important facts 
 
RESOLUTION' rXTO FACTORS. 
 
 51 
 
 (1) Tlio svm of the cuhes of two uumbera is divisible, by 
 the s?tm of tbo inmibors : 
 
 (2) The difference between tlie cvhesi of two iiui^ibers is 
 divisible by tlie difference between the numbers. 
 
 Hence we may resolve into factors expressions in the form 
 of the sum or dilference of the cubes of two numbers. 
 
 Thus u;34-27 = a;3 + 33 = (x + 3)(x'-i-3x + 9) 
 
 2/3 _ 64 = ^3 _ 43 = (y _ 4) ^y'l ^ 4^ ^ 1(3)^ 
 
 Examples.— xxvii. 
 
 Express in factors the followiu;^' expressions : 
 
 T. a^-vh\ 
 
 2. (v" IP. 
 
 a^ - 8. 
 
 /*^-125. 6. x^ + 64ir. 
 
 4. ^'^ + 343. 
 7. a^-216. 8. Rc^ + 27if. 
 9. ()4(i^ - 100061 10. 7'29x^ + 6l2ijK 
 
 Express in four factors each of the following expressions : 
 II. co" - y'''. 1 2. a:« - 1. 1 3. ct« - 64. 1 4. 729 - if. 
 
 105. Before we proceed to describe other processes in 
 Alj^^ebra, we shall give a series of examples in illustration of 
 the principles already laid down. 
 
 The student will tind it of a<lvantage to work every example 
 in the following series, and to accustom himself to read and to 
 explain with facility those examples, in which illustrations are 
 given of what may be called tlie short-hand method of expressing 
 Arithmetical calculations by the symbols of Algebra. 
 
 Examples.— xxviii. 
 
 1. Express the sum of « and 6. 
 
 2. Interpret the expression «-6 + c. 
 
 3. How do you express the double of 03 ? 
 
 4. By how much is a greater than 5 ? 
 
 5. If a; be a whole number, what is the number next 
 above it ? 
 
 6. Write five numbers in order of magnitude, so that x 
 t-hall be the third of the five. 
 
 ONTARIO OOLLtGt Ut feDUUAIlOK 
 
52 
 
 RESOLUTION INTO FACTORS. 
 
 \ 
 
 7. If a be multiplied into zero, wliat is the result ? 
 
 8. If zero be divided by x, what is the result ? 
 
 9. AVhat is the sum of a-\-a-\-a ... written d times % 
 
 10. If the product be ac :ind the multiplier a, what is th« 
 multiplicand ? 
 
 1 1. What number taken from x gives 1/ as a remainder ? 
 
 12. ^ is a: years old, and B is ?/ years old ; how old was A 
 when B was born ? "^ ■ • 
 
 13. A man works every 'lay on week-days for x week:^ iii 
 the year, and during the remaining weeks in the year he does 
 not work at all. During hoAv many days does he rest ? 
 
 14. There are x boats in a race. Five are bumped. How 
 many row over the course ? 
 
 15. A merchant begins trading with a capital of x pounds. 
 He gains a pounds each year. How much capital has he at 
 the end of 5 years ? 
 
 16. A and B sit down to play at cards. A has x shillings 
 and By shillings at first. A wins 5 shillings. How much has 
 each when they cease to play ? 
 
 17. There are 5 brothers in a family. The age of the eldest 
 is X years. Each brother is 2 years younger than the one next 
 above him in age. How old is the youngest ? 
 
 18. I travel x hours at the rate of ij miles an hour. How 
 many miles do I travel ? 
 
 19. From a rod 12 inches long I cut olf x inches, and then 
 I cut off y inches of the remainder. How many inches are 
 left? 
 
 20. If n men can dig a piece of ground in (/ hours, how 
 many hours will one man take to dig it ? 
 
 21. By how much does 25 exceed xl 
 
 22. By how much does y exceed 25 ? 
 
 23. If a product has 2?)i repeated 8 times as a factor, how 
 do you express the product '? 
 
 24. By how much does a + 2& exceed ti - 2ft ? 
 
 2$. A girl is X years of age, how old was she 5 years since ? 
 
RESOLUTION INTO FACTORS, 
 
 53 
 
 How 
 
 26. A boy is ij years of age, how old will he he 7 years 
 hence ? . ^ 
 
 27. Express the difterencc between the scjuares of two 
 numbers. 
 
 28. Express the product arising from the multiplication of 
 the sum of two numbers into the difference between the same 
 numbers. 
 
 39 
 30 
 
 31 
 
 32 
 33 
 34 
 35 
 
 What value of x will make 8x equal to 16 ? 
 What value of x will make 28a; equal to 56 ? 
 
 What value of x will make „ equal to 4 ? 
 
 What Vcilue of x will make a; + 2 equal to 9 ? 
 What value of x will make £C - 7 equal to 16 ? 
 What value of x will make ic- + 9 equal to 34 ? 
 What value of x will make x^-% equal to 92 ? 
 
 How 
 
 Examples.— xxix. 
 
 Explain the operations symbolized in the following expres- 
 eions : - 
 
 I. a + &. 2. a2-62. 3. 4ft2 + 63, 4. 4(^2 + 52), 
 
 5. a2-26 + 3c. 6. a + mx&-r. 7. ((t + m)(6-c). 8. V^ 
 
 9. ^Jx^+y^. 10. rt + 2(3-c). II. (a + 2)(3-c). 
 
 a2 + 62 
 
 12. 
 
 4a6 * 
 
 13- 
 
 x-y 
 
 14 
 
 ^x + y 
 
 Examples.— XXX. 
 
 If a stands for 6, 6 for 5, x for 4, and y for 3, find the value 
 r "''^. following expressions : 
 
 I. a rx-b~y. 2. a + y -h-x. 3. 3a + 4?/ - 6 - 2ic, 
 
 4. 3(fH- ?) - 2(x - 1/). 5. {a + x){b-y). 6. 2(i + 3(. +?/). 
 
 . 62 + 7/ 
 
 7. {2a + 3){x + y). 
 10. a6x. 
 
 8. 2a + 3.t; + 7/. 
 II. ah{x + y). 
 
 a-x 
 12. a^(6 + a;)2. 
 
H' 
 
 kESOLUTJON INTO FACTOkS. 
 
 13. nh{x-yf. 
 16. {Jof. 
 
 1 9. ^±uxy. 
 
 14. Jbh. 
 
 17. (V^ + 7>)--. 
 a^ + h^^ + y 
 
 20, 
 
 15. xV. 
 1 8. sj^hx. 
 
 21. 3rt + (2x-2/)2. 
 
 X + 7/ + 3* 
 
 22. \ft-(h-y)\\a-{x-y)\. 24. 3 (a +- 6 - t/)-'' + 4 (rt + a;)*. 
 
 23. (a-6-7/)- + (rt-a; + 2/)2. 25. 3 (a - 6)2 + (4x - 7/)-'. 
 
 ft 
 
 Examples. — xxxi. 
 
 1. Find the value of 
 
 Zahc - a^ + ¥ + c^, when a = 3,h = 2, c = 1„ 
 
 2. Find the value of 
 
 aj3 ^ ^3 ._ ^^3 ^ •;ixyz, when r = 3, 7/ = 2, s = 5. 
 
 3. Subtract a^ + c^ from (a + cf. 
 
 4. Subtract (x - y^ from a:^ 4. ^2^ 
 
 5. Find the coefficient of x in the expression 
 
 {a + byx-{a + bx)K 
 
 6. Find the continued product of 
 
 2x - m, 2ic + n, a; + 2m, x-2n. 
 
 7. Divide 
 
 ttcr^ + (l)c + ad) r"^ + (hd + ae) r + he by ar + fe ; 
 and test your result by putting 
 
 a = h = c=^d = e = \, and r = 10. 
 
 8. Obtain the product of the four factors 
 
 (a + 6 + c), (& + c - «), (c + a - 6), {a + h- c). 
 
 AVhat does this become when c is zero; when b + c = a; 
 when a = h = c'l 
 
 9. Find the value of 
 
 (a + h) {h + c)-{c + d){d + a) - (a + c){h- cI), 
 where h is cf|ual to d. 
 
 10. Find the value of 
 
 3(1 + (2& - c^ + I (;2 - (2(t + 37>) J + ySc - (2a + Sh) \ -, 
 when (( — (). 6 = 2, c = 4. 
 
RESOL UTION INTO FA C TORS, 
 
 55 
 
 11. If « = 1, & = 2, c = 3, (Z = 4, shew that the numerical 
 values are etjiuil of 
 
 |,;_(c-7, + a,)jj(r? + c)-(6 + a.)|, 
 
 and of (r-'-(c*'^ + 6-) + a"^ + 2(7;c-«fO- 
 
 12. Bracket together the different powers of x in the follow- 
 ing expressions : 
 
 (a) rta;2 + &a;2 + ex + r/x. 
 
 (7) 4a;'' - aj? - ?ix" - hx^ - b.c - ex. 
 
 (S) {ct + it')''-{h-x)\ 
 
 (e) {mx'^ + qx + \f-{nx''- + rix+\f. 
 
 13. Multiply the three factors x-a, x-h, x-c together, 
 and arrange the product according to descending powers of x. 
 
 14. Find the continued product of {x + a) (x + h){x + c). 
 
 15. Find the cube of a + b + c; thence without further 
 multiplication the cubes ofa + 6-c; h + c -a; c + a-b; and 
 subtract the sum of these three cul)es from the first. 
 
 16. Find the product of (3a + 26) (3a 4- 2c - 36). and test the 
 result by making a= 1, 6==c = 3. 
 
 17. Find the continued product of 
 
 a-x, a + x, a^ + x-, a'^ + x^, a^ 4- a^. 
 
 1 8. Subtract (6 -- a) (c - d) from (a -b){c-d). 
 
 AVhat is the value of the result when a = 26 and d = 2c 1 
 
 19. Add together (6 + y){a + x), x-y, ax - by, and a(x + y). 
 
 20. What value of x will make the difference between 
 
 (./: + 1) (x + 2) and (x - 1) (x - 2) ecpuil to 54 i 
 
 2 1 . Add together ax -by, x-y, x{x-y), and (a - x) (6 - y\ 
 
 22. What value of x will make the difference between 
 (2x + 4) (3x + 4) and (3x - 2) (2x - 8) equal to 96 ? 
 
 23. Add together 
 
 2mx - 3?i7/, x + y, 4(7n, + n) (x - y), and mx + vy. 
 
 24. Prove that 
 
 (x + ?/ + rj)2 + x2 + ^/ + ;^2=(x + '//)2 + (?/ + 2;)2 + (a; + ^)2. 
 
w 
 
 H 
 
 !i 
 
 56 
 
 RESOLUTION INTO FACTORS. 
 
 25. Finrl thn proflnct of {Za + 3&) (2a + 3c - 2ft), and test the 
 result by making a = ], /) = 4, c! = 2. 
 
 26. ir (I, h, c, d, e ... denote 9, 7, 5, 3, I, find the values of 
 ^^^'- ; (6c - ad) (bd - ce) ; --^-^■' ; and d" - c\ 
 
 27. Find the value of 
 
 3ahc - a^ + //* + c^ when a = 0, 6 = 2, c = 1 . 
 
 22>. Find the value of . ■' 
 
 7>(t'^ 
 
 2ab'^ 
 
 - v^ when (1 = 4, 6 = 1, c = 2. 
 
 c 6^ 
 
 29. Find the value of 
 
 (a-6-c)2 + (6-a-t;)2 + (c-a-6)2 whena = l, ?> = 2, c = 3. 
 
 30. Find the value of 
 
 {a + h - cf + {a -h + 11)'^ + {b + c - ay when a. = l, 6 = 2, c = 4. 
 
 31. Find the value of 
 
 (a + 6)2 + (6 + c)'^ + {c + a)2 when a= - 1, 6 -. 2, c = - 3. 
 
 32. Shew that if the sum of any two numhers divide the 
 difference of their squares, the quotient is equal to the differ- 
 ence of the two numbers. 
 
 33. Shew that the product of the sum and difference of any 
 two numbers is equal to the difference of their squares. 
 
 34. Shew that the square of the sum of any two consecu- 
 tive integers is always greater by one than four times their 
 product. 
 
 35. Shew that the square of the sum of any two consecutive 
 even whole numbers is four times the square of the odd number 
 between them. 
 
 36. If the number 2 be divided into any two parts, the 
 difference of their squares will always be equal to twice the 
 difference of the parts. 
 
 37. If the number 50 be divided into any two parts, the 
 difference of their squares will always be equal to 50 times the 
 difference of the parts. 
 
 38. If a number n be divided into any two parts, the 
 difference of their squares will always be equal to n times the 
 difference of the parts. 
 
ON SIMPLE EQUATIONS. 
 
 A 
 
 39. If two numlxTs differ by a unit, tlicir product, togetlu?r 
 with the Huui of their squares, is equal to the ditfereiice of the 
 cubes of tlie numbers. 
 
 40. Shew tliat the sum of \\\k\ cubes of auy three consecu- 
 tive whole numbers is divisible l)y three times the middle 
 number. 
 
 !! 
 
 < 'I 
 
 VI, ON SIMPLE EQUATIONS. 
 
 106. An Equation is a statement that two expressions 
 are equal. 
 
 107. An Identical Equation is a statement that two ex- 
 pressions are equal for all numerical values that can be given 
 to the letters involved in them, provided that the same value 
 be given to the same letter in every part of the equation. 
 
 Thus, (jc + a)''^ = ic2 + 2aa; + a2 
 
 is an Identical Equation. 
 
 108. An Equation of Condition is a statement that two 
 expressions are equal for some particular numerical value or 
 values that can be given to the letters involved. 
 
 Thus, ic+l = 6 
 
 is an Equation of Condition, the only number which x can 
 represent consistently with this ec^uation being 5. 
 
 It is of such equations that we have to treat. 
 
 109. The Root of an Equation is that number which, when 
 l)ut in the place of the unknown quantity, makes both sides of 
 the equation identical. 
 
 110. The Solution of an Equation is the process of find- 
 ing what number an unknown letter must stand for that the 
 equation may be true : in other words, it is the method of 
 finding the Root. 
 
 The letters that stand for unknoiun numbers are usually 
 X, y, z, but the student must observe that any letter may 
 stand for an unknown number. 
 
 111. A Simple Equation is one which contains the 
 jlrst 'power only of an unknown quantity. This is also called 
 an Equation of the First Degree, 
 
 
58 
 
 ON SIMPLE EQUATIONS. 
 
 J I 
 
 11 2. Tlie followiiif^ Axioms form the gioimiwork of the 
 Roliition of all w|natioiiB. 
 
 Ax. I. If equal quantities be added to equal quantities, 
 the sums Vill be e(jual. 
 
 Thus, if <:t = J, 
 
 a + c = & + c\ 
 
 Ax. II. If equal quantities be taken from eoual quantitiei*. 
 the remainders will be equal. 
 
 Thus, if 
 
 
 Ax. III. If e(|ual ([uantities be multipUed by equal (pian- 
 lities, the products will be equal. 
 
 rp 
 
 x^hus,if 
 
 (1=0, 
 
 7iia = mh. 
 
 Ax. IV. If e([Ual quantities be divided by equal quantities, 
 the quotients will be equal. 
 
 Thus, if xij = xz, 
 
 y = z. 
 
 ^ 113. On Axioms I. and II. is founded a process of great 
 z utility in the solution of equations, called The TiiANsrosiTiox 
 
 OF Terms from one side of the ecpiation '<;■ the othei', which 
 may be tlius stated : 
 
 \ "Any term of an equation may be tran>*l'<^rred from one side 
 of the equation to the other ij %l8 s iyn be dianqcd^ ' 
 
 For let x-a = 'b. 
 
 Then, by Ax. I., if we add a to both '.ides, the sides remain 
 equal : 
 
 therefore x-a-^a^=h + a, 
 
 that is, t x^h + a. 
 
 ^/ Again, let x + c — d. 
 
 Then, by Ax. II., if wo subtract c Ironi etf'»,h side, the sides 
 remain equal : 
 
 th eri'f ore x + c-c=-d-c, 
 
 that is, x = d-x, 
 
ON SIMPLE EQUATIONS. 
 
 m 
 
 II 
 
 M 
 
 0111 one 
 
 uIgs remain 
 
 114. We may change all the signs of each side of an etj^iia- 
 tion without altering the equality. 
 
 Thus, if a-x = h-c, 
 
 x-a=c-b. 
 
 115. "We may change the position of the tvo sides of the 
 equation, ^eaving the signs unchanged. 
 
 Thus the equation a-h = x-c, may be written thus, 
 
 x-c = a-h. 
 
 116. We may now proceed to our first rule for the solution 
 of a Simple Ejiiation. 
 
 Rule I. Transpose the known terms to the right hand side 
 of the equation and the unknown terms to the other, and com- 
 Itine all the terms on each side as far as possible. 
 
 Then divide both sides of the equation by the coefficient of 
 the unknown quantity. 
 
 This rule ve shall now illustrate by examples, in which x 
 stands for the unknown quantity. - - 
 
 Ex. 1. To solve the equation, 
 
 5x-6 = 3x + 2. 
 Transposing the terms, we get 
 
 5a;-3x = 24-6. 
 Combining like terms, we get 
 
 2a; = 8. 
 Dividing both sides of this equation by 2, we get 
 
 03 = 4, * 
 
 and the value of x is determined. 
 
 Ex. 2. To solve the equation, 
 
 7x1- 4 = 250; -32. 
 Transposing the terms, we get 
 
 7x-25x= -32-4. 
 Combining like terms, we get 
 
 -18a:=-36. 
 Changing the signs on each side, we get 
 
 18x = 3G. 
 
 Dividing both sides by 18, we get 
 
 x = 2, 
 and the value of x is determined. 
 
 /lit 
 
I 
 
 60 
 
 ON SIMPLE EQUATIONS. 
 
 Ex. 3. 
 
 tliat is, 
 
 or, 
 
 therefore, 
 
 Ex, 4. 
 
 that is, 
 or, 
 
 or, 
 therefore, 
 
 Ex. 5. 
 
 that is, 
 
 or, 
 
 or, 
 
 or, 
 
 therefore. 
 
 To solve the equation, 
 
 2a; - 3a; + 120 = 4a: - 6a: 4- 1 32. 
 2x - 3a; - 4a; + 6x = 1 3:4 - I'ztr, 
 
 8x-7a;==I2, , 
 a;=12. 
 
 To solve the equatiGUf 
 
 3a; + 5-8(13-rc) = 0, 
 
 3a; + 5-104 + 8x--0, 
 
 3x-l-8a;=104-5, 
 
 llx = 99, 
 
 x=d. 
 
 To solve the equationy 
 
 6a;-2(4-3a;) = 7-3(l7--iif^ 
 
 6f«;-8 + 6a:=7-r)l+3a-, 
 
 6a; + 6x-3a; = 7-5H-8, 
 
 12x-3.c=15-r)l, 
 
 9a; =-36, 
 
 x= -4. 
 
 EXAMPLES.— XXXll. 
 
 1. 7a; + 5=5a; + ll. 
 
 2. 12a; + 7 = 8a; + 15. 
 
 3. 236x + 425 = 97a; + 564. 
 
 4. 5a;-7 = 3a; + 7. 
 
 5. 12a;-9 = 8a;-l. 
 
 6. 124;B + 19 = 112a; + 43. 
 
 7. l8-2a; = 27-5a;. 
 
 8. 125-7x=145-12x. 
 
 9. 26 -8a; = 80 -14a;. 
 
 10. 133-3a;=:(;-83. 
 
 11. 13-3a;=5x- .5. 
 
 12. 127 + 9a;=12a;+100. 
 
 13. 15-5a;=6-4a;. 
 
 14. 3.r-22 = 7x-^6. 
 
 15. 8 + 4,(j = 12;c-16. 
 
 16. 50 - (3a; - 7) = 4x - (6a; - 35). 
 17. 6a; - 2(9 - 4a;) + 3 (5a; - 7) = 10a; - (4 + 16,r) + 35. 
 
 t8. 9a;-3(5a;-6) + 30 = 0. 
 
 19. 12a; - 5 (9a; + 3) + 6(7 - Sx) + 783 = 0. 
 
 20. a;-7(4a;-ll) = 14(a;-5)-19(8-;r>~t:. 
 
 21. ra;-l-7)(a;-3) = (a;-5)(a;-15). 
 
PROBLEMS LEADING TO SIMPLE EQUATIONS. 6l 
 
 22. (.x-8)(ic + 12) = (cc + l)(a;-6). 
 
 23. {% - 2)(7 - 3J) + (x - 5) (x + 3) - ^{x, - 1) + 12 =0. 
 
 24. (2a: -7) (a; + 5) = (9 -2^;) (4 -a;) + 229. 
 
 25. (7-6a;)(3-2a;) = (4.x--3)(3x-2). 
 
 26. 14 - x - 5 (u; - 3)(x + 2) + (5 - x) (4 - 5x) = 45x - 76. 
 
 27. (a; + 5)2-(4-a;)2=21x. 
 
 28. 5(cc-2)2 + 7(x-3)2 = (3x-7)(4a;-19) + 42. 
 
 29. (3x - 1 7)2 + (4a; - 25)2 - (5^ _ 29)2 = 1 . 
 
 30. (x + 5)(a;-9) + (x + 10)(a;-8) = (2x + 3)(x-7)-113. 
 
 T 
 
 YII. PROBLEMS LEADING TO SIMPLE 
 
 EQUATIONS. 
 
 117. "When we have ca question to resolve by means of 
 Algebra, we represent the number sought by an unknown 
 symbol, and then consider in what manner the conditions of 
 the question enable us to assert tliat tivo eaypressioiis are equal. 
 Thus we obtain an equation, and by resolving it we determine 
 the value of the number sought. 
 
 The whole difficulty connected with the solution of Alge- 
 braical Problems lies in the determination from the conditions 
 of the (question of tivo different exjjressions having the same 
 numerical value. 
 
 To explain this let us take the following Problem : 
 
 Find a number such that if 15 be added to it, twice the sum 
 will be equal to 44. 
 
 Let X represent the number. 
 
 Then a; + 15 will represent the number increased by 15, 
 and 2(x + 15) will represent twice the sum. 
 
 But 44 will represent twice the sum, 
 therefore 2 (ic + 15) = 44. 
 
 Hence - 2a; + 30 = 44, 
 
 that is, V 2x=14, 
 
 or, x = 7, 
 
 and therefore the number sought is 7. 
 
 :it\ 
 
62 P/?OBZEMS LEADING TO SIMPLE EQUATIONS. 
 
 m 
 
 118. We shall now give a series of Easy Problems, in 
 ■which the conditions by which an equality between two expres- 
 sions can be asserted may be readily seen. The student should 
 be thoroughly familiar with the Examples in set xxviii, the use 
 of which he will now find. 
 
 We shall insert some notes to explain the method of repre- 
 senting quantities by algebraic symbols in cases where some 
 difficulty may arise. 
 
 EXAMPLES.— xxxiii. 
 
 1. To the double of a certain number I add 14 and obtain 
 as a result 154. What is the number ? 
 
 2. To four times a certain number I add 16 and obtain as 
 a result 188. What is the number ] 
 
 3. By adding 46 to a certain number I obtain as a result a 
 number three times as large as the original number. Find the 
 Original number. 
 
 4. One number is three times as large as another. If I 
 take the smaller from 16 and the greater from 30, the remain- 
 ders are equal. What are the numbers ? 
 
 5. Divide the number 92 into four parts, such that the first 
 is greater than the second by 10, greater than the third by 18, 
 and greater than the fourth by 24. 
 
 6. The sum of twD numbers is 20, and if tliree times the 
 smaller number be added to five times the greater, the sum is 
 84. What are the numbers ? 
 
 7. The joint ages of a father and his son are 80 years. If 
 the age of the son were doubled he would be 10 years older 
 than his father. What is the age of eaCii ? 
 
 8. A man has six sons, each 4 years older than the one 
 next to him. The eldest is three times as old as the youngest. 
 What is the age of each? 
 
 9. Add ^24 to a certain sum, and the amount will be as 
 much above ^80 as the sum is bslow ^80. What is the sum ? 
 
 10. Thirty yards of cloth and forty yards' of silk together 
 cost ^66, and the silk is twice as valuable as <he cloth. Find 
 the cost of a yard of each. 
 
 22. 
 
4TI0NS. 
 
 roblems, in 
 two expres- 
 dent should 
 :viii, the use 
 
 od of repre- 
 where some 
 
 PROBLEMS LEADING TO SIMPLE EQUATIONS. 
 
 6.^ 
 
 i! 
 
 4 and obtain 
 
 nd obtain as 
 
 as a result a 
 ir. Find the 
 
 lother. If I 
 , the remain- 
 that the first 
 third by 18, 
 
 ee times the 
 r, the sum is 
 
 JO years. If 
 years older 
 
 Ihan the one 
 [he youngest. 
 
 it will be as 
 is the sum \ 
 
 Isilk together 
 Icloth. Find 
 
 11. Find the number, the double of which being added to 
 24 the result is as much above 80 as the number itself is below 
 100. 
 
 12. The sum of ,£500 is divided between A, B, C and D. 
 A and B have together ^280, A and G £260, A and I) £'220. 
 How much does each receive ? 
 
 13. In a company of 266 persons, composed of men, women, 
 and children, there are twice as many men as there are women, 
 and twice as niany women as there are children. How many 
 are there of each i 
 
 14. Divide £1520 between A, B and C, so that A has £100 
 less than B, and B £270 less than C. 
 
 15. Find two numbers, differing by 8, such that four times 
 the less may exceed twice the greater by 10. 
 
 16. A and B began to play with equal sums. A won £5, 
 and then three times yl's money was equal to eleven times J5's 
 money. What had each at first ? 
 
 17. A is 58 years older than B, and ^'s age is as much 
 above 60 as B'a age is below 50. Find the age of each. 
 
 18. A is 34 years older than B, and A is as much above 50 
 as /j is below 40. Find the age of .each. 
 
 19. A man leaves his property, amounting to £7500, to be 
 divided between his wife, his two sons and his three daughters, 
 as follows : a son is to have twice as much as a daughter, and 
 the wife £500 more than all the five children together. How 
 much did each get ? 
 
 20. A vessel containing some water was filled up by pour- 
 ing in 42 gallons, and there was then in the vessel 7 times as 
 much as at first. How many gallon? did the vessel hold 1 
 
 21. Three persons, A, B, C, iiave £76. B has £10 more 
 than J., and G has as much as A and B together. How much 
 has each ? 
 
 22. What two numbers are those whose difterence is 14, 
 and their sum 48 i 
 
 23. A and B play at cards. A has £72 and B lias £52 
 when they begin. When they cease playing, A has three times 
 as much as B. How much did A win i 
 
 f !■ 
 

 I 
 
 I 
 
 j 
 
 
 64 PROBLEMS LEADING TO SIMPLE EQUATIONS. 
 
 Note I. If we have to express algebraically two parts into 
 which a given nur.iber, suppose 50, is divided, and we repre- 
 sent one of the parts by x, the other will be represented by 
 50 -X. 
 
 Ex. Divide 50 into two such parts that the double of one 
 part may be three times as great as the other part. 
 
 Let X represent one of the parts. 
 
 Then 50 - a; will represent the other part. ' 
 
 Now the double of the first part will be represented by 
 2x, and tliree times the second part will be represented by 
 3 (50-a;). 
 
 Hence 2x=3(50-u;), 
 
 or, 2a;=lo0 -3a;, 
 
 or, 5fl3=150; 
 
 .'. x = 3a 
 
 Hence the parts are 30 and 20, 
 
 24. Divide 84 into two such parts that three times one i^art 
 may be equal to four times the other. 
 
 25. Divide 90 into two such parts that four times one part 
 may be equal to five times the other. 
 
 26. Divide 60 into two such parts that one pan is greater 
 than the other by 24. 
 
 27. Divide 84 into two such parts that one part is less than 
 the other by 36. 
 
 28. Divide 20 into two such parts that if three times onu 
 part be added to five times the other part the sum may be 84. 
 
 Note II. When we have to compare the ages of two per- 
 sons at one time and also some years after or before, we must 
 be careful to remember that hoth will be so many years older 
 or younger. 
 
 Thus if X be the age of A at the present time, and 2a; be 
 the age of B at the present time, 
 
 The age of -4 5 years hence will be a; + 5, 
 and the age of B 5 years hence will be 2a; + 5. 
 
 PROP. 
 
 Ex. 
 
 only be t 
 B at the ] 
 
 Let X r 
 
 Then 5 
 
 Now X- 
 and 5s 
 
 Hence 
 
 or 
 or 
 
 Hence ^ 
 
 29. A 
 
 times as 0! 
 
 30. A] 
 
 years will 
 
 31. A ] 
 
 [ times as ol 
 
 32. A ] 
 be only tw 
 
 33. A ] 
 
 [joint ages ( 
 tlie joint a 
 1 uncle % 
 
 Note II 
 
 I after assun 
 
 quantities, 
 
 h'n t]iQ same 
 
 I the sums ir 
 
 Ex. A 
 
 [pences, and 
 lis 78. HoA 
 
 [S.A.] 
 
TIONS. 
 
 ) parts into 
 (1 wc repre- 
 resented by 
 
 uble of one 
 
 PROBLEMS LEADING TO SIMPLE EQUATIONS, 6$ 
 
 
 resented by 
 resented by 
 
 inies one i^art 
 
 mes one part 
 
 in is greater 
 
 is less than 
 
 [•ee times onu 
 may be 84. 
 
 s of two per- 
 ure, we must 
 years older 
 
 le, and 2x be 
 
 Ex. A is 5 times ns old as 7>, and 5 years hence A will 
 only be three tinies as old as B, What are the ac^^'S ol' A and 
 B at the present time ? 
 
 Let X represent the age of 7?. 
 
 Then 5x will represent the age of ^. 
 
 Now CC4-5 will represent J5's age 5 years hence, 
 and 5ic + 5 will represent ^'s age 5 years hence. 
 
 Hence 5a; + 5 = 3(x + 5), 
 
 or 5a5 + 5 = 3x + 15, 
 
 or 2x = 10; 
 
 Hence A is 25 and ^ is 5 years old. 
 
 29. A is twice as old as 7?, and 22 years ago he was three 
 times as old as B. What is yl's age ? 
 
 30. A father is 30 ; his son is 6 years old. In how many 
 years will the age of the father ))e just twice that of the son \ 
 
 31. yl is twice as old as 7?, and 20 years since he was three 
 times as old. What is 5's age ? 
 
 32. A is three times as old as 5, and 19 years hence he will 
 be only twice as old as B. What is the age of each ? 
 
 33. A man has three nephews. His age is 50, and the 
 [joint ages of the nephew^s are 42. How long will it be before 
 
 the joint ages of the nephews will be equal to the age of the 
 uncle ? 
 
 Note III. In problems involving weights and measures, 
 [after assuming a symbol to represent one of the unknown 
 I quantities, we must be careful to express the other quantities 
 \%n the same terms. Thus, if x represent a number of pence, all 
 I the sums involved in the problem must he reduced to pence. 
 
 Ex. A sum of money consists of fourpenny pieces and six- 
 Ipences, and it amounts to £1. 16s. 8d. The number of coins 
 lis 78. How many are there of each sort ? 
 
 [SA.1 1 
 
 i 
 
66 PROBLEMS LEADING TO SIMPLE EQUATIONS, 
 
 Let X be the number of fourpenny pieces. ^ 
 Then 4ic is their vahie in 2)ence. 
 Also 78 — X is the number of sixpences. 
 And 6 (78 — x) is their value in pence. 
 Also .£1. 16s. 8d. is equivalent to 440 pencf. 
 
 
 Hence 
 
 from which ' 
 Hence thei. 
 
 4a; + 6 (78 -a;) = 440, 
 • r 40 + 468 -6a; = 440, 
 
 ^<...- •■!»; = 14. 
 
 i. . fourpenny pieces, 
 and 64 o:.. >ences. 
 
 34. A bill of £100 was paid with guineas and half-crowns, 
 and 48 more half-crowj.s than guineas were used. How many 
 of each were paid ? 
 
 35. A person paid a bill of £3. 145. with shillings and 
 half-crowns, and gave 41 pieces of money altogether. How 
 many of each were paid ] . 
 
 36. A man has a sum of money amounting to £11. 13s. 4d., 
 consisting only of shillings and fourpenny pieces. He has in 
 all 300 pieces of money. How many has he of each sort ? 
 
 37. A bill of £50 is paid with sovereigns and moidores of 
 27 shillings each, and 3 more sovereigns than moidores are 
 given. How many of each are used ? 
 
 38. A sum of money amounting to £42. 8s. is made up of 
 shillings and half-crowns, and there are six times as many 
 half-crowns as there are shilliugf*. How many are there of 
 each sort i 
 
 39. I have £5. lis. M. in sovereigns, shillings and pence. 
 I have twice as many shillings and three times as many pence 
 as I have sovereigns. How many have I of each sort 1 
 
 VIII. 
 
 119. 
 
 expressic 
 
 Thus: 
 
 120. 
 or more ( 
 by the fo 
 
 Thus;: 
 
 J 
 
 121. [ 
 
 sions is t] 
 the formt 
 
 Thus 6 
 6 
 
 Note. 
 named b' 
 
 Common . 
 be given 
 
 122. ': 
 
 thus, H.C. 
 
 123. r 
 
 readily I 
 divide 12 
 
 Now, 
 
 
^UATIONS. 
 
 d half-crowTiP, 
 [. How many 
 
 shillings and 
 )getlier. How 
 
 d£11. 13s. 4d, 
 'S. He has in 
 
 iach sort ? 
 
 • 
 
 d moidores of 
 moidores are 
 
 s made up of 
 Qies as many 
 are there of 
 
 1% and pence. 
 s many pence 
 sort? 
 
 M 
 
 VIII. ON THE METHOD OF FINDING 
 THE HIGHEST COMMON FACTOR. 
 
 119. An expression is said to be a Factor of another 
 expression when the latter is divisible by the former. 
 
 Thus 3a is a factor of 12a, 
 
 bxy of 153;'-Y'*. • 
 
 120. An expression is said to be a Common Factor o"! ty^o 
 or more other (sx-pressions, when each of the latter is divisii 'e 
 by the former. 
 
 Thus 3a is a common factor of 12a and 15a, 
 
 2xy of Ibxhj'^ and 2lx^y^, 
 
 4z of 8a;, I2z^ and 162!^. 
 
 121. The Highest Common Factor of two or more expres- 
 sions is the expression of highest dimensions by which each of 
 the former is divisible. 
 
 Thus 6a^ is the Highest Common Factor of 12a- and 18a'^ 
 
 iSx'^y of lOx^y, I5xhj^ 
 
 and 25x^^3^ 
 
 Note. That which we call the Highest Common Factor is 
 named by others the Greatest Common Measure or the Highest 
 Common Divisor. Our reasons for rejecting these names will 
 be given at the end of the chapter. 
 
 122. The words Highest Common Factor are abbreviated 
 thus, H.C.P. . 
 
 123. To take a simple example in Arithmetic, it will 
 readily be admitted that the highest number which will 
 divide 12, 18, and 30 is 6. 
 
 Now, 12 = 2x3x2, 
 
 vf 18 = 2x3x3, 
 
 30 = 2x3x5. ' • 
 
 i 
 
 n 
 - i 
 
 i: 
 
METHOD OF FINDING THE 
 
 ls;ii 
 
 Having thus reduced the numbers to tlieir simplest factors, 
 it appears that we may determine the Highest Common Factor 
 in the following way. 
 
 Set down the factors of one of the numbers in any order. 
 
 Place beneath them the factors of the second number, in 
 such order that factors like any of those of the first number shall 
 stand under those factors. 
 
 . Ho the same for the third number. 
 
 Then the number of vertical columns in which the numbers 
 are alike v/ill be the number of factors in the h.c.f., and if 
 we multiply the figures at the head of those columns together 
 the result will be the h.c.f. required. 
 
 Thus in the example given above two vertical columns are 
 alike, and therefore there are two factors in the h.c.f. 
 
 And the numbers 2 and 3 which stand at the heads of 
 those columns being multiplied together will give the h.c.f. 
 of 12, 18, and 30. 
 
 124. Ex. 1. To find the h.c.f. of a%h and a'^h^x'^, 
 
 a%^x = aaa .hh ,x, 
 (v^Wx^ = aa . bhh .xx; ' 
 :. h.c.f. = adhhx . . 
 
 = a^b^x. ,,> 
 
 Ex. 2. To find the h.c.f. of 34a26M and Sla^JV, "• 
 ^4a%h'^ = 2 X 17 xaa . bbbbbb . cccc, 
 51a''6'*c^ = 3 X 17 X aaa . bbbb ,cc\ 
 :. B..c.F. = 17 aabbbbcc 
 
 ' =17a^64c2. • -•■ 
 
 EXAMPLES.—XXXiV. 
 
 Find the Highest Common Factor of 
 
 1 . a'^b and a%^. 
 
 2. x^y^z and x^yh'K 
 
 3. 14a;y and 24a:Y 
 
 4. 46m^n^p and QOmHp^. 
 
HIGHEST COMMON FACTOR. 
 
 «9 
 
 \ij)le,si factors, 
 [uinou Factor 
 
 any order, 
 
 L nimiber, in 
 member shall 
 
 the numbers 
 H.C.F., and if 
 inins together 
 
 columns are 
 
 [.C.F. 
 
 the heads of 
 ve the H.c.F. 
 
 aWx^. 
 
 •,%kl 
 
 5. ISahh^d and ^iGa^bcdl 
 
 6. a^h^, aW and a^h\ 
 
 7. 4aft, \()ac and 306c. 
 
 ^.[\1p(f, 34^25 and 51j>Y. > 
 9. ^xhjh^, nxhjh^ and 20x4?/^;:2, 
 10. 3(Xr4/', 90x'-y and UOxh/- 
 
 125. The student mnst he urged to commit to memory the 
 following Table of form?, which can or cannot be resolved into 
 factors. Where a blank occurs after the sign = it signifies 
 that the form on the left hand cannot be resolved into simpler 
 factors. 
 
 x^--y^ = {x + y){x-y) 
 
 x^ + 2/2 = 
 
 x^-y^ = {x — y) (x^ + xy + y~) 
 c(^ + y^ = (x + y) (cc2 — xy + y-) 
 
 rc4-|/4 = (a;2 + 2/2)(cc2-|/2) 
 
 x* + y^ = 
 
 x^ + 2xy + y^=(x + xjf 
 
 x"^ — 2xy + y^ = {x — yY 
 
 x^ + 3x^y + 3k?/2 + y^ = (x + yY 
 
 x^ — 3a;2?/ + ZxAf — y^ = {x~ yY 
 
 a;2-l=(a; + l)(a;-l) 
 
 fc2+l = 
 
 X^ -1={X~\){X'^ + X-lr\) 
 
 a;3 + 1 =: (as + 1) (a;2 - flj + ] ) 
 a;4-l = (x2+l}(a;2-l) 
 
 a;2 + 2a;+l=(;r + l)2 
 «;2-2a; + l=(a;-l)2 
 x^ + 3x2 + 3x+1=(cc + 1)3 
 a;3-3a;2 + 3.x-l = (x-l)3 
 
 The left-hand side of the table gives the general forms, the 
 right-hand side the particular cases in wliich y=l. 
 
 126. Ex. 
 a;2 + 2a;-3. 
 
 To find the h.c.f. of x^-l^ ic2-2x+l, and 
 
 a;2-l = (a;-l)(x4l), 
 a2-2x + l = (x-l) (;/)-!), 
 a;2 + 2a;-3 = (a;-l)(a; + 3), ' '^ 
 
 .. H.C.F. =2/ — 1. - •.. 
 
 m 
 
 ijf'! 
 
 S s: 
 
 I 
 ill 
 
 id 24x^y. 
 and 60wi^?i_p2. 
 
 Examples.— XXXV. 
 
 1 . a2 - &2 and a^ - b^. ' 4. a^ + x^ and (a + xy. 
 
 2. a2 - 62 and a" - 6'*. 5. 9^2 - 1 and (3a; + 1)2. 
 
 3. a2 — 332 and (a — x)^. ' 6. 1 -25^2 and (1 — 5a)2, 
 
 7. cc2 - y"^, (x + yY and x"^ + ^xy + 2y'\ 
 
 8. a;2 — i/2j x^ — y^ and x^ — Ixy + 6?/2. . • . ,, _ - 
 
 9. £c2— 1, jc'— 1 and cc2 + a;-2. 
 
 10. X — a'^j 1 + a^ and ^2 + 5a -f 4. . . . . . o 
 
 r 
 
i 
 
 i! " 
 
 |i|H':l< 
 
 li 
 
 70 
 
 METHOD OF FINDING THE 
 
 127. In large numbers the factors cannot often be deter- 
 mined by inspection, and if we have to find the h.c.f. of two 
 such numbers we have recourse to the following Arithmetical 
 Rule : 
 
 " Divide the greater of the two numbers by the less, and the 
 divisor by the remainder, repeating the process until no re- 
 maindor is left : the last divisor is the h.c.f. required." 
 
 Thus, to find the h.c.f. of 689 and 1573. 
 689; 1573(2 
 
 1378 ' 
 
 
 195; 689 (3 
 585 
 
 104; 195(1 
 104 
 
 
 91; 104(1 
 91 
 
 13;91(7 
 91 
 
 ^ 
 
 /. 13 is the H.C.F. of 689 and 1573. 
 
 
 EXAMPLES.— XXXVi. 
 
 I 
 \ 
 
 Find the h.c.f. of 
 
 
 I. 6906 and 10359. 4. 126025 and 40115. 
 
 
 2. 1908 and 2736. 5. 1581227 and 16758766. 
 
 
 3. 49608 and 169416. 6. 35175 and 236845. 
 
 128. The Arithmetical Rule is founded on the following 
 operation in Algebra, v/hich is called the Proof of the Rule for 
 finding the Highest Common Factor of two expressions. 
 
 Let a and & be two expressions, arranged according to de- 
 scending powers of some common letter, of which a is not of 
 lower dimensions than 6. • 
 
 Let 6 divide a with ^ as quotient and remainder c, 
 
 c 6 2 d, 
 
 d c r with no remainder. 
 
HIGHEST COMMO.V FACTOR. 
 
 n 
 
 be deter- 
 ■P. of two 
 thmetical 
 
 =!, and the 
 til no re- 
 1." 
 
 5. 
 
 58766. 
 
 5. 
 
 Uowing 
 iule for 
 
 to de- 
 3 not of 
 
 er. 
 
 The form of the operation may be sl^jewn thus ; 
 
 d) c{r 
 rd 
 
 Then we can shew , 
 
 I. That d is a common factor of a and b. 
 
 II. That any other common factor of a and 6 is a factor of 
 d, and that therefore d ia the Highest Common Factor 
 of a and b. 
 
 For (I.) to shew that c? is a factor of a and b : 
 
 b = qc + d 
 = qrd + d 
 = (qr + l)d, and /. dia & factor of 6 ; 
 
 and a=i)b-{-c 
 
 =p {qc + d) + c -. 
 
 =pqc+pd + c 
 
 =pqrd+pd + rd 
 
 = {pqr +p + r)d, and /. dia a factor of a. 
 
 And (II.) to shew that any common factor of a and 6 is a 
 factor of d. 
 
 Let 8 be any common factor of a and 6, such that 
 a=m8 and b= n8. 
 
 Then we can shew that 8 is a factor of i. 
 
 For d = b-qc 
 
 = b-q(a-pb) 
 
 = b-qa+pqb 
 
 = n8 - qmS + pqnB 
 
 = {n-qm +pqn) S, and /. 8 is a factor of d. 
 
 Now no ex|)ression higher than d can be a factor of d ; 
 :. d is the Highest Common Factor of a eftid 6, 
 
 ■f n 
 
il 
 
 72 
 
 METHOD OF FINDING THE 
 
 11 
 
 129. Ex. 
 
 To find the h.c.f. of a:''^ + 2x+ 1 and 
 
 a;^ + 2/;2 + 2a;+l. 
 
 a;2 4- 2x + \) :>? + ±f? + 2a; + 1( a; 
 
 x^ + 2x2 + X 
 
 X + 1 j a;H 2x + 1 (^x + 1 
 
 X^ + X 
 
 x + 1 
 
 x+1 
 
 Hence .; + l being the hist divisor is the n.c.F. required. 
 
 130. In the algebraical process four devices are frequently- 
 useful. These we shall now state, and exemplify each in the 
 next Article. 
 
 I. If the sign of the first term of a remainder be negativCf 
 we may change the signs of all the terms. 
 
 II. If a reniiiinder contain a factor which is clearly not a 
 common factoi of the given expressions it may be 
 removed. 
 
 III. We may multiply or divide either of the given expres- 
 sions by any number which does not introduce or 
 remove a common factor. 
 
 TV. If the given expressions have a common factor which 
 can be seen by inspection, we may remove it from 
 both, and find the Highest Common Factor of the 
 parts which remain. If we midtijdy this result hy 
 the ejected factor, we shall obtain the Highest Com- 
 mon Factor of the given expressions. 
 
 131. Ex. I. To find the h.c.f. of 2x2 - x - 1 ^nd 
 
 6x2 -4x- 2. 
 
 2x'»-x-1;Gx2-4x-2(3 
 6x2-3x-3 
 
 - x + 1 
 
HIGHEST COMMON FACTOR. 
 
 w 
 
 Cbiange the signs of the remainder, and it becomes x-l. 
 
 2.c*-23; 
 ^1 
 
 The H.c.F. required is y,-\. 
 
 Ex. II. To find the H.c.F. of a;2 + 3x + 2 and x2 + 5x + 6. 
 
 x2 + 3x + 2 
 2xT4 
 
 Divide the remainder by 2, and it becomes x + 2. 
 x + 2;x'-^ + 3x + 2(,x + l 
 
 a;2 + 2x_ 
 
 aM-2 
 x + 2 
 
 The H.c.F. required is x + 2. 
 
 Ex. III. To liud the H.c.F. of 12x2 + X- land 15x2+ 8x + l. 
 
 Multiply 
 
 by 
 
 ir)x2 + 8x + i 
 
 4 
 
 12x''^ + x-l)60.(;2 + 32x + 4(5 
 60x2+ 5a; -5 
 
 27x + 9 
 
 Divide the remainder by 9, and the result is 3x+l. 
 3a+i;i2x2 + x-U4x-l 
 
 12x2 + 4a; 
 
 l3x-l 
 -3x-l 
 
 The H.c.F. is therefore 3x + 1. 
 
 Ex. IV. To find the H.c.F. of x3 - 5x2 + 6x and 
 
 x^-10x^ + 2lx. 
 
 Remove and reserve the factor x, which is common to both 
 expressioas. 
 
 1 '■ 
 
 
 
m 
 
 |.., 
 
 74 
 
 METHOD OF FlfX>lNG THE 
 
 Then we have remaining ^ — 5x ^- 6 and x^ - lOa; + 21. 
 
 The H.c.F. of these expressions is cc — 3. 
 
 The H.c.F. of the original expressions is therefore ^—Zx. 
 
 Examples.— xxxvii. 
 
 Find the h.c.f. of the following expressions : 
 
 1. a2 + 7a; + 12anda;2 + 9a; + 20. 
 
 2. a;2 + ;[ 2a; + 20 and y? + 1 4a; H- 40. 
 
 3. x2 - 1 7a; + 70 and a;^ - 13a; + 42. 
 
 4. x2 + 5a; - 84 and j? 4- 2 la; + 108. 
 
 5. a;2 + a;-12 anda;2-2a;-3. 
 
 6. '3? -Y 5xy + 6?/2 and x^ + 6xy + 9y^. 
 
 7. a;2 — 6xy + 8?/^ and x^ — 8xy + IG?/^. 
 
 8. a;2 - Uxy - 30?/2 and x'^ -18xy + 46y^ 
 
 9. rc^ — y^ and a;- — 2xy + y^. _, 
 
 10. x^ + y^ and a;^ + 2x'^y + 3a;?/2 + 1/^ 
 
 11. x'^-y* andx^-2c(^ + y^. 
 
 1 2. a;^ + y^ and a;^ + y\ 
 
 1 3. a;* - 7/* and a;^ + 2a;7/ 4- ^2. 
 
 14. a2_j2 + 2&e-c2anda2 + 2a6 + &2_2ac-2&c + c2. 
 
 15. I2x'^ + 7xy + y^ and 28a;2 + Zxy - f. 
 
 1 6. 6a;2 + xy - y^ and 39.^2 _ 22xy + 3y^. 
 
 1 7. 15a;2 - Sxy + 1/ and 40^2 - 3xy - y\ 
 
 18. x^~ 5a;3 + 5.02 - 1 and a;'' 4 x^ - 4a;2 + a; + 1. 
 
 19. ic* + 4a;2 + 16 and a;« + a;* - 2a;3 + 17a;2 - lOx + 20. 
 2Q. u;* + xhf + y^ and a;^ + 2xhj + 3x2?/2 + 2a;?/3 + y^, 
 it. a;0-6ic* + 9a;2-4anda;« + a;6-2a;4 + 3a;2-x-2. 
 
 i 
 
' t: 
 
 HIGHEST COMMON FACTOR. 
 
 75 
 
 22. 1 5a4 + 10a3& f 4a262 + 6a63 _ 354 and Ga^ + 19a26 + Saft^ _ 5^3, 
 
 23. 15x3 - \^x^'\j + 24x1/2 ~ 7i/3 and 27x3 + Z^xhj - 20xy^ + 2i/. 
 
 24. 21x2 _ 83^2/ - 27x + 22y^ + 99y and 12x2 _ ;^r,^y _ g^, 
 
 -331/2 + 22?/. 
 
 25. 3a3- 12a2-a26 + 10a6-262 and Ga^-Va^b + Sah^-h^ 
 
 26. 18a3 - 18a2x + 6ax2 - 6a^ and 60a^ - 75ax + 15x2. 
 
 27. 21x3-26x2 + 8xand6x2-x-2. 
 
 28. 6x* + 29a2x2 + 9a* and 3x3 _ 15^^x2 + a^x - 5a\ 
 
 29. X^ + X^^2 + rjQ2y ^ y3 qj^^ rjA _ ^4^ 
 
 30. 2x3 + 10x2 + 1 4x + 6 and x3 + x2 + 7;c + 39. 
 
 3 1 . 45a3x + 3a2x2 - 9ax3 + 6x* and 18a2x - 8x3. 
 
 132. It is sometimes easier to find the h.c.p. by reversing 
 the order in which the expressions are given. 
 
 Thus to find the h.c.f. of 21x2 + 38x + 5 and 129x2 + 221x + 10 
 the easier course is to reverse the expressions, so that tliey 
 stand thus, 5 + 38x4-21x2 and 10 + 221x + 129x2, and then to 
 proceed by the ordinary process. The h.c.f. is 3x + 5. Other 
 examples are 
 
 (1) 187x3 - 84x2 + 31a; - 6 and 253x3 _ 14^2 + 29a; _ 12, 
 
 (2) 3711/3 + 262/2-50?/ + 3 and 469if + 75?/2 -- 103?/ - 21, 
 of which the h.c.f. are respectively llx-3 and 7y + 2, 
 
 133. If the Highest Common Factor of three expressions 
 a, h, c be required, find first the h.c.p. of a and h. If d be tlie 
 H.C.F. of a and h, theu the h.c.f. of d and c will be the h.c.f. 
 of a, 6, c. 
 
 134. Ex. To find the h.c.f. of 
 
 x3 + 7a;2 _ 03 - 7, x3 + 5x2 - X - ;^ ^.ud x2 - 2x + 1. 
 
 The H.c. F. of x3 + 7x2 - a; - 7 ^^d x3 + 5x2 - x - 5 will be found 
 to be x2 — 1. 
 
 The H.C.F. of x2-l and x2-2x+l will be found to be 
 a-l. 
 
 Hence x— 1 is the h.c.f. of the tki e expressions, 
 
 m 
 
 '4 
 
 ■ M 
 
 
 
 
 wn 
 
 
 

 
 FRACTIONS. 
 
 Examples. — xxxviii. 
 
 Find the Highest Common Factor of 
 
 1 . a;2 + 5ic + 6, a;2 + 7aj + lo, and x"^ + \2x-t 20. 
 
 2. x^ + Ax^-b,oi?-Zx^ 2, and x^ + Ax^ - 8a; + 3. 
 2a;2 + a; - 1, x^ + bx + 4, and x^ + 1. 
 y^-y^-y+l, 2i/-2tj-l,andif-y'^ + y-l. 
 a;3 _ 4^2 + Qx- 10, x"^ + 2x2 -2x + 20, and 
 
 6. x^ - 7a;''* + 16a; - 12, Sx^ - 14x2 + 16x, and 
 
 5x3-10x2 + 7x-14. 
 
 7' 2/^-5i/2 + lli/-15, y^ — y^ + ^y + b, and 
 
 2i/^-7'//2+16?/-15, 
 
 Note, We use the name Highest Com non Factor instoacl 
 of Greatest Common Measure or Highest Common Divisor for the 
 following reasons : 
 
 (1) We have used the word " Measure " in Art. 33 in ;i, 
 different sense, that is, to denote the number of times any 
 quantity contains the vnit of measures "^^t^. 
 
 (2) Divisor does not neceridarily in';.ly a quantity which 
 is contained in another an exact number of times. Thus in 
 performing the operation of dividing 333 by 13, we call 13 
 divisor, but we do not mean that 333 contains 13 an exact 
 number of times. 
 
 IX. FRACTIONS. 
 
 135. A QUANTITY a is ailed an Exact Divisor of a quan- 
 tity ■ , w oci; b contains a tin exact number of times. 
 
 A m.Vi^tifv a is called -i Multiple of a quantity b^ when a 
 contains /' ii exact number of times. 
 
X 
 
 
 )a; + 35. 
 7x - 14. 
 
 61/ -15. 
 
 instead 
 • for the 
 
 1*3 in a 
 les any 
 
 which 
 ?hus in 
 
 call 13 
 m exact 
 
 I a r[uan- 
 Ivvlien a 
 
 FRACTIONS. 
 
 11 
 
 136. Hitherto we have treated of qnantiiies wliich contain 
 the unit of measurement in each case an exact number of 
 times. 
 
 We have now to treat of quantities which conlain some exact 
 divisor of a primary unit an exact number of times. 
 
 137. We must first explain what we mean by a 'primary 
 unit. 
 
 We said in Art. 33 that to measure any quantify we take a 
 known standard or unit of the same kind. Our choice as to 
 tlie quantity to be taken as the unit is at first unrestricted, but 
 when once made we must adhere to it, or at least we must 
 give distinct notice of any change which we make with respect 
 to it. To such a unit we give the name of Primary Unit. 
 
 138. Nextj to explain what we mean by an exact divisor of 
 a primary unit. 
 
 Keeping our Primary Unit as our main standard of mea- 
 surement, we may conceive it to be divided into a number of 
 jjarts of equal magnitude, any one of which we may take as a 
 Subordinate Unit. 
 
 Thus we may take a pound as the unit by which we mea- 
 sure sums of money, and retaining this steadily as ih^ priwary 
 unit, we may still conceive it to be subdivided into 20 equal 
 parts. We call each of the subordinate units in thi case a 
 shilling, and we say that one of these equal subordinate units is 
 one-twentieth part of the primary unit, that is, of a pound. 
 
 These subordinate units, then, are exact divisors of the 
 ^iraary unit. 
 
 139. Keeping the primary unit still clearly in view, we 
 represent one of the subordinate units by the following nota- 
 tion. 
 
 We agree to represent the words one-third, one-^fth, and 
 
 one- twentieth by the symbols r-, ^, -, and we say that if 
 
 the Primary Unit be divided into three equal part*, will 
 represent one of these parts. 
 
 
 mi 
 
 Lfc 
 
 . f ■ I'J 
 
 l\. 
 
ii 
 
 t 
 
 Pi 
 
 I 
 
 l. jl a-' 
 
 78 
 
 FRACTIONS. 
 
 If we have to represent iwo of these subordinate uniLs, we 
 
 2 . 3 
 
 do so by the symbol ^ ; if three, y by the symbol 5 ; if /owr, by 
 
 o o 
 
 the symbol ^, and so on. And, generally, if the Primary Unit 
 be divided into h equal j^arts, we represent a of those parts by 
 the symbol v. 
 
 140. The symbol ^ we call the Fraction Symbol, or, more 
 
 briefly, a Fraction. The number helow the line is called the 
 Denominator, because it denominates the number of equal 
 parts into which the Primary Unit is divided. The number 
 uhove the line is called the Numerator, because it enumerates 
 how many of these equal parts, or Subordinate Units, are 
 taken. 
 
 141. The term number may be correctly applied to Frac- 
 tions, since they are measured by units, but we must be 
 careful to observe the following distinction : 
 
 An Integer or Whole Number is a multiple of the Primary 
 
 Unit. 
 A Fractional Number is a multiple of the Subordinate 
 
 Unit. 
 
 142. The Denominatoi of a Fraction shews what multiple 
 the Primary Unit is of the Subordinate Unit. 
 
 The Numerator of a Fraction shews what multiple the 
 Fraction is of the Subordinate Unit. 
 
 143. The N'aueratoi' ar.d Denominator of a fraction are 
 called the Term., of the Fr;:'tion. 
 
 144. Having thus evjhri.'sd the nature of Fractions, we 
 next proceed to treat ( f tht; operations to which they are sub- 
 jected in Algebra. 
 
 145. Dep, If the qucvutity x be divided into h equal parts, 
 and a of th' iO parts be taken, the result is said to be the 
 
 fraction r of a:. 
 
 
 If X bo the unit, this is called tlie fraction , . 
 
 
 
 r-s 
 
 I 
 
 and 
 
/ 
 
 FRACTIONS. . 79 
 
 146. If the unit be divided into h equal parts, 
 ,- will represent one of the parts. 
 
 r two » 
 
 1 
 
 •T three 
 
 
 
 And generally, 
 
 T will represent a of the parts. 
 
 147. Next let us suppose that each of the h parts is mh- 
 divided into c equal parts : then the unit has been divided 
 into be equal parts, and 
 
 J- will represent one of the subdivisions. 
 
 2 . 
 
 T- two 
 
 And generally, ' 
 
 k " •••■• 
 
 P__ - . n . etc Cb 
 
 148. To shew that -r-=T. 
 Let the unit be divided into h equal parts. 
 Then , will represent a of these parts (1). 
 
 Next let each of the b parts be subdivided into c equal 
 parts. 
 
 Then the primary unit has been divided into be equal parts, 
 
 and r- will represent ac of these subdivisions (2). 
 
 0€ 
 
 Now one of the parts in (1) is equal to c of tue subdivisions 
 in (2), . 
 
 .'. a parts are ej^ual to ac subdivisions ; 
 
 a ac ' 
 
 ''b"^W 
 
 i i" 
 

 80 
 
 FRACTIONS. 
 
 Cor. We draw from this proof two inferences : 
 
 I. If the numerator and denominator of a fraction be 
 multiplied by the same number, the value of the frac- 
 tion is not altered. 
 
 II. If the numerator and denominator of a fraction be 
 divided by the same number, the value of the fraction 
 is not altered. 
 
 149. To make the important Theorem established in the 
 
 preceding Article more clear, we shall give tlie following proof 
 
 4 16 
 that p =9A, by taking a straight line as the unit of length. 
 
 5 
 
 20 
 
 E 
 
 D 
 
 Let the line AG he divided into 5 equal parts. 
 
 Then, if B be the point of division nearest to (7, 
 
 ABkfoiAG. (1). 
 
 5 ^ ^ 
 
 Next, let each of the parts be subdivided into 4 equal parts. 
 
 Then AC contains 20 of these subdivisions, 
 and AB 16 
 
 AB is ^^ of A.O. 
 
 (2). 
 
 Comparing (1) and (2), we conclude thnt 
 
 4_16 
 
 150. From the Theorem established in Art. 148 we derive 
 the following rule for reducing a fraction to its lowest terms : 
 
 Find the Highest Common Factor of the numerator and denomi- 
 nator and divide both hy it. The restdting fraction icill be 
 one equivalent to the original fraction expressed in the simplest 
 terms. 
 
«4M 
 
 FRACTIONS. 
 
 8i 
 
 151. When the numerator and denominator eacli consist of 
 a single term tlie h.c.f. may be determined by inspection, or 
 we may proceed as in the following Example : 
 
 To reduce the fraction :r=.-^iTT-9 to its lowest terms, 
 
 10a^6V _ 2 X 5 X aaahbcccc 
 12a^¥c^ 2 X 6 X aahbbcc ' 
 
 We may then remove factors common to the numerator and 
 
 denominator, and we shall have remaining —^ — j- ; 
 
 ° 6x0 
 
 .*. the required result will be -^j-. 
 
 152. Two cases are especially to be noticed. 
 
 (I) If every one of the factors of the numerator be removed, 
 the number 1 (being always a factor of every algebraical 
 expression) will still remain to form a numerator. 
 
 Za^c 'daac 1 
 
 Thus 
 
 12ft^c^ 3 X 4 X aaacc 4ac 
 
 (2) If every one of the factors of the denominator be removed, , 
 the result will be a whole number. 
 
 Thus 
 
 12(1^0^ 3 X 4 X aaacc 
 
 3a% 
 
 3 X aac 
 
 = 4ac. 
 
 This is, in fact, a case of exact division, such as we have 
 explained in Art. 74. 
 
 Examples. — xxxix. 
 
 Reduce to equivalent fractions in their simplest terms the 
 following fractions : 
 
 I. 
 
 4^ 
 
 12a3' 
 
 18xV^ 
 
 blayh 
 7- 34aV^' 
 
 [S.A.] 
 
 2. 
 
 8. 
 
 8x3 
 
 21a36'ic^- 
 Voah\^^ 
 
 3- 
 6. 
 
 1(«3 
 
 4axy 
 '3abc' 
 
 Sx^yh^ 
 
 ff 
 
 
 ■i 
 
82 
 
 FRACTIONS, 
 
 
 I ' ■ \i 
 
 1 
 
 ml 
 
 lO. 
 
 13- 
 1 6. 
 
 19. 
 
 20. 
 
 21. 
 
 2l0m^n^p 
 
 •> 9. 9 • 
 
 42m'^n^p 
 
 2xy'^ — bxhjz 
 Aa^x + Qd^y 
 
 5x4 + 5a; V 
 
 10a; - lOy 
 Ax^ — 8xy + 47/2* 
 
 aa; + 6i/ 
 7aV-76V 
 
 27a'4^^a; - 48c^ 
 xy - x?/2 
 
 II. - 
 
 a-" 
 
 14. 
 17- 
 
 a'-^ + a6* 
 
 4ax + 2x^ 
 8ax^ — 2x^' 
 
 12 a6^ - 6ah 
 8h'^c - 2c 
 
 24. 
 
 25. 
 26. 
 
 1 2. ;; , — ^. 
 
 I4m^x 
 
 15' 
 
 18. 
 
 21m"'^p - 7ma; 
 
 ay + 1/2 
 
 a6c + hey' 
 
 c^ + 4ac + 4a2 
 
 7ah^x^- la hY 
 \4a?hcx^ - I4a%cy^' 
 
 5a:» + ibdx"^ 
 10cx» + dOcdx"^' 
 
 I0a'^ + 20ah+l0¥ 
 
 y 
 
 X 
 
 27. 
 28. 
 
 5a^ + 5a^6 
 
 4X2 _ Qrj.y ^ 4^2 
 
 "480^-^)2"- 
 
 2mx + 5wx2 
 3m?/ + 6nxy' 
 
 153. We shall now give a set of Examples, some of which 
 may be worked by Resolution into Factors. In others the 
 H.C.F. of the numerator and denominator must be found by 
 the usual process. As an example of the latter sort let us 
 take the following : , - 
 
 T 
 
 ffS _ 4x 1 Sa; ~" 14 
 
 To reduce the fraction nZ3~~Q~2 ZT^ — ~9 1 ^^ '^^'' ^^'^^^^ terms. 
 
 Proceeding by the usual rule for finding the H.C.F. of the 
 numerator and denominator we find it to be a; - 7. 
 
 Now if we divide a;^ — 4x2 -19a; — 14 \yy x-7, the result is 
 a;2 + 3a; + 2, and if we divide 2x3-9a;2-38a; + 21 by a; -7, the 
 result is 2a;2 + 5a; - 3. - ' ' ,• ■■■/ 
 
 a;2 + 3a; + 2 
 
 .3 is equivalent to the proposed 
 iiaction and is in its lowest terms. "■ 
 
 Hence the fraction — „ ^ 
 
 2a;^ + 5x-3 
 
 I. 
 
 a2+7a + 10 
 a2T5a + Q ' 
 
 EXAMPLES.-Xl. 
 
 a;2 - 9a; + 20 
 
 2. 
 
 a;2-2a;-3 
 
 a;2 - '7x + 12' ^' a;2 - lOx + 21* 
 
FRACTIONS, 
 
 »3 
 
 6. 
 8. 
 
 10. 
 
 II. 
 
 12. 
 
 21. 
 
 24. 
 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 30- 
 
 31- 
 
 32. 
 
 JB^— 18a:?/ + 452/^ 
 
 x^ — ^xy- 1057/2* 
 
 a;3-4x2 + 9a;-10 
 ^3 + 20;'^- 3a; + 20* 
 
 a;3-5x2+lla;-15 
 o^ - a;2 + 3a; + 5 
 
 rc^- 8x2 + 21a;- 18 
 3a;3-16x2 + 21x * 
 
 a;3-7x2 + 16a;-12 
 3x3-14.c2 + 16x'* 
 
 a;* + a;"// + xy^ - y* 
 x'^ — x^y — xy-^ — y^' 
 
 5. 
 
 a;* + x^ + 1 
 
 a;" -+ 2^3 j/3 + 2/' 
 
 a;2 + a; + 1 ' ' ofi - y^^ 
 
 m^ + 3m2 — 4m 
 
 a 
 
 M-4a2-5 
 
 a3-3a + 2 
 
 "63-66 + 5 " 
 
 3x2 + 2a;- 1 
 a;3 + a:2 _ ^j _ i* 
 
 14. 
 X 15. 
 
 16. 
 
 17- 
 18. 
 
 19. 
 
 20. 
 
 w^-7m + 6 
 
 a3 + 2a2 + 2a+r 
 
 3rta;2-13ax + 14a 
 7a;3-17a;2 + 6a;~* 
 
 14a;2-34a; + 12 
 9aa;2-39aa; + 42a* 
 
 10a -24a2 + i4a3 
 15-24a + 3a2 + 6a3' 
 
 2aP + a6- - 8a6^+ ba 
 7P~-^262T5r • 
 
 a3-3a2 + 3aj-2 
 3-4rt2 + 6«-4' 
 
 ft 
 
 22. 
 
 a 
 
 ^-a-20 
 
 a^ + a-12 
 
 23- 
 
 X 
 
 3-3a;2 + 4x-2 
 
 a;' 
 
 ;3-x2-2a; + 2 
 
 { x + y + g)2 + (;s - y)2 + (pc -zf + {y - x)' 
 
 ay^ + y^ + z^ 
 2a;^-a;3_ 9;^2 + 13a; -5, 
 "7a;3-19a;2+17x-5~ * 
 
 16a;4 -J'^''f±^x + 6 
 8x4 - 30x3 +"31a;2 _ f 2' 
 4x2-12ax + 9a2 
 . " 8x3-27a3'" • 
 
 6x3-23x2 + 16x-3 
 6x3-17x2+llx-2' 
 
 3-6x2 + llx-6 
 
 33- 
 
 34. 
 
 35. 
 
 36. 
 
 15(i2 + a&-26 2 
 9a2 + 3a6 - 262' 
 
 x2-7x + 10 
 '2x2 -X- 6* 
 
 x^ + 3x2 + 4x + 12 
 x^T4x2"+4x+"3' 
 
 X* - x2 - 2x + 2 
 2x3^x-l * 
 
 X 
 
 a;3-2x2-x + 2 " 
 
 m^ + m2 + m — 3 
 m^ + 3m2 + 5m + 3" 
 
 x^ + 5x4 _ a;2 _ 5aj^ 
 
 x4 + 3x3-x-3 • 
 
 62-26c-c2 <• 
 
 a' 
 
 a- + 2a6 + 62 _ c^* 
 
 37- 
 
 38. 
 
 39- 
 
 40. 
 
 X 
 
 3-2x2-15x+36 
 
 3x2 - 4a; _ 15 
 3a;3 + a;2-5x + 2l 
 
 6x3 + 29x2 + 26x-21' 
 X* — x^ - 4x2 — X + 1 
 
 4x3-3x2-8x-l ' 
 
 a3- 7a2+16ft-1 2 
 3c*3-14a2+16a"' 
 

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84 
 
 FRACTIONS. 
 
 154. The fraction -, is uiid to be a proper fraction, when a 
 is less than b. 
 
 The fraction , is said to be an improper fraction, when a is 
 greater than b. 
 
 155. A whole number x may be written as a fractional 
 number by writing 1 beneath it as a denominator, thus r-. 
 
 156. To prove that r of J = , -J. 
 ^ a ha 
 
 Divide the unit into hd parts. 
 
 Then r oi j = ,- of ,-,- 
 a b bd 
 
 a 
 
 — r of be of these parts 
 
 = T- of he of these parts 
 = ac of these parts 
 
 (Art. 148) 
 
 (Art. 147) 
 
 (Art. 148) 
 (Art. 147). 
 
 But 
 
 ac 
 bd 
 
 = ac of these parts ; 
 
 " l) d bd' ^ 
 
 This is an important Theorem, for from it is derived the 
 Rule for what is called Multiplication of Fractions. We 
 
 ft C 
 
 extend the meaning of the sign x and define , x -. (which 
 
 according to our definition in Art. 36 has no meaning) to mean 
 
 y of -i, and we conclude that r ^^ — 'iZii which in words gives 
 
 us this rule — " Take the product of the numerators to form 
 the numerator of the resulting fraction, and the product of the 
 denominators to form the denominator.^ 
 
 The same rule holds good for the multiplication of three or 
 more fractions. 
 
FRACTIONS. 
 
 »5 
 
 157. To ahew that r-r- j = t-' 
 
 d be 
 
 The quotient, x, of , divided by -^ is such a number that x 
 multiplied by the divisor -, will give as a result the dividend r. 
 
 
 . xc 
 
 " d' 
 
 a 
 
 — _ • 
 
 'b' 
 
 
 • • 
 
 c 
 
 - xc 
 
 Jot 
 
 c 
 
 a 
 6' 
 
 
 xcd 
 
 ad 
 
 
 • 
 
 cd 
 
 ~ be ' 
 ad 
 
 
 x = 
 
 be' 
 
 Hence we obtain ? rule for what ia called Division of 
 Fractions. 
 
 ^. a c ad 
 
 a c a d 
 
 b ' d b c' » 
 
 Hence we reduce the process of division to that of multipli- 
 cation by inverting the divisor. 
 
 158. The following are examples of the Multiplication and 
 Division of Fractions. 
 
 2a; „ _2x 3rt_6aa;_2x 
 
 1. 3^a><'^«-3,i2>< 1 -3a^-^- 
 
 ^x _3.r_^3«_:jx 1 __ 2x _ X 
 
 2. 27/"^ 26 "• r~2/;''3a~6a6"2^* 
 
 4a2 3c _3 X 4 X a-c_2a 
 ^' 9c^^2a"2x9xac'-~3c' 
 
 14x2_^7x_14^2 9j/_9xl4xa;2y_2z 
 ^' 27~y'^ ' dy ~ 2 V ^ 7x ~ 7 x^?^^^* ~ 3y' 
 
 2a 96 5c 2a x 96 x 5c 3 
 
 5' qtx 
 
 X , = 
 
 36 10c 4a 36 X 10c X 4a 4' 
 
 hi 
 
 
 1» 
 
 •■ i 
 
S6 
 
 I .ACTIONS, 
 
 6. ^ 
 
 x^ - 4a; x^ + *lx _x{x — 4) x{x + 7) 
 x^+ 7x2 aj _ 4 "~ x^ (x f 7) ic-4 
 
 _ a;(a; - 4)a;(a! + 7) _ . 
 
 ~a;2(a; + 7)(a;-4)~ 
 
 a2-62 4(a2_fj^,)_ a2_J2 
 
 a'-* 4- a6 
 
 a2 + 2ah + 6^ * a*-* + a6 a''' + 2a6 + 62" 4{a^ _ aft) 
 
 __ (a + 6) (a;;^^) a (a + 6) 
 ~(a + 6)(a + 6) 4a(a-6) 
 _(a + 6)(a-6)a(a + 6) _1 
 ~ (aT65(a+T J4a (a^6) "~ 4* 
 
 ' Examples.— xli. 
 
 Simplify the following expressions : 
 
 3x Ix 
 41/ 9y 
 
 3a 26 
 46 "" 3a 
 
 U%^ 15x1/2 
 
 5- 
 
 45x^1/ 24rt^62* 
 
 3x2l/ 5?/22i 12x2 
 
 4x2;2 6x1/ 20j;;?/2' 
 
 977l29i2 5p2^ 24x2'«/2 
 
 S/?'^^ 2x1/ 90mu* 
 
 9x22/2;3 20a362<. 
 
 10a262c \d>xyh 
 8. 
 
 3- ana 
 
 4x2 3^ 
 
 92/2 X 2|/- 
 
 , 2a 46 5c 
 ^* 56 ''3c'' 6a 
 
 7a^4 20(^ 4ac 
 5c2d3 ^ 42a'»63 '^ 36c«' 
 
 25Pm2 70n3flr 3i?m 
 
 JQ y_ i y^ _£_ -- 1 
 
 14n222 75jp27n, 4/52^ 
 
 Examples.— xlii. 
 
 Reduce to simple fractions in their lowest terms : 
 
 4. 
 
 a-6 a2-62 
 a2 + a6 a2 - a6' 
 
 x 2 + 4x 4x2j-^2x 
 ^' flr2"^=r3^''3x2 + 12x 
 
 x2.4-3x + 2 x2-7x + 12 
 3* ^^6x + 6''" x2 + x • 
 
 x2 + x-2 x2-13a; + 42 
 
 X 
 
 x2 - 7x 
 
 x2 + 2x 
 
 5- 
 6. 
 
 x2-llx + 30 x2-3x 
 x2 - 6x + 9 x2 _ ^x 
 
 x' 
 
 _4 x2-25 
 
 x2 + 5a} a;^+2a:* 
 
 a2-4a + 3 a2-9a + 20 a2-7a 
 
 7. -^i ; X -o ^TT— ;r,- X 
 
 a2-6a + 4 a2-10a + 21 
 
 a* 
 
 5a. 
 
 8. 
 
 62 - 1b + 6 62 + 106 + 24 63^ 862 
 62-^736 -4'' 62 -146 + 48'' 62 + 66* 
 
H 
 
 4' 
 
 42 
 
 FRACTIONS. 
 
 87 
 
 x^-y^ 
 
 xy - 2y^ x^-xy 
 
 {x-yy 
 
 10. 
 
 II. 
 
 12. 
 
 13. 
 
 n^ X 
 
 2 - (n - m)^ 
 
 — — X 
 x^ - 3xy + 2y^ x^ -f- onj 
 
 {a + hy-c'^ c^-{ a-by- 
 
 {x - mf 
 
 {x - nY - m^ '^ x^ - (m - iif' 
 
 { a + by-(c + d)'^ {a-hf-{d- cy 
 {a + cf - (6 + df ^{a- cf - (d -hf 
 
 7? — 2xy + y^ ~z^ x + y-z 
 x^ + 2xy + y^-z^ x-y+z' 
 
 EXAMPLES,— xliii. 
 
 Simplify the following expressions : 
 
 15y . 5i/2 
 
 2a^36 
 I. ~ • - • 
 
 X 5c 
 
 4. — ~3a&. 
 ^ nx 
 
 2. 
 
 14a • 7»* 
 3p . 2p 
 
 8. 
 
 ^ 22)-2>-r 
 1 1 
 
 3 
 
 6. 1-7 
 
 8x*i/ . 2x3 
 
 15a63 • 30aP* 
 4a 
 
 5x* 
 
 1 
 
 32-3X + 2 ' x-1' 
 
 X' 
 
 17a; + 30* x-15" 
 
 158. We are now able to justify the use of the Fraction 
 Symbol as one of the Division Symbols in Art. 73, that is, 
 
 we can shew that y is a proper representation of the quotient 
 
 resulting from the division of a by 6. 
 
 For let X be this quotient. 
 
 Then, by the definition of a quotient, Art. 72, . 
 
 hxx=a» 
 
 But, from the nature of fractions, 
 
 bx-f = a; 
 
 . <*_ 
 
 . . T —Xt 
 
 
 
 1>\ 
 
 
88 
 
 THE LOWEST COMMON MULTIPLE. 
 
 159. Here we may state an important Theorem, which we 
 shall require in the next "hapter. 
 
 If ad = bc, to shew that , = j. 
 * b d 
 
 Since ad = 6c, 
 
 ad he 
 bd~bd* 
 
 
 a c 
 ■'b~d' 
 
 ■ • 
 
 
 X, THE LOWEST COMMON MULTIPLE. 
 
 160. An expression is a Common Multiple of two or 
 more other expressions when the former is exactly divisible by 
 each of the latter. 
 
 Thus 24x^ is a common multiple of 6, Sx^ and 12x3. 
 
 161. The Lowest Common Multiple of two or more 
 expressions is the expression of lowest dimensions which is 
 exactly divisible by each of them. , 
 
 Thus 18x* ia the Lowest Common Multiple of 6a;^, Qx\ 
 and 3a;. 
 
 The words Lowest Common Multiple are abbreviated 
 into L.c.M. 
 
 162. Two numbers are »aid to be imme to each other 
 which have no common factor but unity. 
 
 Thus 2 and 3 are prime to each other. 
 
 a 
 
 163. If a and b be prime to each other the fraction v 
 
 is in its lowest terms. 
 
 a 
 
 Hence if a and 6 be prime to each other, and ^ = j, and 
 if m be the h.c.f. of c and d, 
 
 a= — and 6= — , 
 m m 
 
•■^ 
 
 THE LOWEST COMMON' MULTIPLE. 
 
 89 
 
 a 
 
 
 164. Ill finding the Lowest Common Multiple of two or 
 more expressions, each consisting of a single term, we may 
 proceed as in Arithmetic, thus ; 
 
 (1) To find the l.c.m. of 4a'x and ISax^, 
 
 2 
 
 4a^x, 
 
 I8ax^ 
 
 a 
 
 2a% 
 
 9ax3 
 
 X 
 
 2a% 
 
 9x3 
 
 
 2a^, 
 
 9x2 
 
 L.C.M. = 2 X a X X X 2a2 X 9x2 = 36^3x3. 
 (2) To find the l.c.m. of a6, ac, 6c, 
 
 a 
 
 ah, ac, be 
 
 h 
 
 by c, be 
 
 c 
 
 I, c, c 
 1, 1, i 
 
 L.C.M. = n X & X c = a6c. 
 
 (3) To find the l.c.m. of 12a2c, 14&c2 and 36a62, 
 2 12tt2c, 146c2, 36a62 
 6 ' 
 a 
 b 
 c 
 
 6a\ 
 
 1bc\ 
 
 18a62 
 
 a\ 
 
 7bc\ 
 
 Safe'-i 
 
 ac, 
 
 lhc\ 
 
 362 
 
 ac, 
 
 7c2, 
 
 36 
 
 a. 
 
 7c, 36 
 
 L.C.M. = 2x6xax6xcxax7cx36 = 252a262c2. 
 
 Examples.— xliv. 
 
 Find the l.c.m. of 
 
 I 
 
 2 
 
 3 
 4 
 
 4a3x and Qa'x^. 
 Zxhj and 12x?/2. 
 4a36 and 8a262. 
 ax, a^x and a2x2. 
 2(w;, 4ax2 and x\ 
 
 6. ah, a'^c and 62c3. 
 
 7. a^x, ahj and xhj^. 
 
 8. b\a'^x^, 34ax' and ax*. 
 
 9. ^p^, \0(i^r and 2(ypqr. 
 10. 18ax2, 72a2/2 and 12xi/. 
 
 'i 
 
 I 
 1 
 
90 THE LOWEST COMMON MULTIPLE, 
 
 165. The method of finding the l.c.m., given in the pre- 
 ceding article, may be extended to the case of compound 
 expressions, when one or more of their factors can be readily 
 detennined. Thus we may take the following Examples : 
 
 (1) To find the l.c.m. of a - jc, a' — x^, and a* + oac, . 
 
 a — % 
 
 a + a; 
 
 L.C.M. = (a — a;) (a + a;) a = (a^ — rc^) a = a' — aa;^. 
 
 (2) To find the l.c.m. of a;" - 1, a;-*- 1, and 4a^-4x*, 
 
 a-x, 
 
 a^ — x^, 
 
 a2 
 
 ■\-ajT, 
 
 1, 
 
 a + a;, 
 
 a2 
 
 ■\-ax 
 
 1, 
 
 1, 
 
 
 a 
 
 x^-l 
 
 X" 
 
 1, a;4-l, 4ic0-4x* 
 
 1, x^+l, 4x* 
 
 L.C.M. = (a;2 _ 1) (x2 + 1 ) 4x4 = (x* - 1) 4a;* = 4x8 _ 4a.4. 
 
 166. The student who is familiar with the methods of 
 l[«solving simple expressions into factors, especially those given 
 in Art. 125, may obtain the l.c.m. of such expressions by a 
 process which may be best explained by the following Ex- 
 amples : 
 
 ''*■■■■ 
 
 Ex. 1. To find the l.c.m. of a^ - x^ and a^-tx?. 
 
 a?-x^={a- x) (a + x), 
 a^-x^ = {a — x) (rt^ + ax + x2) 
 
 Now the L.C.M. must contain in itself each of the factors in 
 each of these products, and no others. 
 
 .*. LCM. is {a-x){a + x)(d^ + ax + x^, 
 
 the factor a-x occurring once in each pioduct, and therefore 
 once only in the l.c.m. 
 
 Ex. 2. To find the l.c.m. of 
 
 a2 - 62, a2 - 2ah + &2, and a^ + 2ab + 1\ 
 
 a^-'Zah + b^={a~h){a--b), 
 a^ + 2ab + b'^={a-\-b){a + b)i 
 
 L.C.M. is (a + b)(a- b) (a -b){a + b), 
 
 SjT..„. 
 
e pre- 
 pounc! 
 •eadily 
 
 
 .4 
 
 lods of 
 le given 
 IS by a 
 ng Ex- 
 
 stors 111 
 
 erefore 
 
 TflE LOWEST COMMON MULTIPLE. 
 
 9« 
 
 tlie factor a — h occurring twice in one of the products, and « + 6 
 occurring twice in another of tlic products, and therefore each 
 of these factors must occur twice, in the l.c.m. 
 
 EXAMPLES.— Xlv. 
 
 Find the l.c.m. of the following expressions : 
 
 lo. x^ - 1, a;2 4. 1 and x^ - 1. 
 
 1. re- and ax-vx^. 
 
 2. x^—\ and x^ — x. 
 
 3. a^ — h^ and a^ + ah. 
 
 4. 2x— 1 and 4x2-1. 
 ^. a + b and a^ + h\ 
 6. x + 1, 03 - 1 and x^ — 1. 
 
 11. x^-x, x^—l and x^ + l. 
 
 12. a;2- 1, a;2._a; and x^ — 1. 
 
 13. 2at-l, 4a2_i and Sa^+i. 
 
 14. x + y and 2x^ + 2xy. 
 
 15. (a + 6)2 and a2- 62. 
 
 7. x + l,?"^-! andx2 + a;+ 1. 16. a + ft, a-6 and a2_&2^ 
 
 8. x+l, a;2+i andx3+l. 17. 4(1 + ^),4(1 -x)and 2(1 -cc^) 
 
 9. x-1, x?- 1 and x^-\. 18. x- 1, ic2 + a;-f 1 and aj^-l. 
 
 19. {a- h) {a - c) and (a - c) (6 - c). 
 
 20. (x + 1) (x + 2), (x + 2) (a; + 3) and (x + 1) (x + 3). 
 
 2 1 . a;2 - 1/2, (x + 1/) 2 and (x - 2/)^. 
 
 22. (a + 3) (a + 1), (a + 3) (a - 1) and a"^ - 1. 
 
 23. x^{x - y\ x(;x^ - 1/2) and x + y. 
 
 24. (£C + l)(x + 3), (a; + 2)(a; + 3)(x + 4)and(a;+l)(x-}-2). 
 
 25 . x^- y\ 3 (a; - 2/)2 and 12 (x^ + y^). 
 
 26. 6 (x2 + xy), 8 (xy - 1/2) and 10 (x2 - i/2). 
 
 167. The chief use of the rule for finding the L.C.M. is for 
 the reduction of frf^rtions to connnon denominators, and in the 
 simple examples, wliich we shall have to put before the student 
 ill a subsequent chapter, tlie rules which we have already given 
 will be found generally su(fiv.ient. But as we may have to find 
 the L.C.M. of two or more expressions in which the elementary 
 factors cannot be determined by inspectioi., we must now pro- 
 ceed to discuss a Rule for finding the l.c.m. of two .expressioiia 
 which is applicable to every case. 
 
92 
 
 THE LOWEST COMMON MULTITLE. 
 
 168. The rule for finding the l.c.m. of two expressions a 
 and 6 is this. 
 
 Find a the liighest common factor of a and h. 
 
 Then the l.c.m. of a and i = , x ?>, 
 
 }) 
 or, = , X a. 
 d 
 
 m 
 
 In words, the l.c.m. of two expressions is found by the fol- 
 lowing process : 
 
 Divide one of the expressions hy the h.c.f. and multiply the 
 tiuotient by the other expression. The result is the l.c.m. 
 
 The proof of this rule we shall now give. 
 
 169. To find the l.c.m. of two algebraical expressions. 
 Let a and 6 be the two algebraical expressions. 
 Let d be their h.c.f., 
 X the required l.c.m. 
 
 Now since ic is a multiple of a and 6, we may say that 
 
 x = ma, x = nh; ^ 
 
 :. ma — nh ; 
 
 .-.- = - (Art. 159), 
 
 n a ^ ' 
 
 Now since x is t'lie Lowest Common Multiple of a ancl 6, 
 m and n can have no common factor ; 
 
 .". the fraction — must be in its lowest terms ; 
 
 71 
 
 7/1 = 
 
 d 
 
 ill 
 
 and »i = T (Art. 163). 
 
 Hence, since 
 
 aj = r/ia, 
 
 Also, since 
 
 X = -y X a. 
 d 
 
 x = nh, 
 d 
 
T/fE LOWEST COMMON MUU J PLE. 
 \ 
 
 n 
 
 170. Ex. Find tlie l.c.m. of r^ _ 13,. ^ 42 mid ,,;-• _ ly^ j. 84. 
 First we tiiul the H.C.F. of the two expressions to be a;— 7. 
 
 Then L.C.M. 
 
 (.0- - 13.v; + 42) X (x- - 19a; + 84) 
 
 x-1 
 
 Now each of the factors composing the numerator in divisible 
 by x-1. 
 
 Divide x^ — Wlx -f- 42 by x - 7, and the quotient is x - 6. 
 
 Hence l.c.m. = (,(;-(J)(,c-- 19./; + 84)=x3-25x'-+ 198a; -504. 
 
 i 
 
 
 lat 
 
 Incl 6, 
 
 Examples. — xlvi. 
 
 F^ind the l.c.m. of the folh)\ving expressions : 
 
 1 . X" + 5ic + 6 and x' + 6;<: + 8. 
 
 2. a- -rt - 20 and a'-^ + a- 12. 
 
 3. x?-\- 3.IJ 4- 2 an d .t;^ + 4a: + 3. 
 
 4. .c2 + 1 1 :,; + 30 and x'-^ + 1 2,/; + 35. 
 
 5. .t^ - Ox- - 22 and a;'-' - 1 3.(; 4- 22. 
 
 6. 2x-H 3a; + 1 and a;2 - ./; - 2. 
 
 7. x^ + x-y + xy + y"^ and x^ - ?/*. 
 
 8. x^ - 8x + 15 and x- + 2x - 15. 
 
 9. 2 Irc^ - 26x + 8 and 7a;'' - 4x^ - 2 L^ + 1 2. 
 10. x^ + xhj + xy"^ + y"^ and x^ - x^y + xy^ -y^. 
 
 ir. a^ + 2a^b-ah'^-2Piim\a^-2a^b-ab' + 2lA 
 
 171. To find the l.c.m. of three expressions, denoted by 
 «, b, c, we find m the l.c.m. of a and 6, and then find M the 
 L.C.M. of m and c. iV/ is the l.c.m. of a, b and c. 
 
 The proof of this rule may be thus stated : 
 
 Every common multiple of a and b is a multiple of w, 
 and every multiple of m is a nuiltiple of a and 6, 
 therefore every common multiple of m and c is a common 
 
 multiple of «, b and c, 
 and every common multiple of a, b and c is a common 
 
 multiple of m and c*, 
 and therefore the l.c.m. of m and c is the l.c.m. of «, 6 
 
 and c. 
 
94 
 
 OlSr ADDITION AND SUBTRACTION 
 
 Examples.— xlvii. 
 
 Find the l.c.m. of the following expressions : 
 
 1. a;2-3x + 2, .'k2-4x + 3 anda;2-5a; + 4. 
 
 2. a;2 + 5a; + 4, a:2 + 4x + 3 and a;2 + 7x4- 12. 
 
 a;2 - 9a! + 20, a;2 - 12x + 35 and x^ - llx + 28. 
 6x2 - ac - 2, 21a;!^ - 17x + 2 ^mi 14^8 ^ 5^ _ x, 
 
 x2 - 1, a;2 + 2x - 3 and 6x2 _ a; _ 2. 
 
 3- 
 
 4- 
 
 5- 
 
 6. 'J? - 27, x2 - 15x + 36 and x^ - 3x2 _ 2x + 6. 
 
 XI. ON ADDITION AND SUBTRACTION 
 OF FRACTIONS. 
 
 172. Having established the Rules for finding the Lowest 
 Common Multiple of given expressions, we may now proceed 
 to treat of the method by which Fractions are combined by 
 the processes of Addition and Subtraction. 
 
 173. Two Fractions may be replaced by two equivalent 
 fractions with a Common Denominator by the following 
 rule : 
 
 Find the l.c.m. of the denominators of the given fractions. 
 
 Divide the l.c.m. by the Denominator of each fraction. 
 
 Multiply the first Numerator by the first Quotient. 
 
 Multiply the second Numerator by the second Quotient. 
 
 The two Products will be the Numerators of the equivalent 
 fractions whose common denominator is the L.C.M. of the 
 original denominators. 
 
 The same rule holds for three, four, or more fractions. 
 
 174. Ex. 1. Reduce to equivalent fraction^ with the 
 lowest common denominator, 
 
 2x + 5 , 4x-7 
 ___and--^. 
 
noN 
 
 le Lowest 
 proceed 
 bined by 
 
 [uivaleiit 
 following 
 
 ictions. 
 ion. 
 
 Itient. 
 
 univalent 
 [. of the 
 
 Ins. 
 ritb the 
 
 OF FRACTIONS. 
 
 95 
 
 Denominators 3, 4. 
 
 Lowest Common Multiple 12. 
 
 Quotients 4, 3. 
 
 New Numerators Sx + 20, I2x - 21. 
 
 8a; + 20 12x-2I 
 
 Equivalent Fractions 
 
 12 ' 12 * 
 
 Ex. 2. Reduce to equivalent fractions with the loweot 
 common denominator, 
 
 66 + 4c 6a -2c 3a -5& 
 ab ' ac * be ' 
 Denominators ab, ac, be. 
 Lowest Common Multiple abc. 
 Quotients c,b, a. 
 New Numerators 56c + 40^, 6a6 - 26c, Sa^ - 5a6. 
 
 -c . , , T, , . 56c + 4c- 6a6 - 26c Sa^ - 5a6 
 
 iiquivalent Fractions — -, , - . — , , . 
 
 a6c a6c ' a6c 
 
 Examples.— xlviii. 
 
 Reduce to equivalent fractions with the lowest common 
 denominator : 
 
 3x , 4a; 
 
 1. 4-and-- 
 
 3a;-7 , 4a;-9 
 
 2. - - and ^g-. 
 
 3- 
 
 4- 
 
 ■ 
 5. 
 
 2x-4w , 3x-8v 
 — — -^ and ,/."• 
 
 4a + 56 , 3a -46 
 
 4a -5c , 3a~2c 
 bac 12a^c ' 
 
 r a -b , a^-ah 
 ^' a^6^"^-ar- 
 
 3 , 3 
 
 7. , and . 
 
 1 + X 1 - a; 
 
 Q 2 , 2 
 
 8. - — and ^ :;. 
 
 5 , 6 
 
 9. - — and , „. 
 
 ^ 1 - a; l-x^ 
 
 a , 6 
 
 10. - and -TT ^.. 
 
 c c(6 + x) 
 
 ' '• (a- 6) (6- c) ^"^ (a-6) (a-c)' 
 
 12. 
 
 1 
 
 and 
 
 1 
 
 a6 (a - 6) (a - c) ac (a - c) (6 — c) 
 
 I 
 
 I 
 
o6 
 
 ON ADDITION AND SUBTRACTION 
 
 ^': 
 
 175. To shew that v -i- ,= - , , • 
 
 b a bd 
 
 Suppose the unit to be divided into bd equal parts. 
 
 7 
 
 Then ^-i will represent ad of these parts, 
 
 he 
 and -j'j will represent &c of these parts. 
 
 , c he 
 and j= T-i' 
 
 Hence r+i will represent ^>xl + he of the parts. 
 
 ad + 6c 
 "bd 
 
 Bu 
 
 will represent art + be of the parts. 
 
 -, „ a c ad + bc 
 Tnereforej- + ^^=-^^. 
 
 By a similar process it may be shewn that 
 
 a c _ ad - be 
 I'd 
 
 bd 
 
 ,►,« in' (^ <^ ad + be 
 176. Since ^ + -^=-^^, 
 
 our Rule for Addition of Fractions will run thus : 
 
 "Reduce the fractions to equivalent fractions having the 
 Lowest Common Denominator, Then add the Numerators of 
 the equivalent fractions and place the result as the Numerator 
 of a fraction, whose Denominator is the Common Denominator 
 of the equivalent tractions. 
 
 The fraction will be equal to the sum of the original frac- 
 tions." 
 
 The beginner should, however, generally take two fractions 
 at a time, and then combine a third with the resulting fraction, 
 as will be shewn in subsequent Examples. 
 
 ...acad — bc 
 Also, since ^--^=^j-^^-, 
 
 the Rule for Suhtraeting one fraction from another will be, 
 
ring the 
 orators of 
 Uineratoi' 
 )iuiniitor 
 
 [iial frac- 
 
 llractions 
 fraction, 
 
 I be, 
 
 " Eediice the fractions to eq^uivalent fractions having the 
 Lowest Common Denominator. Then subtract the Numerator 
 of the second of the equi\'alent fractions from the Numerator 
 of the first of the equivalent fractions, and place the result as 
 the Numerator of a fraction, whose Denominator is the Common 
 Denominator of tlie equivalent fractions. This fraction will be 
 equal to the difference of the original fractions." 
 
 These rules we shall illustrate by examples of various degrees 
 of difficulty. 
 
 Note. When a negative sign precedes a fraction, it is best 
 to place the numerator of that fraction in a bracket, before 
 combining it with the numerators of other fractions. 
 
 177. Ex. 1. To simplify 
 
 4a; -3// . 3.7; + 7?/ 5x-2// 9x + 2ij 
 ' "~21 "^~"42~* 
 
 y---14 
 
 Lowest Common ^lultiple of denominators is 42. 
 
 Multiplying the numerators by 6, 3, 2, 1 respectively, 
 
 24.x -ISy 9./: + 2ly _ 10x-4y 9^ +^2// 
 _.___._•.. + - ~^-- 4g; + ~42~ 
 
 24x- - 1 8>/ -i- 9x + 21?/- (1(U - 4?/) + 9x + 2y 
 24x - 1 8?/ + 9a' + 2 1 7/ - lOx-\-4y + 9.0 + 27/ 
 
 _ 32x + 9 v 
 Ex. 2. To simplify 
 
 42 
 
 2x + 1 4x + 2 1 
 
 3:- 
 
 i)X 
 
 + : 
 
 Lowest Common Multiple of denominators is 105j;. 
 Multiplying the numerators by 35, 21, 15x, respectively, 
 
 70.*; + 35 _ 84x+ji2 15x_ 
 
 iobx 105x lOoa; 
 
 _ 70a; + 35 - (84a; + 42) + ISic 
 
 10 
 
 xc 
 
 7(>x + 35 - 84.'(; - 42 + 15a; a; - 7 
 
 105.C 
 
 lOox' 
 
 fs.A.] 
 
 G 
 
 
 if 
 
 "li 
 
98 
 
 ON ADDITION AND SUBTRACTION 
 
 3- 
 
 4- 
 
 5 
 6 
 
 7 
 8 
 
 lO 
 
 II 
 
 Examples.— xlix. 
 
 4a; + 7 3a; - 4 
 
 ____+ — 
 
 2. 
 
 3a-46 2rt-& + c 13rt-4r 
 
 15 7 
 
 4a; - 3?/ 3a; + 7y _ 5x - 2?/ O.x + 2j/ 
 
 14 
 
 21 
 
 42 
 
 3a; - 27/ 5a; - 7i/ 8a; + 2|/ 
 5a; 10a; 25 
 
 4a;2-7i/2 3^ ^8?/ 5-2|/ 
 3x2 -+ W/'^~'\^~' 
 
 4a2 + 562 3a + 26 7-2tR 
 262""^ 56 "^ 9 • 
 
 4a; + 5 3a; -7 _9_ 
 ~3 5¥~'^12x^" 
 
 5a + 26 _ 4c -36 6a6 - Ihc 
 3c 2a 14ac 
 
 2a + 5c 4ac - Zi? 5ac - 2c''* 
 a-c ac^ a^c^ 
 
 3a:7/-4 5i/2 + 7 6a;2-ll 
 
 xhj" 
 
 xy" 
 
 x'^y 
 
 a 
 
 -6 4a - 56 3a - 76 
 
 + — 
 
 a^6 a26c 
 
 + 
 
 62c2 
 
 3 
 
 + 
 
 12 
 
 I. 
 
 178. Ex. To simplify 
 
 + b a ^' 
 
 L.c.M. of denominators is a'^ - 6 
 Multiplying the niinie 
 
 we get 
 
 rators by a - 6 and a + 6 respectively, 
 
 a'-'-2a6 + 62 a2 + 2a6 + 62 
 
 a- 
 
 - 6'^ 
 
 + 
 
 a' 
 
 62 
 
 a2-2(f6 + 62 + a2 + 2a6+62 
 
 a2-6^ 
 
 2a2+^6''' 
 
 ff2-62'* 
 
 b 
 
OF FRACTIONS. 
 
 ^ 
 
 13fe-4r 
 
 )ectively, 
 
 Examples.— 1. 
 
 1 
 
 I. — 
 
 ic - 6 a; + 5* 
 
 a; + ?/ 55-1/ 
 x - 2/ a; + y* 
 
 a; a; 
 
 7- — + 
 
 2. 
 
 X - 7 X - 3* 
 
 5 
 
 1 1 
 
 •^ 1+a; 1 -a; 
 
 a (ad - be) x 
 
 ' l-x 1-x'' 
 
 6. 
 
 c c{c + dx) ' 
 
 x+y x-y 
 2 3a 
 
 • h 7 w. 
 
 x + a (x + a)*^ 
 
 8. 
 
 lo. 
 
 1 
 
 X 
 
 x-y (x- y) 
 
 N2' 
 
 2a (a + x) 2a (a - x) ' 
 
 179. Ex. 1. To simplify 
 
 3^5 6^ 
 
 1 + // 1-7/ l+l/'''' 
 
 Taking the first two fractions 
 
 3 5 
 
 1+y 1-y 
 
 _8 + 2y 
 
 we can now combine with this result the third of the original 
 fractions, and we have 
 
 3 5 6 
 
 \-\-y 1-y 1+1/2 
 
 6 
 
 ^8_+% 
 
 1-/ l+y^ 
 
 ^ 8 + 27/ + 8?/2 + 2?/3 _ 6 _ 6?/2 
 
 1-2/' 1-2/* 
 
 _^8 + 2]/jf87/- + 2//'' - 6 + 6i/' 
 1-2/4 
 
 9.y^ + Uy'^ + 2y + 2 
 1-2/* 
 
 f I 
 
100 
 
 ON ADDITION AND SUBTRACTION 
 
 Ex. 2. To simplify 
 
 ,— v + ,- 
 
 (a-6)(6-c) (a-6)(c-rt) (T^(c-a)' 
 L.C.M. of first two denominators being (a - &) (6 - c) (c - a) 
 
 2c -2a 
 
 + 
 
 26 -2c 
 
 — .+ 
 
 (a — b)(h- c) (c - a) {a- b) (/> - c) (c — a) (6 -c) {c- a) 
 2b -2a 2 
 
 -x + 
 
 {a -b){b- c) (c - a) (h-c) (c - a)' 
 L.C.M. of the two denominators being {a - b) (h -c) (c - a) 
 2b-2a + 2a-2b' 
 
 > {a-b){b- c) (c - a) {a - b) {b -c) {c- a) 
 
 =0. 
 
 Examples.— li. 
 
 l+a I -d l-a^' 
 
 1 1_ 2b 463 
 
 2. 
 
 1 2x 
 
 + 
 
 a;2 
 
 1 - X l + x i +x'^' 
 
 a;2 
 
 X X"" X 
 
 ^' l^x~l^:c^'^\ + ¥' 
 
 ^ x-1 x-2 x-3 
 
 :*; - 2 x-Z x-4 
 
 5a2 
 
 a; w 
 •' 2/ ^ + 2/ x^ + xy 
 , ic + 3 a;-4 a; + 5 
 
 O. 7 H n H f^. 
 
 a: + 4 x~Z x + 7 
 
 8. _?-.- + -^ . 
 
 x-a {x-af {x-a)'^' 
 
 1 1 8 
 
 ^' x-\ x + 2 (x+l)(a; + 2)* 
 
 1 » 
 
 lo. 
 
 (..■ + 1) {x + 2) {x + 1) (u; + 2) {x + 3)' 
 
 .'- X X 
 
 II. .. Y + - -T + — . V 
 
 12. 
 
 (a + c) (rt 4- d) {a + c) (a + e)' 
 
 a-b ^~^__ J. __ ^^^ 
 
 '^* (6 + c){c-^ay (c + a) (a ■♦- 6) "^ (^+ 6) (6 Tc)' 
 
OF FRACTIONS. 
 
 463 
 
 X - a x -h {a -Iff 
 
 x-b x-a (x -a){x- bj' 
 
 x + y 2x x-ij-x^ 
 
 ^ a + b h-\c c + a 
 
 {b-c){c — a) {c-a){a-b) (a-6)(6-c)' 
 
 X 2X71 
 
 17 4- ^ — 
 
 x^ + xy + if x-^ — y^ 
 
 i8. 
 
 _2_ _2_ _2_ (a-?))2 + (6-c)2 + (c- a)^ 
 « - 6 6 - c c - rt (a - 6) (6 — c) (c - a) 
 
 
 20. 
 
 21. 
 
 1 
 
 (h + 1) {n + 2) (71 + 1) {n + 2) (n + 3) {n + 1) (n + 3)' 
 
 a^ — ftc b'^-ac 
 
 + 77— --,—— V + 
 
 a& 
 
 (a + 6)(a + c) (6 + a)(6 + c) (c + 6)(c + a)" 
 
 180. Since 
 
 -r=0', and -3-t- = «) Art. 77, 
 ab _ — ab 
 
 From this we learn that we may change the pign of the 
 denc^ninator of a fraction if we also change the sign of tlie 
 numerator. 
 
 Hence if the numerator or denominator, or both, be expres- 
 sions with more than one term, we may change tl;e sign of 
 every term in the denominator if we also change the sign of 
 every term in the numerator 
 
 a-b _-{a-b) 
 c-d -{c-dj 
 
 _ —a + b 
 ~ -T'+d ' 
 
 For 
 
 or, writing the terms of the new fraction so that the positive 
 
 terms may stand first, 
 
 _b — a 
 
 ~d-G' 
 
I03 
 
 OlSr ADDITION- AND SUBTRACTION 
 
 181. Ex. To simplify i^^^-^-^) -5^^^^'. 
 
 a-x 
 
 x-a 
 
 Changing vhe signs of the numerator and denominator of the 
 second fraction, 
 
 x(a + x) —5ax + x^ 
 a-x a—x 
 
 _ ax + x^-( — 5ax + x-) _ ax + x^ + finx -x^ _ Qax 
 
 a — x 
 
 a — x 
 
 a-x 
 
 F 
 
 fracl 
 the 
 also I 
 
 the 
 
 \ 
 
 182. Again, since -ah= the product of -a and &, 
 and ah = the product of + a and b, 
 
 the sign of a product will be changed by changing the signs of 
 one of the factors composing the product. 
 
 Hence (a — h)(h- c) will give a set of terms, 
 
 and (h-a) {b-c) will give the same set of terms wi'h dif- 
 ferent signs 
 
 This may be seen by actual multiplication : 
 
 {a-h){b-c) = ab-ac-b'^ + bc, 
 {b - a) {b - c) = - ab + ac + b"^ - be. 
 
 Consequently if we have a fraction 
 
 1 
 
 {a-b){b-cy • 
 
 and we change the factor a -h into b ~ a, we shall in effect 
 change the sign of every term of the expression which would 
 result from the multiplication of (a - h) into (6 - c). 
 
 Now we may change the signs of the denominator if we also 
 change the signs of the numerator (Art. 180); . 
 
 1 -J; 
 
 " {a-b){b-c)~{b-a){b-cy 
 
 If we change the signs of two factors in a denominator, the 
 sign of the numerator will remain unaltered, thus 
 
 _ J 1 
 
 (a'-b){b-c)~{h-a){c-by 
 
 I. 
 
OF FRACTIONS. 
 
 io3 
 
 183. Ex. Simplify ' 
 
 i + 1 ' ^ \ 
 
 {a-b)(b-c) {b-a){'i-c) (c -a){c- b)' 
 
 First change the signs of the factor. (6 - a) in the second 
 fraction, changing also the sign of the numerator ; and change 
 the signs of the factor (c - a) in the third fraction, changing 
 also the sign of the numerator, 
 
 trip rfimilii m .■, , ^ , _ — ^-_— 
 
 (a-6)(6-c) {a-b){a-c) (a-c) (c-b)' 
 
 Next, change the signs of the factor (c - 6) in the third, 
 changing also the sign of the numerator, 
 
 the result is 
 
 (a - 6) (6 - c) (a -b){a- c) (a - c) (6 - c)' 
 L.C.M. of the three denominators is {a -b) {b- c) (a - c), 
 _ CT- c -b+c a-b 
 
 ~ {a-b){b-c){a~c) '^(a-b)(a- c) {b - c) ~ (a - 6) (a'^c) "(6^) 
 _ a-c- b + c- {a -b) _ 
 
 (a - 6) (6 - c) (a - c) {a- b) (6 -c){a- c) 
 
 = 0. 
 
 Ex \MPLES.— lii. 
 
 I. 
 
 X x-y 
 
 + — K 
 
 x-y y-x 
 
 w X 
 
 2. 
 
 + 
 
 x^ 
 
 3 + 2a; 2 - 3x 16a; - g^ 
 
 2-x 2 + a; "^ x^-i'' 
 
 114 
 
 •5' a;+l 1-a; a;2_i- 
 
 ey + 6 2y-2^3-'3y'^' 
 
 1 2 1 
 
 5" (m-2)(m-3)'**(m-l)(3-w)'*"(m-l)(m-2)* 
 
 . 1 1 g^ + ftg 2a62 Ja26 
 
 ^' (a-6)(a; + 6)'^(6-a)(a; + a)' 7- ^2.52 a^-fts'^aa + i^' 
 
 
 I i 
 
 8. 
 
 -T^ + 
 
 4(I+a;) 4(a;-l) 2(l + a;2)* 
 
 1 1 ^ 1 
 
 10. 
 
 + 
 
 + 
 
 1 
 
 (*(a-fc)(a-c) 6(6-a)(6-c)^c(c-a)(c-6y 
 
I04 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 184. Ex. To simplify 
 
 _._.!__ + _1 
 
 Here the denominators may be expressed in factors^ and we 
 have 
 
 (x-5)(x--'G)'^(x-5)(x-V)* 
 
 The L.c.M. of the denominators is (:c - 5) (x — 6) (a; - 7), and 
 we have 
 
 a-7 
 
 + , ^..-r- 
 
 a — 6 
 
 (a; - 5) U - 6) (x - 7) (x - 5) {x - G) (x - 7) 
 
 ^ • 2x-13 
 
 ~(x-5)(;c"-6)>-7)' 
 
 Examples.— liii. 
 
 I. 
 
 2. 
 
 :r.-V 
 
 x^ + 9.-C + 20 a;2 I- 1 2x + 35* 
 1 1 
 
 tt;2 - 13x + 42 a;2 - 15x + 54* 
 ^' a;2 + 7^_44"fa;2-2a;-143' 
 
 1 
 
 + 
 
 2x 
 
 + ..., 
 
 • r'-2 + 3x + 2 rc2 + 4x + 3 :>■? + 5x + 6* 
 m 2m 2m7i 
 
 6. 
 
 + — - / -J- 
 
 n m + n {vi + 7ij^ 
 
 1 +x 1 -X 
 
 i I 
 
 1+X + X2 l-X + X~ l+X^ + X*' 
 
 7- .: 
 
 2 7x 
 
 + 
 
 7x 
 
 3 ( 1 - x) 1 + X 3x2 H- v 3^,2 _ 3' 
 
 8. 
 
 1 
 
 + 
 
 1 
 
 + 
 
 1 
 
 + 
 
 8(x-l) 4(3-x) 8{x-b) (l-x)(x-3)(x-5) 
 
 9. 1 - X + 
 
 x^ — x' -t- 
 
 x^ 
 
 l+» 
 
 or. 
 
 or. 
 
\ 
 
 \l 
 
 XII. ON FKACTIONAL EQUATIONS. 
 
 185. We shall explain in this Chapter the method of 
 solving, first, Equations in which fractional terms occur, and 
 secondly. Problems leading to such Equations. 
 
 186. An Equation involving fractional terms may be 
 reduced to an equivalent Equation withort fractions hy mul- 
 tiplying every term of the equation by the Lowest Common 
 Multiple of the denominators of the fractional tcrm^s. 
 
 Tliis process is in accordance with the principle laid down 
 in Ax. III. page 58 ; for if both sides of an equation be multi- 
 plied by the same expression, the resulting products will, by 
 that Axiom, be equal to each other. 
 
 187. The following examples will illustrate the process of 
 clearing an Equation of Fractions. 
 
 2 b 
 The L.c.M. of the denominators is 6. 
 Multiplying both sides by 6, we get « 
 
 6^ 6x ,„ 
 
 0^1 
 
 3x + a; = 48, 
 4a; = 48; 
 
 X x+ 1 
 
 Ex.2. ^^ + -:--=a;-2 
 
 The L.c.M. of the denominators is 14. 
 Multiplying both sides by 14, we get 
 
 14.0 14a; + 14 
 
 2 
 
 + 
 
 = 14a; -28, 
 
io6 
 
 ON FRACTIONAL EQUATIONS. 
 
 or. 
 
 or. 
 
 or, 
 
 7a; + 2x + 2 = 14a:-28, f 
 7x + 2a;-14x= -28-2, * 
 -5a;=-30. ^ 
 
 Changing the signs of both sides, we get 
 
 5x=30; 
 .*. as=6. 
 
 188. The process may be shortened from the followin;^ 
 considerations. If we have to multiply a fraction by a multiple 
 of its denominator, we may first divide the multiplier by the 
 denomitiator, and then multiply the numerator by the quotient. 
 The result will be a whole number. 
 
 Thus, 
 
 X 
 
 3 
 
 x-\ 
 
 X 12 = ccx 4 = 4a;, 
 
 x56 = (a5-l)x8 = 8x--8. 
 
 Ex. 1. 1 + 1 + ^ = 39. 
 2 3 4 
 
 (•i 
 
 The L.c.M. of the denominators being 12, if we multiply the 
 numerators of the fractions by C, 4, and 3 respectively, and the 
 other side of the«equation by 12, we get 
 
 6x + 4a; + 3a; = 468, 
 
 or, 13x = 468; 
 
 .-. X:^36. 
 
 ^^•^" ^"2^^3a;~12' 
 
 The L.C.M. of the denominators is 12a:. Hence, if we mul- 
 tiply the numerators by 12, 6, 4, and x respectively, we get 
 
 96-90 + 28 = 17ic, 
 
 or. 
 
 or. 
 
 34 = 17a;, 
 17x = 34; 
 
= 8. 
 
 ON FRACTIONAL EQUATIONS. 
 
 tc7 
 
 EXAMPLES.—liV. 
 
 3a; 
 
 9. 
 
 X X 
 
 + ^ = 8. 
 
 3 5 
 
 X X 
 
 t 4 7 
 
 2a; ^ 7x - 
 7- "3+^ = 12 + ^ 
 
 5. 36-^ = 8, 
 
 2a; 176 -4a; 
 
 9 
 
 V' -^ 
 
 x+2 x-l x-2 
 
 2 
 
 2a; 
 
 4a; 
 
 8. -.,+12 = - +6 
 •3 
 
 •«• hh^-l 
 
 i 
 
 3a; 
 
 5x 
 
 9- T^^^T"*"^ 
 
 -5-^"^ 8 
 -5-^^-8, 
 
 lo. 
 
 7x 
 8 
 
 19- 
 
 a; + 9 2x 3x-6 „ 
 
 20. 
 
 17 -3x 29- 11a; 28a;+14 
 
 + 
 
 21 
 
 II. 
 
 12. 
 
 5a; 
 
 9 
 
 =^-8 = 74- 
 
 7x 
 12' 
 
 X 
 
 -4 = 24-|. 
 
 21. - 
 
 2x-10 
 
 = 0. 
 
 3x + 4 4a; -51 ^ 
 
 3a; 
 
 13. 56--j- = 48- 
 
 5x 
 
 8 
 
 23- 
 
 X 
 
 3=--l. 
 
 X 
 
 ml- 
 
 3a; 180 -5a; 
 
 14. -.- + 
 
 3a; 
 
 6 
 
 = 29. 
 
 15. ^-11 = 
 
 33-8 
 
 , a; a; a;_13 
 ^^' 2''"3'^4~12* 
 
 24. 
 
 12+a; 
 
 X 
 
 — 5 = 
 
 6 
 
 X 
 
 '5- 4^ + ro^ + 20^=40. 
 
 26. 2^. + ii:^=3|x-43l. 
 
 23 3_1 325 
 
 28. 2 
 
 4 a; a; 100* 
 
 1 . 18-a; 1 1 3-2,x 2 
 
 = l«a' + o + — ,^T^ + 
 
 9 3 
 
 10 
 
 29. 
 
 30. 
 
 a; a; 5a; 
 3"^4~¥ 
 
 7a; + 2 
 10 
 
 -12 = l;ra;-58. 
 
 3.x 3a; + 13 17a; 
 
 12- ^ = 
 
 / 
 
loR 
 
 OX FRACTIONAL EQUATIOXS. 
 
 180. It must next be observed that in clearing an eciuation 
 of fractions, whenever a fraction is prectded by a negative sign, 
 we must phice the result obtained by multiplying that nume- 
 rator i)i a bracket, after the removal of the denominator. 
 
 For example, we ought to proceed thus : — 
 
 ■pp.-- I x + 2 x-2 x-l 
 Ex.1. .__.= -^— .^.. 
 
 Multiply by 70, the l.c.m. of the denominators, and we get 
 14x + 28 = 35j;-70-(10x-10), 
 or 1 4x 4- 28 = 35a; - 70 - lOx + 10, 
 
 from which we shall find a; = 8. 
 Ex.2. y-?:'':_.l5±?=i. 
 
 5,7; 3rc 
 
 Multiplying by 15.6', the L.C.M. of the denominators, we get 
 51-6x-(20aj + 10) = 15.?;, 
 or 51 - 6;^ - 20a; ~ 10 = 15,^, 
 from which we shall find u;= 1. 
 
 Note. It is from want of attention to this way of treating 
 fractions preceded by a negative sign that beginners make so 
 many mistakes in the solution of equations. 
 
 1. 5a; -— = 71. 
 
 3-.x*_.2 
 
 2, X 3— O3. 
 
 5 - 2a; _ 
 
 3. — ^- + 2=a;- 
 
 7- 
 8. 
 
 9- 
 
 EXAMPLES.— Iv. 
 
 
 5x 5a; 9 3 -a; 
 ^ 2 44 2 • 
 
 
 _ 5a; -4 ^ l-2a; 
 
 5. 2. ^ =7 g . 
 
 6a;-8 
 
 , x + 2 14 3 + 5a: 
 
 4 • - - 2 • 
 5a; + 3 3-4a; x 31 9-5x 
 
 2 9 4 • 
 
 "*'2 2 
 
 8 3 
 
 a; + 5 a;-2 _a; + 9 
 
 "T^ 5~~ iV 
 
 x + l ^j4_^jf4 
 
 6 
 
 10. a;- 3 
 
 a; + 2 X 
 
 II. 
 
 8 3 
 
 oc+_5 ^^+^_^^ 
 ~7 "T" 3~* 
 
latiou 
 1 sign, 
 lume- 
 
 ^eget 
 
 we get 
 
 treating 
 Lake BO 
 
 OX FA' ACTIONAL EQUATIONS. 
 
 T09 
 
 
 13' 
 
 .r + 2 a; - 2 a; - 1 
 
 2 
 
 7 • 
 
 a; + 9 3x-6 _ 2a; 
 
 14. ^^ 5-=3-y 
 
 a;+l a;-3_a; + 30 
 '5-2 3 13 • 
 
 ,6 !:;-^|^=3.-2i. 
 
 2.1; + 7 9.>;-8_ y-ll 
 17. 7— 11— 2 ■ 
 
 18. 
 
 19- 
 
 20. 
 
 7a:^31 _ 8 4-15.T _ Ix -8 
 4 26 ~" 22 • 
 
 " 3 7 ~ 13 • 
 
 7.V + 9 S-r+l 9.»;-13 249 -9a; 
 
 8 
 
 14 
 
 % 
 
 ,-, a; a; x 10 -a; -_3 • 
 
 190. Literal equations are those in Mliich known quantities 
 are represented hy letters, usually the first in the alphabet. 
 The following are examples : — 
 
 ElXi 1. To solve the equation - 
 
 that is. 
 
 or. 
 
 therefore. 
 
 ax + hc = hx + ac, 
 
 ax-hx = ac~ he, 
 
 (a-h)x = (a — b)c, 
 
 x — c. 
 
 2x 
 
 Ex. 2. To solve the equation 
 
 that is. 
 
 or. 
 
 a^x + hx- c = h'^x + cx-d, 
 
 ci^x + bx-b^x-cx = c-d, 
 
 (a^ + b-b^-c)x = c — d, 
 
 therefore. 
 
 x= 
 
 a 
 
 c — d 
 
 Examples.— ivi. 
 
 I. ax+bx = c. 
 
 2. 2a — ca;=3c— 56a;. 
 
 3. bc + iix-d = a% -fx, 
 
 4. dm-bx = bc-ax. 
 
 5. abc — a^x = ax — a%. 
 
 6. 3a ex - 6bcd =12cdx + abc 
 
no 
 
 ON FRACTIONAL EQUATIONS. 
 
 8. — ac^ + hh + obex = ahc + cmx — ac-x + h^c — mc. 
 
 9. (a + a; + 6)(a + 6-aj) = (a + a;)(6-cc)-a6. 
 
 10. {a — x){a + x) = 2a^ + 2ax-x^. 
 
 11. (a2 + a;)2=a;''^ + 4a2 + a*. 
 
 12. {a^-x){a^ + x) = a* + 2ax-x^. 
 ax-h 
 
 13- 
 
 a:4-ac 
 
 a= . 
 
 c 
 
 3a - 6x 1 
 
 14. ax 2"~ = 2- 
 
 4ax - 26 
 
 15. 6a 5 =a;. 
 
 m Tp^ic + x^) mx^ 
 
 17. — ~ ^=mqx + - — 
 
 ' })X P 
 
 18. — 6=j — x. 
 a a 
 
 16. ax- 
 
 6x+l_a(^-l) 
 
 19. 
 
 20. 
 
 cc^ — a a -a; 2a; a 
 ks k b X 
 
 3 dh-T? _^x — ac' 
 c hx " ex ' 
 
 21. 
 
 22. 
 
 ah + x h^-x x — h ab-x 
 
 b'^ 
 
 a^b 
 
 a' 
 
 62" 
 
 3aa; — 26 ax — a ax 2 
 
 36 
 
 "26 
 
 6 3' 
 
 
 y ax X ^ 
 23. am — 6--r-H — =0. 
 
 "^ b m 
 
 2a263 
 
 24- 7:7-riA-r7: 
 25. 
 
 62a; 
 
 3a2c 3acx 6^ - 2a62a; 
 
 (a + 6) a(a + 6) a + 6 6 (a + 6) ' 
 
 ax^ 
 
 ax -. 
 , ra + — = 0. 
 b — cx c 
 
 26. -— i - 
 
 ax 
 
 ax 
 ac-^-T, 
 a 
 
 ab , - 1 
 
 27. — = bc + d + -, 
 
 ' X X 
 
 - m(a — x) 
 
 28. c = a + -- A 
 
 Sa + x 
 
 a^c 
 
 29. (a + a;)(6 + x)-a(6 + c) = -v-4-a;2. 
 
 ace (a + 6)2. a; , _, 
 
 m — r — ^^ ^ bx—ae — 36.r. 
 
 2. 
 
 I9t. In the examples already given the l.c.m. of the 
 (lenoniinators can generally be determined by inspection. 
 When compound expressions appear in the denominators, it 
 is sometimes desirable to collect the fractions into hvo^ one 
 
 12. 
 
 »3- 
 
 6 
 
 (x 
 
ON FRACTIONAL EQUATIONS 
 
 III 
 
 of the 
 pection. 
 [tors, it 
 loOj one 
 
 on each side of the equation. When tliis hua been done, we 
 can clear the equation of fractions by multiplying the nu- 
 merator on the Ujt \ij the denominator on the nr//i^, and the 
 numerator on the Hijlit by the denominator on the Ujt^ and 
 making the produ ts equal. 
 
 a 
 
 For, if T = -j, it is evident that acZ = he. 
 ' of 
 
 „ 4a; + 5 13x-6 2x-3 
 
 iliXi 
 
 4a; + 5 
 
 "~i(r 
 
 4x + 5 
 
 13x-6 
 
 2x-3_ 
 5 ~~ 
 
 5 
 
 13x-6 
 
 10 5 7u: + 4 ' 
 
 4a; + 5-(4x-6)_13x-6. 
 
 10 
 11 
 
 rx+4 
 
 10 
 
 1 3x - 6 , 
 
 "7a: + 4 ' 
 
 whence we find 
 
 11 (7a; + 4) = 10 (13a; -6)? 
 104 
 
 x = 
 
 53' 
 
 Examples.— ivii. 
 
 3 
 
 4' 
 
 5 
 II 
 
 12 
 
 »3 
 
 3a; + 7_3a; + 5 
 4^T5~4a; + 3' 
 
 03 + 6 _ X 
 
 2xT5~2a;-5' 
 
 2a; + 7 _ 4a;-l 
 "^+2'~2a;-l' 
 
 5a; - 1 _ 5x - 3 
 
 2a; + 3~2x'^* 
 
 1 2 
 
 6. — 
 7 
 
 l-5a; 
 1 
 
 + 
 
 1 -2a; 
 1 
 
 =0. 
 3 
 
 8. 
 
 a;-l 'a;fl x^-l' 
 
 + 
 
 = 0. 
 
 lo. 
 
 9 
 a; 
 
 3~i3^ 
 3a; + 2 
 
 18 
 
 5a; 
 
 :/;2 - 5a; 2 
 
 7~3- 
 
 2a;-4 ^ 
 a;-l x + 2 
 
 3a;-2 4a;-3 
 
 i(a; + 3)-i(ll-a;) = ^(a;-4)-2\(a;-3). 
 
 (a; + l)(2a; + 2)_ . _iL_?+l 
 
 (x-3)(a; + C) '' '^ a; + l x-\ 
 
 2 8 
 
 . -I -^.. : 
 
 1 — X 1 + X 
 
 (2x + 3) % 1 _ 
 
 ■ 2x+l "^3a;~^'^^' 
 
 14. 
 15. 
 
 x^ 
 
 _45_ 
 1 -»'-«• 
 
 i| 
 
 rl 
 
T12 
 
 ON FRACTIONAL EQUATIONS. 
 
 l6. 
 
 17. 
 
 4 3 
 
 + 
 
 -iiL=.^A. 
 
 cc-S 2x-16 24 3:/;- 24" 
 a;4_(4^2_20a; + 24) 
 
 x'-* — 2a; + 4 
 
 = a;2 + 2x - 4. 
 
 „ 2x^ + 2x3-23:^2 + 3]^ -, 2 ^ 
 
 18. 7, --. ; = ^^ - 4x - 3. 
 
 a;'^ + 3x - 4 
 
 (A 2\ 1 3x-(4-5x) 
 
 20. o - X I 3„ - - I = TT X \ \ 
 
 V 2 x/ 2 4 
 
 192. Equations into wliicli Decimal Fractions enter do not 
 present any serious difficulty, as may be seen from the follow- 
 ing Examples : — 
 
 Ex. 1. To solve the equation 
 
 •5x = -03x4-1 '41. 
 
 Turning the decimals into the form of Vulgar Fractions, 
 we get 
 
 5x_^x_ 141 
 
 Io"ioo'*"ioo' 
 
 Then multiplying Loth sides by 100, we get 
 
 50x = 3x+141; 
 therefore 47x = 141 ; 
 
 therefore x = 3. 
 
 Ex.2. l-2x-^i^— =-4a; + 8-9. 
 
 •5 
 
 First clear the fraction of decimals by multiplying its 
 numemtor and denominator by 100, and we get 
 
 .\ 
 
 1 
 
 18x-5 
 
 
 
 l-2x tr. -•4X + 8-9; 
 
 
 * 
 
 60 
 I2x 18x-5 4x 89 
 
 
 therefore 
 
 l(). 50 ""lO'^'lO' 
 
 
 therefore 
 
 60.r- 18x4-5 = 20x + 445; 
 
 ^ # 
 
 therefore 
 
 22x = 440; 
 
 
 therefore 
 
 x = 20. 
 
 
 10. 
 II. 
 
 13- 
 14. 
 
 15- 
 
 16. 
 
. IJWH"J ' L« '-^. 
 
 Oy FRACTIONAL EQUA 770 XS. 
 
 tt.^, 
 
 lo not 
 ollow- 
 
 ctioiis, 
 
 m its 
 
 Examples.— Iviii. 
 
 1 . 'bx - 2 = •25.r -f '2,/: - 1. 
 
 2. 3-25x - 5-1 + .V - -"ibx = 3*9 + 'bic. 
 
 3. •125x + -01x=13--2,<:+-4. 
 
 4. •3r<j + l-305.(; + -S.-; = 22-9", - -IQbx. 
 
 5. -ire- •01x+-005x-= 11-7. 
 
 6. 2 -40; = -8.'; + 8-9. 
 
 •5 
 
 7. 2'4.c- 10-75 = -25,«. 
 
 4-05 
 9. -77- + 3 8/ 5 = 4-025. 
 ^ 9a; 
 
 8. -50; + 2 -'750; = -4.0- 11. 
 
 2 + a/l ^\ . 5a; + 3 
 
 „^ 2 + a/l _\ . i 
 
 10. 2-oi(; j---( -r-2j = -o- 
 
 8-5 -2 ,1 \-'\x 
 
 11. -^ = 4 . 
 
 2 a; 4 x 
 
 8 
 
 •48./; 3 — 4j; 
 12. -g —-==im^,. 
 
 2-3x 5;); 2a;-3 a;-2 „7 
 13- ——-+.— .-—^-=--— + 2... 
 
 14. 
 
 1-5 1-25 
 24-08 1 
 
 X 
 
 9 1-8 9 
 
 + -.-04 (a; +-9) = 241 '2. 
 
 X 
 
 15. •5x + 
 
 •45a;- -75 1-2 •3.x- --6 
 
 6 
 
 •9 
 
 16. -5- 
 
 3-5.(; 24 - 3a; 
 
 X 
 
 -2 
 
 8 
 
 — •i'rn 
 
 i75x-. 
 
 17. 
 
 •15,(; + 
 
 •135X--225 -36 -090;- -18 
 
 •6 
 
 •9 
 
 193. To .<!/ie?y that a simple equation can only have one root. 
 
 Let x = a\)ii the equation, a lonii to which all equations of 
 the first degree may be reduced. 
 
 Now suppose a and (^ to he two roots of the equation. 
 Tlien, by Art. 109, 
 
 a = rt , 
 
 an( 
 
 1 
 
 P = 
 
 a 
 
 a, 
 
 P 
 
 therefore 
 
 in other words, the two supposi'd root.'? are identical. 
 Ts.A.l 
 
 H 
 
 ■» -\l 
 
 tl 
 
XIII. PROBLEMS IN FRACTIONAL 
 EQUATIONS. ^ 
 
 I 
 
 194. We shall now give a series of Easy Problems resulting 
 for the most part in Fractional Equations. 
 
 Take the following as an example of the /o>m in which such 
 Problems should be set out ^y a beginner. 
 
 " Find a number such that the sum of its third and fourth 
 parts shall be equal to 7." 
 
 Suppose X to represent the number. 
 
 Then ^ will represent the third part of the number, 
 o 
 
 and 2; ^vill represent the fourth part of the number. 
 
 Hence - + : will represent the sum of the two parts. 
 But 7 will represent the sum of the two parts. 
 Therefore 
 
 X X ^ 
 
 3 + 4 = ^- 
 
 Hence 4x + 3:£ = 84, 
 
 that is, 7a; = 84, 
 
 that is, a; = 12, 
 
 and therefore the number sought is 12. 
 
 Examples.— lix. 
 
 1. What is the number of which the half, the fourth, and 
 the fifth parts added together give as a result 95 ] 
 
 2. What is the number of which the twelfth, twentieth, 
 and fortieth parts added together give as a result 38 ? 
 
 3. What is the number of which the fourth part exceeds 
 the lifth part by 4 ] 
 
, i 
 
 and 
 leth, 
 eeds 
 
 PROBLEMS IN FRACTIONAL EQUATIONS. 115 
 
 4. What is the number of which the twenty-tifth part 
 exceeds the thirty- fifth part by 8 ? 
 
 5. Divide 60 into two such parts that a seventh part of one 
 may be equal to an eighth part (jf the oclier. 
 
 6. Divide 50 into two such parts that one-fourth of one 
 part being added to five-sixths of the other part the sum may 
 be 40. 
 
 7. Divide 100 into two such parts that if a third part of the 
 one be suljtracted from a fourth part of the other the remainder 
 may be 11. 
 
 8. What is the number which is greater than the sum of its 
 third, tenth, and twelfth parts by 58 ? ^ 
 
 9. When I have taken away from 33 the fourth, fifth, and 
 tenth parts of a certain number, the remainder is zero. What 
 is the numl)er ? 
 
 10. What is the number of which the fourth, fifth, and 
 sixth parts added together exceed the half of the number 
 by 112 \ 
 
 11. If to the sum of the half, the third, the fourth, and the 
 twelfth parts of a certain number I add 30, the sum is twice as 
 large as the original number. Find the number. 
 
 12. The difference between two numbers is 8, and the 
 quotient resulting from the division of the greater by the less 
 is 3. What are the numbers ? 
 
 13. The seventh part of a man's property is equal to his 
 whole property diminished by £1626. What is his property ? 
 
 14. The difference between two numbers is 504, and the 
 quotient resulting from the division of the greater by the less 
 is 15. What are the numbers ? 
 
 15. The sum of two numbers is 5760, and their difference 
 is equal to one-third of the greater. What are the numbers ? 
 
 16. To a certain number I add its half, and the result is as 
 much above 60 as the number itbeU' is below 65. Find the 
 number. 
 
 1:1 
 
 
 i- 'A 
 
1x6 PROBLEMS IN FRACTIONAL EQUATIONS. 
 
 17. The ditlerence between two iminbers is 20, and one- 
 seventh of the one is equal to one-third of the other. What 
 are the numbers ] 
 
 18. The sum of two numbers is 31207. On dividing one 
 by the other the quotient is found to Ije 15 and the remainder 
 1335. What are tlie numbers ? 
 
 19. The ages of two brothers amount to 27 years. On 
 dividing the age of the elder by that of the younger the quo- 
 tient is 3^. What is the age of eacli ? 
 
 20. Divide 237 into two sucli parts that one is four-fifths of 
 the other. 
 
 21. Divide i>'180() between A and i>\ so that ^'s share may 
 b(ftwo-sevenths of yJ's share. 
 
 22. Divide 46 into two such parts that the sum of the 
 quotients obtained by dividing one j^art by 7 and the other by 
 3 may be equal to 10. 
 
 23. Divide the number a into two such parts that the sum 
 of the quotients obtained by dividing one part by in and the 
 other by n may be equal to h. 
 
 24. The sum of tAVO nundDers is a, and their difference is h. 
 Find the numbers. 
 
 V^ 
 
 25. On multii>l}'ing a certain number by 4 and dividing 
 the product by 3, 1 obtain 24. W^hat is the numljtr ? 
 
 26. Divide £864 between y1, T>. and 6^ so that A gets — 
 
 of wliat B gets, and Cs share is equtd to the sum of the shares 
 of A and B. 
 
 Tj. A man leaves the half of his property to his wife, a 
 sixth part to each of his two children, a twelfth part to liis 
 brother, and the rest, amounting to i,'600, to charitable uses. 
 What was the amount of his property ? 
 
 28. Find two numbers, of which the sum is 70, such that 
 the first divided by the second gives 2 as a quotient and 1 as 
 a remainder. 
 
 29. Find two numbers of which the difference is 25, such 
 that the second divided by the first gives 4 as a quotient and 
 4 as a remainder. 
 
 
 hour 
 
a I 
 
 PROBLEMS IN FKA C TIONA L EQUA TIONS. 1 1 7 
 
 his 
 
 (ses. 
 
 Ihat 
 as 
 
 licli 
 lud 
 
 30. Divide the numher 208 into two parts such that the 
 sum of the fourtli of the Ljreater and the tliird of the less is 
 less by 4 than four times the difference between the two part^.. 
 
 31. There are thirteen days between division of term and 
 the end of the first two-thirds of the term. How many days 
 are there in the term ? 
 
 32. Out of a cask of wine of which a fifth part had leaked 
 away 10 gallons were drawn, and then the cask was two-thirds 
 full. How much did it hold ? 
 
 33. The sum of the ages of a father and son is half what it 
 will be in 25 years : the difference is one-third what the sum 
 will be in 20 years. Find the respective ages. 
 
 34. A mother is 70 years old, her daughter is exactly half 
 that age. How many years have passed since the mother was 
 3J times the age of the daughter ? 
 
 35. A is 72, and B is two-thirds of that age. How long is 
 it since A was 5 times as old as 5 ? • 
 
 Note T. If a man can do a piece of work in x hours, the 
 part of the work which he can do in one hour will be repre- 
 sented by -. • 
 
 Thus if A can reap a field in 12 hours, he will reap in one 
 
 hour r^: of the field. 
 
 12 . * . 
 
 Ex. A can do a piece of work in 5 days, and B can do it 
 in 12 days. How long will A and B working together take to 
 do the work \ 
 
 Let X represent the number of da}s A and B will take. 
 Then - will represent the part of the work they do daily. 
 
 Now - represents the part A does daily, 
 and -.-o represents the part B does daily. 
 
 
Il8 PROBLEMS IN FRACTIONAL EQUATIONS. 
 
 4y 
 
 Hence p + y^ will represent the part A and B do daily. 
 Consequently ^ + th = "• 
 
 Hence 
 
 or 
 
 12 X 
 
 12x + 5a; = 60, 
 17a;=60; 
 60 
 17* 
 9 
 
 a; = 
 
 That is, they will do the work in 3— days. 
 
 36. A can do a piece of work in 2 days. B can do it in 3 
 days. In what time will they do it if they work together ? 
 
 37. A can do a piece of work in 50 Jays, B in 60 days, 
 and G in 75 days. In what time will they do it all working 
 together? . 
 
 38. A and B together finish a work in 12 days ; A and G 
 in 15 days ; B and G in 20 days. In what time will they 
 finish it all working together ? 
 
 39. A and B can do a piece of work in 4 hours ; A and G 
 
 3 1 
 
 in 3^ hours ; B and G in 5= hours. In what time can A do 
 
 it alone ? 
 
 40. A can do a piece of work in 2- days, B in 3., days, 
 
 3 
 
 and G in 3^ days. In what time will they do it all working 
 
 togetlier? 
 
 3 . 
 
 41. A does v^ of a piece of work in 10 days. He then calls 
 
 D 
 
 in J5, and they finish the work in 3 days. How long would B 
 take to do one-third of the work hy himself ? 
 
 Note II. If a tap can fill a vessel in x hours, the part of 
 the vessel filled by it in one hour will be represented by . 
 
 Ex. Three taps running separately will fill a vessel in 20, 
 30, and 40 minutes respectively. In what time will they fill it 
 when they all run at the same time \ 
 
 ■:*l.' 
 
PROBLEMS IN FA' A CTIONAL EQUA TIONS. 1 19 
 
 Let % represent the number of minutes they will take. 
 
 Then - will represent the part of the vessel filled in 1 
 minute. 
 
 Now l^ n 
 
 (presents the part filled 
 
 by the 
 
 1 
 
 
 * 
 
 30" 
 
 
 
 1 
 
 
 
 40 •• 
 
 
 
 Hence 
 
 111 
 20'*'iBT)'^4()' 
 
 I 
 
 or, multiply] 
 
 ing both sides by 120x, 
 
 
 
 6x + 4x + 3x = 
 
 •120, 
 
 that is, 
 
 13x = 
 
 120; 
 
 
 /. x = 
 
 120 
 13* 
 
 second , 
 third.. 
 
 L'king 
 
 ,rt of 
 
 Hence they will take 9 y^ minutes to fill the vessel. 
 
 42. A vessel can be filled by two pipes, running separately, 
 in 3 hours and 4 hours respectively. In what time will it be 
 filled when both run at the same time ? 
 
 43. A vessel may be filled by three different pipes : by the 
 
 first in I5 hours, by the second in 3- hours, and by the third 
 
 in 5 hours. In what time will the vessel be filled when all 
 three pipes are opened at once ? 
 
 44. A bath is filled by a pipe hi 40 minutes. It is emptied 
 by a waste-pipe in an iiour. In what time will the bath be 
 full if both pipes are opened at once ? 
 
 45. If three pipes fill a vessel in a, h, c minutes running 
 separately, in what time will the vessel be filled when all three 
 are opened at once ? 
 
 i 
 
 's!8 
 
 i 
 
120 PI^OnLEMS IiV rA\l CTIOXA L EQl 'A TJOXS. 
 
 > 
 
 46. A vessel coiitainiiig 755 . gallons can he filled by three 
 pipes. The first lels in 12 gallon!^ in :i nn"iiutes, the second 
 
 15.J gallons in 2 minutes, the tliird 17 gallons in 3 minutes : 
 
 in what time will the vessel be filled by the three pipes all 
 running together ? 
 
 47. A vessel can be filled in 15 minutes by three pipes, 
 one of which lets in 10 gallons more and the other 4 gallons 
 less than the third each minute. The cistern holds 2400 gallons. 
 How much comes through each pipe in a minute \ 
 
 Note III. In questions involving distance travelled over in 
 a certain time at a certain rate, it is to be observed that 
 
 Distance 
 
 Rate 
 
 :Time. 
 
 That is, if I travel 20 miles at the rate of 5 miles an hour, 
 
 20 
 number of honrs I take = -, . 
 
 o . 
 
 Ex. A and b set out, one from Newmarket and the other 
 from Cambridge, at the same time. The distance between the 
 towns is 13 miles. A walks 4 miles an hour, and B 3 miles an 
 hour. Where will they meet ? ' 
 
 Let X represent their distance from Cambridge when they 
 meet. 
 
 Then 13 -.t will represent their distance from Newmarket. 
 Then ■.^=time in hours that 7i> has been walking, ,; 
 
 13-35 
 
 A 
 
 And since both liave been walking the ^ianie time, 
 
 ' 
 
 
 X V^-x 
 
 / 
 
 
 
 
 3~ 4 ' 
 
 
 
 
 or 
 
 4.c = 39-3a;, 
 
 - 
 
 
 ' 
 
 or 
 
 7a: = 39; 
 
 ■ , 
 
 
 ^- 
 
 # 
 
 39 
 
 ••• -=y • •• 
 
 
 • 
 
 48. 
 
 49. 
 
 of 2i 
 
 50. 
 
 54. 
 
PROBL RMS TN FRACTION A L EQUA TIONS. 12 1 ^ 
 
 That is, they meet at a distance of 5- miles from Cam- 
 bridge. 
 
 48. A person starts from Ely to walk to C\'mil)ridge (which 
 
 4 
 is distant 16 miles) at tlie rate of 4^ miles an hour, at the 
 
 same time that anotlicr person leaves Cambridge for Ely 
 walking at the rate of a mile in 18 minutes. AVhere will they 
 meet? • 
 
 49. A person walked to the toj) of a mountain at the rate 
 of 2-- miles an hour, and down the same way at the rate of 
 
 3^ miles an hour, and was out 5 hours. How far did he walk 
 2 
 
 altogether ? ' 
 
 50. A man walks a miles in h hours. Write down 
 
 (1) The number of niiles he will walk in c hours. 
 
 (2) The number of hours he will be walking d miles. 
 
 51. A steamer which started from a certain place is fol- 
 lowed after 2 days by another steamer on the same line. The 
 first goes 244 miles a day, and the second 286 miles a day. In 
 ho\s' many days will the second overtake the first 1 
 
 52. A messenger who goes 31- miles in 5 hours is followed 
 
 after 8 hours by another who goes 22- miles in 3 hours. When 
 will the second overtake the first ? 
 
 53. Two men set out to walk, one from Cambridge to 
 London, the other from London to Cambridge, a distance of 
 60 miles. The former walks at the rate of 4 miles, the latter 
 
 3 
 
 at the rate of 3- miles an hour. At what distance from Cam- 
 4 
 
 bridge will the}'" meet ? " 
 
 54. A sets out and travels at the rate of 7 miles in 5 hours. 
 Eight hours afterwards B sets out from tlie same place, and 
 travels along the same road at the rate of 5 miles in 3 hours. 
 After what time will B overtake A ? 
 
 ill 
 
 ^ 
 
 I 
 
122 PROBLEMS IN FRACTIONAL EQUATIONS. 
 
 Note IV. Tn problems relating to clocks the chief point to 
 he noticed is that the minute-hund moves 12 times as fast as 
 the hour-hand. 
 
 The following examples should be carefully studied. 
 
 Find the time between 3 and 4 o'clock when the hands of a 
 clock are 
 
 (1) Opposite to each other. 
 
 (2) At right angles to each other. 
 
 (3) Coincident. 
 
 (1) Let ON represent the position of the minute-hand in 
 Fig. I. 
 
 0I> represents the position of the hour-hand in Fig. I. 
 M marks the 12 o'clock point. 
 T 3 o'clock 
 
 The lines OM^ OT represent the position of the hands at 
 3 o'clock. 
 
 Now suppose the time to be x minutes past 3. 
 
 Then the minute-hand has since 3 o'clock moved over the 
 arc MDN. 
 
 And the hour-hand has since 3 o'clock moved over the 
 arc TD. 
 
 Hence arc MDN= twelve times arc TJ), 
 If then we represent MDN by x, 
 we shall represent TD by y^. 
 
 Also W( shall represent MT by 15, 
 
 and DN h\ 30, 
 
 
PROF/.RnrS IN" FPACTIOyAL F.QUATTOh'S. 123 
 
 Now 
 
 MDN^MT^TD^DN, 
 
 
 that is, 
 
 12 ' 
 
 or 12x=180 + a; + 3GO 
 or lla; = 54(); 
 540 
 
 
 Hence the time is 49pj minutes past 3. 
 
 (2) In Fig. II. the description given of the state of the 
 dock in Fig. I. api)lies, except that UN will he represented hy 
 15 instead of 30. 
 
 Now suppose the time to be x minutes past 3. 
 
 Then since 
 
 MT)N=MT+TD-\-DN, 
 
 X 
 
 from which we get 
 
 a; = 15+yx + 15. 
 
 x = 
 
 8 
 
 360 
 11' 
 
 that is, the time is 32-- minutes past 3. 
 
 (3) In Fig. III. the hands are both in the position ON. 
 Now suppose the time to be x minutes past 3. 
 Then since 
 
 MN=MT+TN, 
 
 ■1!- ^ 
 
 ^ = 15^-^2, 
 
 or 12x = 180 4-x, 
 180 
 
 or x = 
 
 11 
 
 .4 
 
 that is, the time is 16 — minutes past 3, 
 
 55. At what time are the hands of a watch opposite to 
 each other, . 
 
 (1) Between 1 and 2, 
 
 ' (2) Between 4 and 5, 
 
 (3) Between 8 and 9 ? 
 
 
 
1 24 PROBLEMS IN FRA CTTONAL EQUA TIONS. 
 
 ¥.■, 
 
 I'M 
 
 H9: 
 
 iii: 
 
 56. At wliat time are the hands of u watch ut riglit angles 
 to each otlier, " 
 
 (1) Between 2 and 3. 
 
 (2) Between 4 and 5, 
 
 (3) Between 7 and 8 ? 
 
 57. At what time are the liands of a watch together, 
 
 (1) Between 3 and 4, 
 
 (2) Between 8 and 7, 
 
 (3) Between 9 and 10 ? 
 
 58. A person buys a certain number of apples at the rate 
 of five for twopence. He sells half of them at two a penny, 
 and the remaining half at three a penny, and clears a penny 
 by the transaction. How many does he buy ? 
 
 59. A man gives away half a sovereign more than half as 
 many sovereigns as he has : and again half a sovereign more 
 than half the sovereigns then remaining to him, and now has 
 nothiiii^ left. How much had he at first 1 
 
 2a + 7i 
 3?i + 69a 
 
 'o 
 
 60. What must be the value ol n in order that - 
 
 may be ec|ual to -^ when a is - ? 
 
 61. A body of troops retreating before the enemy, from 
 which it is at a certain time 25 miles distant, marches 18 miles 
 a day. Tlie enemy pursues it at the rate of 23 miles a day, 
 but is first a day later in starting, then ter 2 days is forced 
 to halt for one day to repair a bridge, and lliis they have to do 
 again after two days' more marcliing. After how many days 
 from the beginniLg of the retreat will the retreating force be 
 o/e-rtalven? 
 
 62. A person, after p.aying an income-tax of sixpence in the 
 pound, gave away one-thirteenth of his remaining income, and 
 had .£540 left. What was his original income ? 
 
 63. From a sum of money I take away ^50 more than the 
 lialf, then from tlie remainder £'M^ more than the fifth, then 
 from the e^cond remainder .£20 more than the fourth j)art : 
 and at last only £10 remains. What was the original sum 1 
 
 69. 
 
n 
 
 64. I bQiight a certain number of eggs at 2 a penny, and 
 the same number at 3 a penny. I sold them at 5 for twopence, 
 and lost a penny. How many eggs did I buy ? 
 
 65. A cistern, holding 1200 gallons, is filled by 3 pipes' 
 Ay B, G in 24 minutes. The pipe A requires 30 minutes moio 
 than G to fill the cistern, and 10 gallons leys run through i)er 
 minute than through A and J^'^ogether. What time would 
 each pipe take to fill the cistern by itself? 
 
 '^ 66. A, B, and G drink a barrel of beer in 24 days. A and 
 
 4 
 B drink f:rds of what G does, and B drinks twice as much as A. 
 o 
 
 In what time would each separately drink the cask ? 
 
 67. A and B shoot by turns at a target. A puts 7 buUets 
 out of 12 into the centre, and B puts in 9 out of 12. Between 
 them they put in 32 bullets. How many shots did each fire ? 
 
 68. A farmer sold at market 100 head of stock, horses, 
 oxen, and sheep, selling two oxen for every horse. He obtained 
 on the sale ^2, 7«. a head. If he sold the horses, oxen, and 
 sheep at the respective prices ^22, ^12, lO.s-., and £1, 10s., how 
 rjany horses, oxen, and sheep respectively did he sell ? 
 
 69. In a Euclid paper A gets 160 marks, and 2> just passes. 
 A gets full marks for book-work, and twice as many marks 
 for riders as B gets altogether. Also B, sending answers 
 to all the questions, gets no marks for riders and half marks 
 
 for book- work. Supposing it necessary to get y of full marka 
 
 in order to pass, find the. number of marks which the paper 
 carries. 
 
 70. It is between 2 and 3 o'clock, but a person looking at 
 the clock and mistaking the hour-haad for the riuute-hand, 
 fancies that the time of day is 55 minutes earlier than tho 
 reality. What is the true time ] 
 
 71. An army in a defeat loses one-sixth of its number in 
 killed and wounded, and 4000 prisoners. It is reinforced by 
 3000 men, but retreats, losing a fourth of its number in doing 
 so. There renuiiu 18000 men. What was the ori-iinal forced 
 
 72. The national debt of a country was incnased by one- 
 fourth in a time of war. During twenty years vl' peace whicli 
 
 « 
 
 
126 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 followed .£25,000,000 was paid off, and at the end of that time 
 the interest was reduced from 4^ to 4 per cent. It was then 
 found that the interest was the same in amount as before the 
 war. What was the amount of the debt before the war ? 
 
 73. An artesian well supplies a brewery. The consump- 
 tion of water goes on each week-day from 3 a.m. to 6 p.m. at 
 double the rate at which the water flows into the Avell. If 
 the V ell contained 2250 gallons when the consumption began 
 on Monday morning, and if^was just emptied wlien the con- 
 sumption ceased in the evening of the next Thursday but one, 
 what is the rate of the influx of water into the well in gallons 
 per hour ? 
 
 XIV. ON MISCELLANEOUS FRACTIONS. 
 
 195. In this Chapter we shall treat of various matters con- 
 nected with Fractions, so as to exhibit the mode of applying 
 the elementary rules to the simplification of expressions of a 
 more complicated kind than those which have hitherto been 
 discussed. 
 
 196. The attention of the student must first be directed 
 to a point in which the notation of Algebra diff'ers from that of 
 Arithmetic, namely when a whole number and a fraction stand 
 side by side with no sign betireen them. 
 
 3 3 
 
 Thus in Arithmetic 2- stands for the sum of 2 and =. 
 
 V V 
 
 But in Algebra a;- stands for the product of x and -. 
 
 So in Algebra 3— • stands for the product of 3 and - - ; 
 c c 
 
 that is, 3 
 
 a f 6 3a -f- 36 
 
 
ON MISCELLANEOUS FRACTIONS. 
 
 1^7 
 
 Examples.— Ix. 
 
 Simplify the following fractions : 
 
 I. a + aj + S-. 
 
 a^ -Vax ^x-ob 
 
 X' 
 
 2 -— . 
 
 X 
 
 3. ^--1 + 2 J-, 
 a + /; a^ — 52 
 
 107. A fraction of which the Numerator or Denominator 
 
 is itself a fraction, is called a Complex Fraction. 
 y X 
 
 Thus -, i and - are complex fractions. 
 a a m 
 
 t 6 n 
 
 A Fraction whose terms are whole numbers is called a 
 Simple Fraction. 
 
 All Complex Fractions may be reduced to Simple Fractions 
 by the processes already described. We may take the follow- 
 ing Examples : 
 
 ^ ^ m~b~ n~b m~bm' 
 
 (3) 
 
 a c 
 h 
 m 
 n q 
 
 h d (a c\ /m p\_ ad-hc ,'mq-np 
 ^ ^ m p~\b d/ ' \n q/~ bd ' nq 
 
 _ad — be nq _ 7iq (ad - be) 
 "~ bd mq - np bd {mq — np)' 
 
 x+l^ 
 
 X 
 
 1+x^ X _a;(l+^_ 
 '""l'"''^l~^l+x ~* 
 
128 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 (4) 1-^"1+^^(J___1,V/ ^_^_1 \ 
 ^' X 1 Vl-a; \ + xJ '\l-x \+x/ 
 
 \-x \+x 
 
 2a; l-x^ 
 
 r^'i 
 
 \-x 
 
 ^_ 
 
 Y-x^^V+^^~\+x^' 
 
 (5) 
 
 3 
 
 3 
 
 3 
 
 1 + 
 
 3 
 
 1 + 
 
 3_ 
 
 \-x 
 
 1 + .-J- i+^Hi^) 
 
 l-a; + 3 ^l-x + 3 
 
 1-35 
 
 3(4-a;) ^12- 
 
 , 3-3ic 
 
 1 + - 
 
 4-a; 
 
 3x 
 ' 4-a; + 3-"3^"'4-a; + 3-Ta;~ 7 -4a;* 
 4 — ic 
 
 Ex AMPLES.— ixi. 
 
 Simplify the following expressions : 
 
 I. 
 
 5 X 
 
 2. 
 
 2/ 
 
 S-) 
 
 y a? 
 a;-!/' 
 
 5 + 03 + 
 
 l-a;2 
 
 1 + 
 
 r 
 
 X 
 
 X'' 
 
 a; + 
 
 x" 
 
 K-D' 
 
 r 
 
 2-a; + -2 
 
 "■ITT' 
 
 X 
 
 X 
 
 a a 
 
 1 + - 
 
 X 
 
 8. 
 
 x+a x-a 
 2x 
 
 2x 
 
 x^-a^ 
 
 x^ + 
 
 x+y x-y 
 II. y i" . 
 
 '±-y _ 5_ti 
 
 x+y x-y 
 
 l-T-xS 
 
 198. 
 tions eq 
 
 Split 
 foUowin 
 
 I. 
 
 2. 
 
 aM 
 
 a2?„ 
 
 a;-^ 
 
 199. 
 
 fraction 
 
 Thus 
 
 200. 
 
 a . 
 
 r 13 a 
 b 
 
 [sj 
 
ON MISCELLANEOUS FRACTIONS. 
 
 12. 
 
 13. 
 
 1 
 
 1 +a: + a;^ 
 
 a + h b 
 a + 6 
 
 - + /: 
 a 
 
 14. 
 
 IS- 
 
 129 
 
 2m 
 
 -3 + 
 
 1 
 
 m 
 
 2w-l 
 
 
 
 m 
 
 
 ab 
 
 1 
 
 ■-- + 
 ac 
 
 1 
 
 be 
 
 a2- 
 
 ■{b + cy^' 
 
 
 ab 
 
 
 198. Any fraction may be split up into a number of frac- 
 tions eq^ual to the number of terms in its numerator. Tliua 
 
 x^ + x^ + x + l x^ x"^ X 1 
 «*~ x^'^x^'^x^'^^' 
 
 1111 
 
 X x^ rr x^ 
 
 Examples.— ixii. 
 
 Split up into fou^ fractions, each in its lowest terms, the 
 
 following fractions : 
 
 I. 
 
 2. 
 
 a^ + 3ft3 + 2a^ + 5(1 
 
 
 2a4 
 
 • 
 
 It^bi 
 
 ! + ab-d + abc^ 
 
 + bcd^ 
 
 
 abed 
 
 
 x^- 
 
 3a;2i/ + 3,T?/- 
 
 ■y^ 
 
 x^y^ 
 
 4- 
 
 5- 
 6. 
 
 9(1^-1 2^2 + 6a -3 
 108 
 
 dpcp'S 
 
 10a;-^- 25a;2 + 75a;-125 
 1000 • 
 
 .rr2 
 
 
 199. The quotient obtained by dividing the unit by any 
 fraction of that unit is called The Reciprocal of that fraction. 
 
 a 
 
 Thus -, that is, , ia the Reciprocal of p 
 
 ';!! 
 
 200. We have shewn in Art. 158, that the fraction symbol 
 
 a . 
 
 J- is a proper representative of tlje Division of a by 6. In 
 
 Cs.A.] 
 
 I 
 
I30 
 
 ON MISCELLANEOUS FACTIONS. 
 
 Chapter IV. we treated of cases of division in which tlie divisor 
 is contained an exact number of times in the dividend. We 
 now proceed to treat of cases in which the divisor is not con- 
 tained exactly in the dividend, and to shew the proper method 
 of representing the Quotient in such cases. 
 
 Suppose we have to divide 1 by \-a. We may at once 
 
 represent the result by the fraction . But we may 
 
 actually perform the operation of division in the following 
 way. 
 
 X-a) 1 (1 +cH-a2 + a34-.„ 
 •^ l-« 
 
 a 
 
 
 
 
 a- 
 
 -a2 
 
 
 
 
 a2 
 
 
 
 a2- 
 
 -a3 
 
 
 
 
 a3 
 
 
 
 
 a3- 
 
 -a* 
 
 a* 
 
 « 
 
 The Quotient in this case is interminable. We may carry 
 on the operation to any extent, but an exact and terminable 
 Quotient we shall never find. It is clear, however, that the 
 terms of the Quotient are formed by a certain law, and such 
 a succession of terms is called a Series. If, as in the case 
 before us, the series may be indefinitely extended, it is called 
 an Infinite Series. 
 
 If we wish to express in a concise ioftti the result of thft 
 operation, we may stop at any term of the quotient and write 
 the result in the following way. 
 
 "=1+1 » 
 
 I -a I -a 
 
 1 , a2 
 
 I -a I -a' 
 
 = = 1 +a + a^ + z. , 
 
 -^ = l + a + a^ + a^ "^^ 
 
 alwa_^ 
 whic 
 divisi 
 divisc 
 
 l-a 
 
 I -a' 
 
e divisor 
 id. We 
 not con- 
 method 
 
 at once 
 we may 
 jllowing 
 
 ly carry 
 minable 
 that the 
 [id such 
 le case 
 called 
 
 of tbft 
 d write 
 
 ON MIS^LLANEOUS FRACTIONS. 
 
 13J 
 
 always being careful to attach to that term of the quotient, at 
 which we intend to stop, the remainder at that point of the 
 division, placed as the numerator of a fraction of which the 
 divisor is the denominator. 
 
 Examples.— Ixiii. 
 
 Carry on each of the following divisions to 5 terms in the 
 quotient. 
 
 1. 2byl+a. 7. 
 
 2. m by m + 2. 8. 
 
 3. a- 6 by a + 6. 9. 
 
 4. a^ + x^ by a^ - x\ i o. 
 
 5. ace by a- X. 11. 
 
 6. 6 by a + x. 12. 
 
 13. If th>- divisor be x-a, the quotient x^-^ax^ and the 
 remainder AaJ^, what is the dividend ? 
 
 1 4. If the divisor be m - 5, the quotient m^ + bim? + 15m + 34, 
 and the remainder 75, what is the dividend 1 
 
 1 by 1 + 2ic - 2a;2. 
 1 + cc by 1 - a: + a;2. 
 1 + 6 bv 1 - 26. 
 a;^ - 6^ by 05 + b. 
 a'^hy X- h. 
 a^ by (a 4- a;)''. 
 
 201. If we are required to multiply such an expression as 
 
 x^ X 1 , X 1 
 Y + 3 + 4 ^y 2~3' 
 
 we may multiply each term of the former by each term of the 
 latter, and combine the results by the ordinary methods of 
 addition and subtraction of fractious, thus 
 
 X' X i 
 
 y+3'^4 
 
 X 1 
 
 2 3 
 
 \ 
 
 X^ X^ X 
 
 "4 '^y^S 
 
 
 x'^ X 
 6 9 
 
 1 
 
 12 
 
 •JC* X 
 
 ^ "*'72 
 
 1 
 
 ONTAl<lOCOllt6tll^>^3lJUAIIUl; 
 
 1 
 
 i 
 
 11' 
 
 
132 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 Or we may first recliice the multiplicand and the multiplier 
 to single fractions and proceed in tlie following way : 
 
 / x^ X 1 
 (-2- + ..+ 
 
 3 4 
 
 Hl-\) 
 
 _6x^ + 4a;4-3 3a; -2_18xH cc-B " 
 12 ^ 6 ~ 72 
 
 72 "^72 72~ 4^"*" 72 12' 
 This latter process will be found the simpler by a beginner. 
 
 EXAMPLES.—lXiV. 
 
 Multiply 
 
 X'' 
 
 X 1 
 
 X 1 
 
 '' ¥ + 2 + 5^^3 + 4- 
 
 a' 
 
 a 1 
 
 a 1 
 
 ^' y~6"^3 ^ 4~5' 
 ■ . 11, 1 
 
 3. ^+^+- + '^i^'y^-~' 
 
 7. l+- + -2-byl 
 
 4. a;2-H-l byx2+l+i-. 
 
 X^ '' X^ 
 
 L 1 h i _ L 
 
 5- n2 + 7,2 "^y n'i A2 • 
 
 a^ ^¥ 
 
 a^ 62 
 
 /; 1 I 1 v 1 1 1 
 
 0. - - - + - by -+,+-. 
 
 a c '' a c 
 b &2 
 
 ■- + -0-. 
 
 8. 1 + 2^ +- ^a;^ by 1 - ^a; + ga;2 - --x\ 
 5 3 7,211 
 
 2ic2 ' X 3 
 
 X 2' 
 
 202. If we have to divide such an expression as 
 
 x^ + 3x + - + -^- 
 
 X X"^ 
 
 1 
 
 by a; + -, we may proceed as in the division of whole numbers, 
 
 carefully observing that the order of descending powers of x 
 is 
 
 111 
 
 /y»d /^<o /y» __ 
 
 ,x , X , ^, ^, ^2 ) ^3 
 
 
x" 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 133 
 
 Any isolated digits, ao 1, 2, 3 ... will stand between x 
 
 and -. 
 
 X 
 
 Thus the expressioii 
 
 2jd rj^i 
 
 X 
 
 arranged according to descending powers of x, will stand thus, 
 
 ic5 + 3a;2 + 5a; + 4 + _ + _ + 
 
 X X' x^ 
 
 The reason for this arrangement will be given in the Chapter 
 on the Theory of Indices. 
 
 Ex. 
 
 x + - ja;3 + 3a; + - + -„( x^ + ^^--^ 
 x/ X x' ^ x^ 
 
 x^-^- X 
 
 2x + ? 
 
 X 
 
 
 2x + - 
 
 X 
 
 • 
 
 1 
 
 - + 
 
 X 
 
 1 
 
 - U 
 
 X 
 
 1 
 
 x^ 
 
 II 
 
 
 Or we nfay proceed in the follov/ing way, which will be 
 found simpler by the beginner. 
 
 \ X x^/ \ xl - 
 
 a;6 + 3a;* + 3a;2+i a;2 + l 
 
 x^ 
 
 OS 
 
 a;« + 3a;4 + 3x2 + 1 x 
 
 : X 
 
 X- 
 
 x'-^ + l 
 
 x^ x^ x^ x^ 3;^ 
 
 
134 
 
 OlSr MISCELLANEOUS FRACTIONS. 
 
 |i 
 
 EXAMPLES.— IXV. 
 
 Divide : 
 
 « 2 It. 1 
 
 3. m3 + -obym + -. 
 
 4. c^--^.hjc--j. 
 
 dp 
 
 d' 
 
 OC II CC 1J 
 
 5. -2 + 2 + sTt)y- + ^. 
 
 
 7. -„-^„-3- + 3^by — ^. 
 ' y^ x^ y X y X 
 
 8. 
 
 3a^ 
 
 77 , 43 
 
 33 
 
 cc^ 
 
 4a;H— x3-— -x3-— x + 27by 2--a; + 3. 
 
 a^ 6^ , a b 
 9. r5 + -^byT + -. 
 
 1 1 1_^ |. 1 1 1 
 a^ ¥ c^ abc ah c' 
 
 203. In dealing with expressions involving Decimal Frac- 
 tions two methods may be adopted, aa will be seen from the 
 following example. 
 
 Multiply 'lx — '2y by "030;+ '4y. 
 
 We may proceed thus, applying the Rules for Multiplication, 
 Addition, and Subtraction of Decimals. 
 
 •lic--2y 
 '03x + '4y 
 
 •003x2 --006x1/ 
 
 + •04 xy-'08y^ 
 
 •003x2 + -034x1/ -'08^2 
 Or thus, 
 
 __x — 2y 3x + 40i/ 
 
 io~"^~roor~ 
 
 ^3x2 + 34xy-80y2 
 
 1000 
 = -003x2 + •034x?/ - -OSt/^. 
 
 The latter method will be found the simpler for a beginner. 
 
 Ml 
 
 I. 
 
 3- 
 
 S- 
 
 6. 
 

 ON MISCELLANEOUS FRACTIONS. 
 
 '35 
 
 Examples.— ixvi. 
 
 2. -OSa; + 7 by '2^ - 3, 
 
 4. 4-3a; + 5 -21/ by •04x - '06!/. 
 
 Multiply : 
 I. -la; - '3 by •5x + '07, 
 3. •3x--2//by •4x + -7i/, 
 
 5. Find the value of 
 
 a' - 6H c^ + 3a6c when a = -03, & = -1, and c = -07. 
 
 6. Find the value of 
 
 flj^ - 3ax2 + 3a2x - a^ when a; = '7 and a — •03. 
 
 204. When any expression E is put in a form of which / is 
 
 K 
 a factor, then -j is the other factor. 
 
 Thus 
 
 So 
 
 dh-\-ac-v'bc=^ahc 
 
 and 
 
 a& + ac -;■ he 
 ahc 
 
 =,..(n.?2+2i). 
 
 Examples.— ixvii. 
 
 1. Write in factors, one of which is a^x, the series 
 
 aiX + a^x^ + a^xi^ + a^x^ + . . . 
 
 2. Write in factors, one of wliich is xyz, the expression 
 
 xy-xz + yz. 
 
 3. Write in factors, one of which is x^, the expression 
 
 x^ + xy + y"^. 
 
 4. Write in factors, one of which is a + 6, the expression 
 
 {a + b,'^-c{a + b)^-d{a + b) + e. 
 
 M • 
 
 I ( 
 
i 
 
 136 
 
 ON MISCELLANEOUS FRACTIONS, 
 
 205. We shall now give two exainplea of a process by 
 whicli, wlien certain fractions are known to be equal, other 
 relations between the quantities involved in them may be 
 determined. 
 
 This process will be found of great use in a later part of 
 the subject, and the student is advised to pay particular 
 attention to it. 
 
 -.a c 
 
 (1) If , = ,, shew that 
 
 a — b c — d' 
 
 Let 
 Then 
 
 = X. 
 
 X; 
 
 • Now 
 and 
 
 h 
 
 c 
 d' 
 
 and c = \d. 
 a + h ^\b + h _h{\+l)_\ + l' 
 a-h~ \h - h ~ h{\- 1) ~ x" P 
 
 c + d ^\d + d_d,{\+^_\ + l 
 c-~d \d -d ~ d {\^) ~ X~-T' 
 
 a + h 
 
 c + rf 
 
 A + 1 
 
 Hence ~-^ and — -^ being each equal to ~^ are equal to 
 one another. 
 
 (2) If 
 
 Let 
 
 ^-T=F:7=:r-'^' '^"^ thatm + n + r=0. 
 ^^ -X ' 
 
 
 then 
 
 -»?. 
 
 n = \h-\c, 
 r=Xc — \a; 
 + n + r=\a-\b + \h ~\c + \c- \a=0. 
 
ON MISCELLANEOUS FRACTIONS. 
 
 137 
 
 Examples.— Ixviii. 
 
 a c 
 
 I. If T = 7 prove the following relations : 
 . . a-h c-d 
 
 (0 -7r-=-d- 
 
 (2) 
 (3) 
 
 (4) 
 
 6 
 
 a _ c 
 
 a-\-b~ c + d' 
 
 2a _3c^ 
 
 4rt - 56 ~ 4c - 5(Z' 
 
 a^ - b'^ c^ - d^' 
 
 . ^n + h _ 8c + d_ 
 ^^^ 4a + 7b~~4c + 7oC 
 
 .,. a^-h'^_ab 
 
 2. If r = T = 
 
 a-o b-c c-a 
 
 (? - d^ cd' 
 
 najf^6_13a + 6 
 iic+'d'Uc + d' 
 
 a^-ab + b '^ ^ c2-cd + d> 
 , then l + mi-n=0. 
 
 (7) 
 
 (8) 
 
 ace 
 
 j^a + b b + c c + a ., , , 
 
 1. It —V— = = , prove that a = b=c. 
 
 I> a a ^ 
 
 5. If^^ = ~2 = ^^ shew that 
 
 6i~26;+362 + 468' 
 
 T, -y, 3: be in descending order of magnitude, shew 
 
 , -rn a c e 
 
 D. II Tj 
 
 that 
 
 a + c + e 
 
 a 
 
 7 ZTZf^^ ^^^® ^^^^^^ A ^^^ greater than ^. 
 a'i_'^2 „i .i,.^4xi + 5i/, 4x2 + 5i/i 
 
 7. If - = - ^, shew that 
 
 2/1 2/a 
 
 a c 
 
 7xi + diji 7x^ + 9i/a 
 
 8. If r — -i, shew that 
 
 If T = ^, shew that 
 d 
 
 n^ -h no a 
 
 6-62 
 
 C-' + cd cd - d^ 
 
 7a + b _ 7c^+ d 
 3a + 56 ~ 3c + 5ci* 
 
 M 
 
138 
 
 ON- MISCELLANEOUS ER ACTIONS. 
 
 a 
 
 lo. If ^ be a 2->roper fraction, shew that ftf ig greater 
 
 than p c being a positive quantity. 
 
 h + c 
 
 a 
 
 II. If ^ be an improper fraction, shew that ?-^- is less 
 
 h + c 
 
 than ^, c being a positive quantitj^ 
 
 206. We shall now give a series of examples in the working 
 of which most of the processes connected with fractions will 
 be introduced. 
 
 EXAMPI.ES.— Ixix. 
 
 I. Find the value of 3aH?^'_^ when 
 
 3. Simplify f^t^_^LJ?'j^/''i+i',«-j3\ 
 
 4. Add together 
 
 ^2 ,^2 ,2 y2 ^2 ^2 2 aj!i 2 
 
 4 6 +8' 4 -"6+ 8 ^^^4 -^6 + 8' 
 and subtract z^-x'^^. |' from the result ' 
 
 5. Find the value of '^t^^rf+Soft 
 
 when 
 
 6. Multiply 5a;2 + 3ax - L^ by 2a;2 -ax~ ^. 
 
 a3-63 
 
 7 Shewthat^.„,26.-??i. 
 
ON MISCELLANEOUS FRACTIONS. 
 
 139 
 
 
 8. Simplity^l + ^?L + ^:-^; 
 
 X x-y sr-xy 
 
 9. Shew that , ., ^ ;:; = 12iC-25 + 
 
 bx^ + 9a; - 2 
 
 10 Sim lif «''*-9'''''' + 7a:2 + 9a;-8 
 i"^P ^ y x^ + 7x^ - 9a;2 - 7a; + 8' 
 
 a; + 2' 
 
 Tl. Simplify 
 
 x' 
 
 x^-l 
 
 +' 
 
 l-x — — 
 
 1 + a; 
 
 l-x 
 
 [ 2. Si niplify a + nh + h'^(a + ah + &Y--r ). 
 
 [3. Multiply together {^ + ]){l'' + l>){l-i)' 
 
 14. Add together 
 
 1 
 
 a+V h+V c+l 
 sum be etjual to 1, then ahc = a + o + c + 2. 
 
 .- J.- -. X ^ b ¥ b b'^ , 
 
 15. Divide -1 ^^4-_ + _ by x-a. 
 
 , and shew that if their 
 
 16. Simplify 
 
 a h 
 
 ^-^C+— 7-rt + 
 
 't 
 
 6 c a , 
 
 a c 
 
 and shew that it is equal 
 
 to 
 
 s (g - g) + (.s - 6) (s - c) 
 
 if 2s = rt + & + 
 
 17. Shew that 
 
 + 
 
 1+-^ 
 
 a-T-x 
 
 1- 
 
 a-^x 
 
 1 + 
 
 _4a4 
 a^ - a;* 
 
 a^-^-ic'-^ 
 
 rr 
 
 18. Simplify + 
 
 + ?) rf - /) 
 
 a~h~^ a + b a^ + b'^' 
 
 19. Siuiplify 
 
 20. Simplify 
 
 /> a + b a^-j- b 
 
 a 
 
 + 
 
 + b 2a 2a'(ci - b) 
 
 „2-ab + b^ 
 
 a2-fe2 
 
 
140 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 \\\ 
 
 21. Simplify 
 
 (^-Xf 2x2 -4a; + 2 i_a;2 
 
 _,. ,.-. a2^.52 + 2aft_c2 a + 6 + c 
 
 22. Simplify-,-_-^^,_^2a6-^H-7-a 
 
 23. Simplify /-^+ 1 -^^\ - /^^-a.--l-^y 
 
 24. Find the value of I r I ^j, when a; = —?,—. 
 
 _ \a;-6/ a; + a-26 2 
 
 25. Simplify ^-^-^y,-r j2 + (^ + 5). _ ,2 + (ftT^^— ««• 
 
 26. Simplify 
 
 (a;2-4x)(^^-4) 
 
 (x"2-2x)2 
 
 \? 
 
 27. Simplify ^^^^/^^-.^-^^ 
 
 28. Simplify ^ + ^-^---^^2^-^2 + J|^ -^^^2^7^ 
 
 29. Divide ~o 1 -0 by . 
 
 30. Simplify 
 
 31. Simplify 
 
 a + 6 
 
 a-h 262 )a-5 
 
 + 
 
 
 2 (a -6) 2(rt + 6) 
 ( a + 64-c)^ + (6 - c)2 + (c - a)2 + (ct-6)2 
 
 32. Take 77^ — i.— ,7^0 from 
 
 l + 3x2 + 2a;3 
 
 34. si.p,i.y(^-i)(^-i).(;;:-i)(^,^,-i) 
 
 35. Simplify 
 
 
 / 2a3__ W 2«6^ \ 
 
 
 47- 
 
-0 
 
 \}- 
 
 ON MISCELLANEOUS FRACTIONS. 
 
 141 
 
 36. Simplify 
 1 
 
 -<-V-r 
 
 2 (a;- 1)2 4(a;-l) 4(x+l) (x- l)2(a; + l)* 
 
 37. Prove that 
 
 + . 
 
 ahx a{a — b){x-a) h(h — a) {x - h) x {x - a) {;r - /■)* 
 
 38. Tf s = a 4 & + c + . . . to «. terms, shew that 
 
 111 \ 
 
 ... I - n. 
 
 s — a s~h s-c 
 a b c 
 
 ... =s(- 
 
 b c 
 
 / x^ V^ \ (x^ — y-)^ 
 
 39. Multiply ( ^^ - ^~--,) by ^-^^--^^J^-^ 
 
 40. Simplify 
 
 , a-x , a'- — x^ 
 
 1 + — — l + -2-r ^ 
 a + x a^-^ X 
 
 a-x' a^ — a;-' 
 a + x a^ + x^ 
 
 41. Divide x^ + 3 - 3^ 2 - x-j + 4yx + -J by x + \ 
 
 42. li s = a + h + c+ ...to n terms, shew that 
 s-a s — b s-c 
 
 — + + ■ 
 
 S 8 
 
 + ...^71-1. 
 
 43. Divide ( — ^ v - ) hy ( 0^--, + -J- >\ 
 
 ^■^ \x-y x + yJ '' xx-' + y^ x-~y^I 
 
 ■y x + y> 
 2xy 
 
 1- ^^y /l-'^\ 
 (x-v)'^ ^ a;/ 
 
 45. If 
 
 (x-y) 
 a + b c + d 
 
 l-ab cd- 1 
 
 , prove that 
 
 a + h + c + ' 
 
 1111 
 
 abed 
 
 abed. 
 
 46. Simplify 
 
 
 47. Reduce 
 
 _ l^^X X4l 1 
 
 3 («2~- xTl) "^ 2V + 1) "^ 6F+ 0' 
 
 •■Ml 
 
142 
 
 STMUL TAN ROUS EQUA TIONS 
 
 48. Simplify 
 
 1 
 
 1 
 
 1 y 
 
 y + - 
 
 1 X 
 
 + -, — 
 
 y 
 
 C-. ,.„ a-x a-y (a — c>)-' (a-yY 
 49. Simplify — • ^— ^^ ' — ]— ". 
 
 6c ca ah . 
 
 a\ 
 
 , a 
 
 5 1. Simplify p (a« - W), 
 
 a — 
 
 1- 
 
 a 
 
 XV. SIMULTANEOUS EQUATIONS OF 
 THE FIRST DEGREE. 
 
 207. To determine several uiiknoAvn quantities we must 
 have as many independent equations as there are unknown 
 quantities. 
 
 Thus if we had this equation given, 
 
 £C + ?/ = 6, 
 
 we could determine no definite values of ic and ?/, for 
 
 or other values might be given to x and y, consistently with 
 the equation. In fact we can find as many pair.'^ of values of 
 K and y as we please, which will satisfy the equation. 
 
 or 
 
ith 
 
 of 
 
 OF THE FIRST DEGREE:, 
 
 143 
 
 We must have a second equation independent of the first, 
 and then we may find a pair of values of x and y which will 
 satisfy both equations. 
 
 Thus, if besides the equation x + y = 6, we had another 
 equation x-y = 2, it is evident that the values of x and y 
 which will satisfy both equations are 
 
 x = 4 
 
 y = 2 
 since 4 + 2 = 6, and 4-2 = 2. 
 
 Also, of all the pairs of values of x and y which will satisfy 
 one of the equations, there is but one pair which will satisfy 
 the other equation. 
 
 We proceed to shew how this pair of values may be found, 
 
 208. Let the proposed equations be 
 
 2ic + 7i/ = 34 
 bx + 9y = 6l. 
 
 Multiply the first equation by 5 and the second equation by 
 2, we then get 
 
 I0x + S5y = l70 
 10x-hl8i/=102. 
 
 The coefficients of x are thus made alike in both equations. 
 
 If we now subtract each member of the second equation 
 from the corresponding member of the first equation, we shall 
 get (Ax. II. page 58) 
 
 35^-18?/ = 170 -102, 
 or 17?/ = 68; 
 
 ••• 2/ = 4. 
 We have thus obtained the value of one of the unknown 
 symbols. The value of the other may be found thus : 
 
 Take one of the original equations, thus 
 
 2x + 7i/ = 34. 
 
 Now, since y=4,7y = 28; 
 
 :. 2a; + 28 = 34; 
 
 .'.05 = 3, 
 
 Hence the pair of values of x and y which satisfy tlje 
 equations is 3 and 4. 
 
 I 
 
 I 
 
 
If 
 
 144 
 
 SIMULTANEOUS EQUATIONS 
 
 Note. The process ot' tniis obtaining from two or more 
 equations an equation, from which one of the unknown quanti- 
 ties has disappeared, is called Elimination. 
 
 209. We worked out the steps fully in the example given 
 in the last article. We shall now work an example in the form 
 in which the process is usually given. 
 
 Ex. 2'o solve the equations 
 
 V 5x + 4y = 5S. 
 
 Multiplying the first equation by 5 and the second by 3, 
 
 16a; + 35?/ = 335 
 .I5a;fl2?/=174. 
 
 23?/= 161, . 
 
 ?/ = 7. 
 
 3x + 7// = 67, 
 3a; + 49 = 67, 
 .'. 3a;=18, 
 .'. x = 6. 
 
 Hence x = 6 and i/=«7 are the values required. 
 
 210. In the examples given in the two preceding articles 
 we made the coefficients of x alike. Sometimes it is more con- 
 venient to make the coefficients of y alike. Thus if we have 
 to solve the equations 
 
 29a; + 2?^ = 64 
 
 13x4- y = '2d, 
 
 we leave the first equation as it stands, and multiply the 
 second equation by 2, thus 
 
 29a;4-2^ = 64 
 26x4-2^ = 58. 
 
 Subtracting, 
 and therefore 
 
 Now, since 
 
 Subtracting, 
 and therefore 
 
 Now, since 
 
 3a; = 6, 
 
 a;=2. 
 
 VSx + y = 29, 
 26 + 2/ = 29, 
 - .-. 2/ = 3. 
 
 Hence x=2 and i/ = 3 are the values required. 
 
 
 r 
 
 I 
 
 I. 
 
 4. 
 
 7- 
 
the 
 
 OF THE FIRST DEGREE. 
 
 M5 
 
 I. 2aj + 7y = 41 
 3x + 4?/ = 42. 
 
 4. 14a; + 9?/ =156 
 
 7a; + 2y = 58. 
 
 7. 6a; + 4i/ = 236 
 3.c+15i/ = 573. 
 
 EXAMPLES — IXX. 
 
 2. 5x + 8?/ = 101 
 9a; + 2y = 95. 
 
 5. x+15y = 49 
 3x+ 7i/ = 71. 
 
 8. 39.x + 277/ = 105 
 52a; + 29«/ = 133. 
 
 3. 13.7j-<-17?/=189 
 2a;+ 2/ = 21. 
 
 6. 15x4- 19*/ =132 
 35.*;+ 177/ = 226. 
 
 9. 72a; + 147/ = 330 
 63x+ 77/ = 273. 
 
 211. We shall now give some examples in which negative 
 signs ocL:ir attached to the coefficient of y in one or both of 
 the etj^uat, jns. 
 
 Ex. To solve the equations: 
 
 6a; + 357/ = 177 
 8a;-2l7/= 33. 
 
 Multiply the first equation by 4 and the second by 3. 
 
 24a; + 1407/ = 708 
 24a;- 637/= 99. 
 
 Subtracting, 
 and therefore 
 
 2037/ = 609, 
 
 7^ = 3. 
 
 The value of x may then be found. 
 
 I. 2a; + 7?/ = 52 
 3a; -5?/= 16. 
 
 4. 4a; + 97/ = 79 
 7a;- 177/ = 40. 
 
 Examples. —Ixxi. 
 
 2. 7a;- 47/ = 55 3. a; + 7/ = 96 
 15a; - 137/ = 109. x-y = 2. 
 
 5. a; +197/ = 97 6. 29a;- 147/= 175 
 
 7a; -53?/= 121. 87a;-56y = 497. 
 
 7. 171a; -213?/ = 642 8. 43a; + 2// = 266 9. 5a; + 97/ =188 
 114a; -326?/ = 244. 12x-17i/ = 4. 13a;-2^ = 57. 
 
 Fs.A.l K 
 
 tfi 
 
 
 ir n 
 
Ill 
 
 212. We liave liitlierto taken examples in which the 
 coelticients of x are both positive. Let us now take the follow- 
 ing equations : 
 
 5x-1y = (i 
 
 9i/-2x=10. 
 
 Change all the signs of the second equation, so that we g«i 
 
 5x — 7y = 6 
 2x-9y=-lO. 
 
 Multiplying by 2 and 5, 
 
 lOx-Uy = U 
 10x-45?/=-50. 
 
 Subtracting, 
 
 -142/-}- 45?/ = 12 + 50, 
 or, 31?/ = 62, 
 or, 2/ = 2. 
 
 The value of x may then be found. 
 
 Examples.— Ixxii. 
 
 I. 4a;- 7?/ = 22 
 7y-dx = l. 
 
 2. 9x — 5y = 52 
 8y-^x = 8. 
 
 3. I7x + 3y=67 
 I6y-3x=23. 
 
 4. 7y + 3x = 78 
 19i/-7x = 136. 
 
 6. 3x + 2y=S9 
 3y-2x=U. 
 
 7. 5?/-2x = 21 
 13a; -4?/= 120. 
 
 9. 12a; + 7y = l7Q 
 3y-19a; = 3. 
 
 5. 5.1; -3^ ==4 
 12^^' -7a; =10. 
 
 8. 9i/-7a;=13 
 15a; -7?/ = 9. 
 
 213. In the preceding examples the values of x and y have 
 been positive. We shall now give some equations in which x 
 or y or both have negative values. 
 
 Ex. To solve the equations : 
 
 2x-9'y = 11 * 
 
 3x-4y = 7. 
 
 Multiplying the equations by 3 and 2 respectively, we get 
 
 6a; -27?/ = 33 
 Qx- 81/ = 14. 
 
OF THE FIRST DEGREE. 
 
 M7 
 
 Subtracting, ^ 
 
 -19«/ = 19, 
 
 or, l%=-b), 
 
 or, ?/=-!. 
 
 Now since 9?/ = - 9, 
 
 Sx - 9?/ will be equivalent to 2a3 - ( - 9) or, 2x + 9. 
 
 Hence, from the first equation, 
 
 2a; + 9 = ll, 
 
 Examples.— Ixxiii 
 
 I. 2x + 3i/=8 
 3a; + 7^ = 7. 
 
 4. 72/-3x=139 
 2x + 5?/ = 91. 
 
 7. 17a; + 122/ = 59 
 19ic- 42/ = 153. 
 
 2. 5a;-2y = 51 
 19x-3y = 180. 
 
 5. 4a; + 9?/ = 106 
 8a;+17i/=198. 
 
 8. 8a; + 3?/ = 3 
 12a; + 9j/ = 3. 
 
 3. 3a3-5y = 51 
 
 2a;-}-7i/ = 3. 
 
 6. 2a;-7i/ = 8 
 4?/ -9a; = 19. 
 
 9. 69i/-17x=l()3 
 14a;-13»/=-4I. 
 
 214. We shall now take the case of Fractional Equations 
 involving two unknown quantities. ' 
 
 Ex. To solve the equations, 
 
 5 
 
 3^ = 9 
 
 x-2 
 
 Fii jt, clearing the equations of fractions, we gef; 
 
 l0x-y + '3 = 20 
 9y = 27-x + 2, 
 from which we obtain, 
 
 10a;-2/=l7 
 x + 9y = 29, 
 and hence we may find x = 2, ?/ = 3. 
 
 I 
 
 |i 
 
148 
 
 SIMULTANEOUS EQUATIONS 
 
 Examples.— ixxiv. 
 
 35 1/ H, 
 
 I - + - - = 7 
 3"^2~^- 
 
 ?/. 
 
 X 
 
 2. 10a; + | = 210 3. „ + 7i/ = 251 
 
 X 
 
 102/ -2 = 290. 
 
 ?/ 
 
 + 7«=299. 
 
 cr + 1/ _ 
 3 
 
 4. --r^+5 = 10 
 
 y 
 
 5. 7. + ^^=413 6. ^:^^^=io-? 
 ^--?^ + 7 = 9i. 39x = 14y-1009 l?r_?5 = ?;- + 1. 
 
 7. ^ -^ o 
 
 4j,--3 3. 
 
 8. | + 8 = |-12 
 
 , ic + 'J/ v/ 205-?/ „_ 
 
 3a;-57/ 2x + i/ 
 
 ). ____ + 3-— — 
 
 „ a; — 2?/ a; 7/ 
 
 «-- 4--2n- 
 
 10. ^- + 8i/ = 31 
 
 -^4— + 10a; = 192. 
 4 
 
 11. — ^-^- + 3a: = 22/-6 
 
 5 6 
 
 12. 
 
 a;- 2 _ 10^05^^10 
 "5 3 "~ 4 
 
 2i/ + 4_ 4a; + ? / + 13 
 ~^3 ^8 • 
 
 5.x -6?/ , „ . a 
 
 \ 
 
 14. 
 
 5a3 + 6?/ 3x - 2ij 
 5a;-3 3a;-19 
 
 37/ -a; 
 
 15. 
 
 - = 4- 
 2 2 3 
 
 2a; + ^ _ 9x j-^ _ 3^ J- 9 _ 4« + 5^ 
 
 "2 8 ~ 4 16 • 
 
 4a; + 5i/ 
 2x-y , ^ 1 
 
OF THE FinST DEGREE. 
 
 149 
 
 y 
 
 3 
 
 215. We have now to explain the niethod of solving Literal 
 Equations involving two unknown quantities. 
 
 Ex. To solve the equations, 
 
 ax + by = c 
 px + qy = r. 
 
 Multiplying the firyt equation by p and the second by a, we 
 get 
 
 apx + bpy = cp 
 apx + aqy = ar. 
 
 Subtracting, hpy-aqy^cp-ar, 
 
 OT, {hp — aq)y = cp-ar; 
 
 cp-ar 
 ^ hp — aq' 
 
 We might then find x by substituting this value of y in one 
 of the original equations, but usually the safest course is to 
 begin afresh and make the coeffi<'ients of y alike in the original 
 equations, multiplying the first by q and th(.j second by &, 
 
 w^hich gives 
 
 Subtracting, 
 
 aqx + hqy = cq 
 hpx + hqy = br. 
 
 aqx — hpx = cq-hry 
 
 or, {aq — bp)x = cq-br; 
 cq-hr 
 
 * rp — _ ±_ 
 
 aq-op 
 
 Examples.— Ixxv. 
 
 I. mx + ny = e, 
 
 2. 
 
 ax + hy = G 
 
 3. 
 
 ax - hy = m 
 
 px + qy^f. 
 
 
 dx-ey=f. 
 
 
 cx + ey = n. 
 
 4. ex =dy 
 
 5. 
 
 mx-ny—r 
 
 6. 
 
 x + y=^a 
 
 x + y = e. 
 
 
 m'x + n'y = t 
 
 
 x-y — b. 
 
 7. ax + hy = c 
 
 8. 
 
 ahx + cdy = 2 
 
 7 f 
 
 9' 
 
 a b 
 h + y~3a + x 
 
 dx+fy=c'^. 
 
 
 d-h 
 ax-cy= ^^ . 
 
 
 lu + 2by = d. 
 
I50 
 
 SIMULTANEOUS EQUATIONS 
 
 lo. bcx + 2,b -cij = 
 
 11. {b + c){x + c-h)+a{ii + a) = 2a'^ 
 
 be 
 
 ay _^{b + cy 
 {b-c)x 
 
 •• a^ 
 
 12. 3x- + r)?/ = 
 
 (8ft -2m) 6m 
 
 ft'i - m^ 
 
 b^x 
 
 bcm^ 
 
 - , + (?) + c + m) my-=m'^x + (6 + 2m)6m. 
 
 216. We now proceed to the solution of a particular class 
 of Simultaneous Equations in whicli the unknown symbols 
 a[)pear as tlie denominators of fractions, of which the following 
 are examples. , 
 
 Ex. 1. lo solve the equationSy 
 
 a b 
 X y 
 
 m n 
 X y 
 
 = d. 
 
 or. 
 
 Multiplying the first by m and the second by a, we get 
 
 Subtracting, 
 
 am bm 
 
 - +- =cm 
 
 » y 
 
 am an , 
 
 =aa. 
 
 » y 
 
 bm an , 
 1 = cm- ad. 
 
 y y 
 
 or, 
 
 bm + an 
 
 "7" ' 
 
 : cm - ad, 
 
 or, bm + an = (cm-ad)yj 
 bm + an 
 
 ••• 2/ = 
 
 cm - ad' 
 
 ■•f 
 
 .-.»fe 
 
 
 I. 
 
 Then the value of x may be found by substituting this value 
 of y in one of the original equations, or by making the terms 
 containing y alike, as in the example given in Art. 215. 
 
OF THE FIRST DEGREE. 
 
 151 
 
 Ex. 2. To solve the cquationn: 
 
 25 4 
 
 I 1^11 
 
 Multix)lying the second equation hy 8, we ;:< t 
 
 2 _ n _ 4 
 X 3// "27 
 2 8 11 
 X y d 
 
 _ 5 _8_J _11 
 'Sy y-'27 d' 
 
 Subtracting 
 
 o> 
 
 ^, . . 5 8 11 4 
 
 Changing signs, 3^ + ^^ = -^-27, 
 
 or. 
 
 whence we find 
 
 5 + 2433-4 
 3y - 27"' 
 
 and then the value of x may he found hy substituting 9 for y 
 in one of the original e(|uations. 
 
 Examples.— Ixxvi. 
 
 '>i 
 
 ^■n^' 
 
 1 + ?=10 
 
 X y 
 
 X y 
 
 a b 
 - + - = m 
 
 X y 
 
 a h 
 
 = n. 
 
 X y 
 
 7» 
 
 ax by 
 
 5 
 ax 
 
 by 
 
 = 3. 
 
 1 2 
 
 -+ ~ = rt 
 X y 
 
 3 4 , 
 
 - 4 - = ?>. 
 iC 7/ 
 
 19 
 
 7 5 
 
 _ + -: 
 iC 7/ 
 
 .?-? = 7 
 
 8. 
 
 3. 
 
 a 
 
 -4 
 
 X 
 
 - = c 
 
 y 
 
 
 b 
 
 - + 
 
 X 
 
 2/ 
 
 6. 
 
 5 
 3a; 
 
 by 
 
 
 7 
 
 1 
 
 
 6x 
 
 107/ 
 
 m n 
 
 -- + --- 
 7<a; my 
 
 = m + 71 
 
 n 
 
 - + 
 
 X 
 
 7/ 
 
 
 I 
 
 m 
 
m 
 
 STMtJL TAN'ROUS EQUA TIONS 
 
 217. There are two other methods of solving Simultaneous 
 Equations of which we have hitherto made no mention, because 
 they are not generally so convenient and simple as the method 
 which we have explained. They are 
 
 I. The method of Substitution. 
 If we have to solve the equations 
 
 a; + 3?/= 7 
 2a; + 4)/ = 12 
 
 we may find the value of ic in terms of ij from the first equa- 
 tion, thus 
 
 and substitute this value for x in tlie second equation, thus 
 
 2{7-3y) + 4y = l2, 
 from which we find 
 
 We may then find the value of a; from one of the original 
 equations. 
 
 II. The method of Comparison. 
 
 If we have to solve the equations 
 
 6x + 2y = l6 
 
 7x-3y= 5 
 
 we may find the values of x in terms of y from each equation, 
 thus 
 
 x = — K i from the first equation. 
 x= — =--, from the second equation. 
 
 Hence, equating these values of x, we get 
 
 16-% 5 + 3^ * 
 
 an equation involving only one unknown symbol, from which 
 we obtain 
 
 2/ = 3, 
 
 and then the value of x may be found from one of the original 
 e(|uations, . 
 
 fit 
 
OF THE FIRST DEGREE, 
 
 153 
 
 218. If there be three unknown symbols, their values may 
 be found from three independent equations. 
 
 For from two of the equations a third, which involves only 
 two of the unknown symbols, may be found. 
 
 And frc^ the remaining equation and one of the others 
 a fourth, containing only the same two unknown symbols, may 
 be found. 
 
 So from these two equations, which involve only two un- 
 known synibol«^ the value of these symbols may be found, and 
 by substituting ihese values in one of the original equations 
 the value of the third unknown symbol may be found. 
 
 Ex. 5x-6y + 4z=ib 
 
 7fc + 4?/-3,^'=19 
 
 2.C+ y + (j:: = 4(y. 
 
 Multiplying the first ])y 7 and the second by 5, we get 
 
 35a; -42?/ + 28.-; =105 
 35x + 20)/ -15:3 = 95. 
 Subtracting, 
 
 -62?/ + 432 = 10 (1). 
 
 Again, multiplying the first of the original equations by 2 
 and the third by 5, we get 
 
 I0x-l2y + 8z = 30, 
 10:c + 5?/ + 30;;! = 230. 
 
 Subtracting, - 17y-22«= -200 (2). 
 
 Then, from (1) and (2) we have 
 
 G2y-4'3z=-lO 
 
 17y + 22;3 = 200, 
 from which we can find 1/ = 4 and 2J = 6. 
 
 Then substituting these values for y and z in the first equa- 
 tion we find the value of x to be 3. 
 
 I. 
 
 2. 
 
 ^ EXAMPLES.- 
 
 -Ixxvii. 
 
 bx + 1y- 2: = 13 
 
 3. r)x~3y + 2z = 2l 
 
 Rc + 'Mj+ z = 17 
 
 Hx- 7/ -3,-3= 3 
 
 x-4y + lOz=:2'3. . 
 
 2a; + 3//-t-2;2 = 39. 
 
 5x + 3y-6z = 4 
 
 4. 4x-by + 2z= G 
 
 3x- 2/ + 22 = 8 ? 
 
 2x + 3y~ 2! = 20 
 
 x-2y + 2z = 2. 
 
 7x-4?/ + 3«=35. 
 
 
 ■ fit 
 
154 
 
 PROBLEMS RESULTING IiV 
 
 5 x-\ y+ z= Q 
 
 15a;+ 10// + 6:^ = 53. 
 
 6. 8a; + 4?/ -3,'^ = 6 
 
 x^-Zy- z = 7 
 4x-6y + 4z = 8. 
 
 7. x+ y+ z = ^0 
 8:c + 4?/ + 2^ = 50 
 
 27a; + 9?/ + 32 = 64. 
 
 8. 
 
 10. 
 
 4x-3y+ s = 
 
 9x + y -5z = 
 
 X - 4y f 3;i' = 
 
 9 
 
 16 
 
 2. 
 
 I2x + 5y — 4z 
 I3x-2y + bz: 
 
 17a;- 7/- ?;: 
 
 = 20 
 = 58 
 = 15. 
 
 y-x + z= - 
 z-y-x= -i 
 a; 4-^ + ." = 35. 
 
 5 
 
 25 
 
 XVI. PROBLEMS RESULTING IN SIMUL- 
 TANEOUS EQUATIONS. 
 
 219. In tlie Solution of Problems in which we represent 
 two of the numbers souglit by unknown symbols, usually x and 
 y, we must obtain two independent equations from the condi- 
 tions of the question, and then we may obtain the values of 
 the two unknown symbols by one of the processes described in 
 Chapter XV. 
 
 Ex. If one of two numbers be multiplied by 3 and the 
 other by 4, the sum of the })roducts is 43 ; and if the former be 
 multiplied by 7 and the latter by 3, the difference between the 
 results is 14. Find the numbers. 
 
 Let X and y represent the numbers. 
 
 Then 3a; + 4?/ = 43, 
 
 and . 7a; -3?/ = 14. 
 
 From these equations we have 
 
 21a; + 28?/ = 301, 
 21a;- 9(/ = 42. 
 
 Subtracting, 37y = 259. . , 
 
 Therefore ^ 2/ = 7, 
 
 and then the value of x may be found to be 5. 
 
 Hence the numbers are 5 and 7. 
 
Examples.— ixxviii. 
 
 1. The sum of two numbers is 28, and their clifFerence is 4, 
 find the numbers. 
 
 2. The sum of two numbers is 256, and their dift'erence is 
 10, find the numbers. 
 
 3. The sum of two numbers is 13"5, and their difference is 
 1, find the numbers. 
 
 4. Find two numbers such that the sum of 7 times the 
 greater and 5 times the less may be 332, and the product of 
 tlieir difference into 51 may he 408. 
 
 5. Seven years ago the age of a father was four times that 
 of his son, and seven years hence the age of the father will be 
 double that of the son. Find their ages. 
 
 6. Find three numl)ers sucli that the sum of the first and 
 second sliall be 70, of the first and third 80, and of the second 
 and third 90. 
 
 7. Three persons A, B, and C make a joint contribution 
 which in the whole amounts to i,'400. Of this sum B contri- 
 butes twice as much as A and £20 more ; and G as much as A 
 and B together. WJiat sum did each contribute ? 
 
 8. If A gives B ten shillirigs, B will have tliree times as 
 much money as A. *If B gives A ten shillings, A will have 
 twice as much money as B. What has each ? 
 
 9. The sum of £760 is divided between A, B, G. The 
 shares of A and B together exceed the share of G by .£2140, 
 and the shares of B and C together exceed the share of A hy 
 £360. What is the sliare of each ? 
 
 10. The sum of two numbers divided by 2, gives as a quo- 
 tient 24, and the difference between them divided by 2, gives 
 as a quotient 17. What are the numbers? 
 
 11. Find two numbers such that when the greater is 
 divided by tlie less the quotient is 4 and the remainder 3, aiul 
 when tlie sum of tlie two numbers is increased by 38 and tlie 
 result divided by the greater of the two numbers, the quotient 
 is 2 and the remainder 2. 
 
 12. Divide the number 144 into three such parts, that 
 when the first is divided by the second the (juotient is 3 and 
 the remainder 2, and when the third is divided by the sum 
 of the other two parts, the quotient is 2 and tlie leniaimKir 6, 
 
 A, 
 
 /L 
 
 [I. 
 
 n. 
 
 
 r-i 
 
 m 
 
 f ■ 
 
 i-st-r 
 
 N^" 
 
 i 
 
156 
 
 PROBLEMS RESULTING IN 
 
 :l 
 
 V- 
 
 
 K 
 
 ( 
 
 13. A and B Ijiiy a horse for ^120. A can pay for it if ^ 
 will advance half tlie money he has in his jiocket. 11 can pay 
 for it if A will advance two-thirds of the nionev he has in his 
 pocket. How nmch has eacli? 
 
 14. "How old are you?" said a son to his father. The 
 father replied, "Twelve years hence you will he as old as I was 
 twelve years ago, and I slmll he three times as old as you were 
 twelve years ago." Find the age of each. 
 
 15. Required two numl)ers such that three times the 
 greater exceeds twice tlie less hy 10, and twice the greater 
 together with three times the less is 24. 
 
 16. The sum of the ages of a father and son is half what it 
 will he in 25 years. Tlie difference is one-tlrird what the sum 
 will he in 20 vears. Find their ages. 
 
 17. If I divide the smaller of two numhers hy the greater, 
 the quotient is "21 and the remainder "0157. If I divide the 
 greater number hy the smaller, the quotient is 4 and the 
 remainder '742. Find the numhers. 
 
 18. The cost of 6 harrels of heer and 10 of porter is £51 ; 
 the cost of 3 harrels of heer and 7 of porter is .£32, Is. How 
 much heer can he hought for £30? 
 
 19. The cost of 7 Ihs. of tea and 5 Ihs. of coffee is £1, 9s. Ad. : 
 the cost of 4 Ihs. of tea and 9 lbs. of coffee is £1, 7*'. : what is 
 the cost of 1 lb. of each ? , 
 
 20. The cost of 12 horses and 14 cows is £380 : the cost of 
 5 horses and 3 cows is £130 : what is the cost of a horse and a 
 cow respectively? 
 
 21. The cost of 8 yards of silk and 19 yards of cloth is 
 £18, 4s. 2(i. : the cost of 20 yards of silk and 16 yards of cloth, 
 each of the same quality as the former, is £25, ICs. Sd. How 
 much does a yard of each cost ? 
 
 22. Ten men and six women earn £18, 18s. in days, and 
 four men and eight wonuai earn £(5, Os. in 3 days. What are 
 the earnings of a man and a woman daily? 
 
 23. A farmer hought 100 acres of land for £4220, part at 
 £37 an acre and part at £45 an acre. H(jw many acres had 
 he of each kind ? . v 
 
 from 
 
SilMUL TANEOUS EQUA TIONS, 
 
 157 
 
 it if 7i 
 ail pay 
 1 in liis 
 
 The 
 s I was 
 u were 
 
 es the 
 greater 
 
 what it 
 lie sum 
 
 greater, 
 ide the 
 iiid the 
 
 is £51 ; 
 How 
 
 9s. 4<^ : 
 >\hat is 
 
 cost of 
 ,e and a 
 
 cloth is 
 :)f cloth, 
 . How 
 
 lys, and 
 /hat are 
 
 , part at 
 ;r^8 had 
 
 Note I. A iiiimbor consisting of two digits may be repre- 
 sented algebraically by 10j; + i/, where x and ij represent the 
 significant digits. 
 
 For consider such a number as 76. Here the significant 
 digits arc 7 and 6, of which the former has in consequence of 
 its position a local value ten times as great as its natural 
 value, and the number represented by 76 is equivalent to te7i 
 times 7, increased by 6. 
 
 So also a number of which x and y are the significant digits 
 will be represented by ten times x, increased by y. 
 
 If the digits composing a number lOx + y be inverted, the 
 resulting number will be lOy + x. Thus if we invert the digits 
 composing the number 76, we get 67, that is, ten times 6, in- 
 creased by 7. 
 
 If a nnniber be represented by lOx + y, the sum of the 
 digits will be represented by :»; + y. 
 
 A number consisting of tJiree digits may be represented 
 
 algebraically by 
 
 100:i;+107/ + ;3. 
 
 Ex. The sum of the digits composing a certain number is 
 5, and if 9 be added to the number the digits will be inverted. 
 Find the numl)er. . 
 
 Let lOx + y represent the number. • 
 
 Then x + y will represent the sum of the digits, 
 and lOy + x will represent the number with the digits inverted. 
 Then our equations will be 
 
 x + y = 6, 
 lOx + y + d^ lOij-i-x, 
 from which we may find x = 2 and 1/ = 3 ; 
 
 .•. 23 is the number required. 
 
 24. The sum of two digits composing a number is 8, and if 
 36 be abided to the number the digits will be inverted. Find 
 the number. 
 
 25. The sum of the two digits composing a number is 10, 
 and if 64 be added to the number the digits will be inverted. 
 What is the number ? ' 
 
 
 ■I1J 
 
 :!:|ll 
 
 M 
 
 
 5»i 
 
 .'■•■r-j 
 
 1 . 'f 
 
 I' 
 
 .If- 
 v] 
 
 4 
 
 lli 
 
IS^ 
 
 PROBLEMS RESULTING IN 
 
 n 
 
 26. The sum of the digits of a number less than 100 is 9, 
 and if 9 be added to the number the digits will be inverted. 
 What is the number ? 
 
 27. The sum of the two digits composing a number is 6, 
 and if the number be divided by the sum of the digits the 
 quotient is 4. What is the number ? 
 
 28. The sum of the two digits composing a number is 9, 
 and if the number be divided by the sum of the digits the 
 .quotient is 5. What is the number / 
 
 29. If I divide a certain number by the sum of the two 
 digits of which it is composed the quotient is 7. If I invert 
 the order of the digits and then divide the resulting number 
 diminished by 12 by the difference of the digits of the original 
 number the quotient is 9. AVhat is the number ? 
 
 30. If I divide a certain number by the sum of its two 
 digits the quotient is 6 and the remainder 3. If I invert the 
 digits and divide the resulting number by the sum of the digits 
 the quotient is 4 and the remainder 9. Find the number. 
 
 ' 31. If I divide a certain number by the sum of its two 
 digits diminished by 2 the quotient is 5 and the remainder 1. 
 If I invert the digits and divide the resulting number by the 
 sum of the digits increased by 2 the quotient is 5 ar.d the re- 
 mainder 8. Find the number. 
 
 32. 
 
 Two digits which form a number change T)iaces on the 
 
 addition of 9, and the sum of these two numbers is 33. Find 
 the numbers. 
 
 33. A number consisting of three digits, the absolute value 
 of each digit being the same, is 37 times the square of any 
 digit. Find the number. 
 
 34. Of the three digits composing a number the second is 
 double of the third : the sum of the first and third is 9 : the 
 sum of all the digits is 17. Find the number. 
 
 35. A number is composed of three digits. The sum of the 
 digits is 21 : the sum of the first and second is greater than the 
 third by 3; and if 198 l)e added to the number the digits will 
 be inverted. Find the number. 
 
sntUL TANEOUS EQUA TIOKS. 
 
 l^^ 
 
 W is 9, 
 verted. 
 
 cr is 6, 
 ^its the 
 
 er is 9, 
 
 Tits the 
 
 the two 
 [ invert 
 number 
 original 
 
 its two 
 vert the 
 tie digits 
 iher. 
 
 its two 
 
 under 1. 
 
 by the 
 
 the re- 
 
 » on the 
 Find 
 
 ite value 
 of any 
 
 lecond is 
 s 9 : tlie 
 
 ni of the 
 than the 
 gits will 
 
 Note IT, A fraction oi' which the tenns arc unknown may 
 
 be represented by --. 
 
 1 
 
 Ex. A certain fraction l)ecom(*s - when 7 is added to its 
 
 denominator, and 2 when 13 is added to its numerator. Find 
 the fraction. 
 
 Let - represent the fraction 
 
 Then 
 
 JK_ _1 
 
 2/ + 7~2' 
 
 are the equations; from which we may find a; = 9 and i/ = ll. 
 
 9 
 
 That is, the fraction is , ,. 
 
 36. A certain fraction becomes 2 when 7 is added to its 
 numerator, and 1 when 1 is subtracted from its denominatoi-. 
 What is the fraction ? 
 
 37. Find Huch a fraction that when 1 is added to its 
 numerator its value becomes , and when 1 is added to the 
 
 denominator tlie value is - . 
 
 4 
 
 38. What fraction is that to the numerator of which if 1 l)e 
 added the value will be : but it 1 be added to the denominator, 
 
 the value will be . ? - 
 
 o 
 
 39. The numerator of a fraction is made equal to its 
 d(!nominator by the addition of 1, and is half of the deno- 
 minator increased by 1. Find the fraction, 
 
 40. A certain fraction becomes - when Ij is taken from tlie 
 numerator and the denominator, ard it becomes - when 5 
 
 '' '!ll 
 
rdo 
 
 PROBLEMS RESUL TING IN 
 
 is added to the numerator and the 'denominator. What is the 
 fraction ? 
 
 7 
 41. A certain iraction becomes when the denominator is 
 
 20 
 
 increased by 4, and - when the numerator is dimiiished by 
 15 : determine the fraction. 
 
 42. What fraction 
 1 
 
 ' .> \ 
 
 added it becomes „, an 
 M 
 
 1 
 
 added it becomes - ? 
 o 
 
 : to tiie numerator of which if 1 be 
 iLiO denominator of which if 17 be 
 
 Note III. In questions relating to money put out at 
 simple interest we are to observe that 
 
 Principal x Rate x Time 
 
 Interest = 
 
 where Rate means tlie number of pounds paid for the use (jf 
 ^100 for one year, and Time means the number of years for 
 which the money is hait. 
 
 ^ 43. A man ])uts out £2000 in two investments. For the first 
 he gets 5 per cent., for tlie second 4 per cent, on the sum 
 invested, and bv the first investment he has an income of 
 i>10 more than on the second. Find liow much he invests in 
 each case. 
 
 44. A sum of money, put out at simple interest, amounted 
 in 10 months to £5250, and in 18 months to €5450. What 
 was the sum and the rate of interest ? 
 
 45. A sum of money, put out at simpie interest, amounted 
 in 6 years to £5200, and in 10 years to £6000. Find the sum 
 and the rate of interest. ^^ 
 
 Note IV. Wlien tea, spirits, wine, beer, and such com- 
 modities are mixed, it must be observed that 
 
 quantity of ingredients = quantit3' of mixture, 
 cost of ingredients = cost of mixture. 
 
 Ex. I mix wine which cost 10 shillings a gallon with 
 another sort which cost 6 shillings a gallon, to make 100 
 
 / 
 
i is the 
 
 lator i.s 
 
 lied by 
 
 if 1 l)e 
 
 if 17 be 
 
 out at 
 
 e use of 
 ears for 
 
 the first 
 he smii 
 ome of 
 vests in 
 
 lounted 
 What 
 
 lounted 
 ;he sum 
 
 h coni- 
 
 )n with 
 
 i];c 100 
 
 SIMULTANEOUS FQUATTONS. 
 
 l6l 
 
 / 
 
 gallons, which I may sell at 7 shilliiip;s a gallon without profit 
 or loss. How much of each do I t^ike % 
 
 Let X re]>resent the number of j^allons at 10 shillin.ys a L,'allou, 
 an<l ?/ 6 
 
 Then a' + // = 100, 
 
 and 10.v; + 6?/ = 700, 
 
 are the two eijuations from which we may find the values of 
 X and \j to be 25 and 75 respectively. 
 
 46. A wine-merchant has two kinds of wine, the one costs* 
 36 pence a quart, the other 20 pence. How much of each list 
 he put in a mixture of 50 quarts, so that the cost price ■ f i 
 may be 30 pence a ipiart ? 
 
 47. A grocer mixes tea which cost him l.s. 2r/. per lb. vith 
 tea that cost him Is. 6(^. per lb. He has 30 ll)s. of tlu^ "uixture, 
 and by selling it at the rate of l.s\ 8t?. per H). he g .ned as 
 much as 10 lbs. of the cheaper tea cost him. How many lbs. 
 of each did he put in the mixture ? 
 
 Note V. If a man can row at the rate of x miles an hour 
 ill still water, and if he be rowing on a stream that runs at the 
 rate of \j miles an hour, then 
 
 ic + ^ will represent his rate down the stream, 
 x — ij UJ) 
 
 48. A crew which can pull at the rate of twelve miles an 
 hour down the stream, finds that it takes twice as long to come 
 up a river as to go down. At what rate does the stream flow '? 
 
 49. A man sculls down a stream, which runs at the rate of 
 4 miles an hour, for a certain distance in 1 hour and 40 minutes. 
 In returning it takes him 4 hours and 15 minutes to arrive at 
 a point 3 miles short o-f his starting-place. Find the distance 
 he pulled down the stream, and the rate of his pulling. 
 
 50. A dog pursues a hare. The hare gels a start of 50 of 
 her own leaps. The hare makes six leai)S while the ilog makes 
 5, and 7 of the dog's leaps are equal to 9 of the hare's. How 
 iiwiuy leaps will the hare take before she is caught ? 
 
 fs.A.] I, 
 
 I 
 
 fr 
 
l62 
 
 PROBLEMS RESULTLYG IN 
 
 K 
 
 \ 
 
 t;i. A greyhovmd starts in pursuit of a liaie, at the distance 
 ol" 50 of his own leaps from ^ler. He makes 3 l(!aps while tlie 
 hare makes 4, and lie covers as much <froiind in two leaps as 
 the hare docs in three. How many leaps does each make 
 hefore the hare is caught? 
 
 52. I lay out half-a-crown in apples and pears, Ijuying the 
 apjjles at 4 a penny and tlie pears at 5 a penny. I then sell 
 half the apples and a thii'd of the f)ears for thirteen pence, 
 which Avas the price at which I bought them. How many of 
 each did I buy ? 
 
 53. A company at a tavern found, when they came to pay 
 their reckoning, that if there had been 3 more persons, each 
 would have paid a shilling less, but had there been 2 less, 
 each would have paid a shilling more. Find tlie number of 
 the company, and each man's share of the reckoning. 
 
 54. At a contested election there are two members to be 
 returned and- three candidates. A, B, and G. A obtains 1056 
 votes, B, 987, C, 933. Now 85 voted for B and C, 744 for 
 B only, 98 for C only. How many voted for A and C, for 
 A and B, and for A only 1 
 
 55. A man walks a certain distance : had his rate been 
 half a mile an hour faster, he would have been H hours less 
 on the road ; and had it been half a mile an hour slower, he 
 would have been 2^ hours more on the road. Find the distance 
 and rate. 
 
 56. A certain crew pull 9 strokes to 8 of a certain other 
 crew, but 79 of the latter are equal to 90 of the former. Which 
 is the faster crew 1 
 
 Also, if the faster crew start at a distance equivalent to 
 four of their own strokes behind the other, how manv strokes 
 will they take before they bump them ? 
 
 57. A person, sculling in a thick fog, meets one barge and 
 overtakes another which is going at the same rate as the 
 former ; shew that if a be the greatest distance to which he 
 can see, and &, h' the distances that he sculls between the 
 times of his first seeing and passing the barges, 
 
 21 1 
 
 I 
 I 
 
SIMUL TANEOUS EQUA TIOXS. 
 
 163 
 
 ill stance 
 idle the 
 leaps as 
 li make 
 
 >'ing the 
 lion sell 
 1 pence, 
 many of 
 
 e to pay 
 ns, each 
 I 2 less, 
 niber of 
 
 I's to l»e 
 
 ins 1056 
 
 744 for 
 
 x\ C, for 
 
 ite been 
 3urs less 
 Dwer, he 
 distance 
 
 in other 
 Which 
 
 alent to 
 r strokes 
 
 arge and 
 as the 
 rhich he 
 laan the 
 
 / 
 
 / 
 / 
 
 58. Two trains, D2 fi-i.-t long and 84 feet long respectively, 
 are moving with nnifoiiu velocities on parallel rails in opposite 
 directions, and are observed to pass eacii other in one second 
 and a half; but when liiey are moving in the same direction, 
 their velocities being the same as before, the faster train is 
 observed to })as.s the other in six seconds; find the rate in 
 miles per hour at which each train moves. 
 
 59. The fore-wheel of a carriage makes six revolutions 
 more than the hind-wheel in 120 yar<ls ; but only four revolu- 
 tions more when the circumference of the fore-wheel is increased 
 one-fourth, and that of the hind-wlivel oue-tifth. Find the 
 circumference of each wheel. 
 
 60. A person rows from Cambridge to Ely (a distance of 
 20 miles) and back again in 10 hours, and tinds he can row 
 
 2 miles against the stream in the same time that he rows 
 
 3 miles with it. Find the rate of the stream, and the time of 
 his going and returning. 
 
 6r. A number consists of 6 digits, of whicli the last to the 
 left hand is 1. If this number is altered by removing the 1 
 and putting it in the unit's place, the new number is three 
 times as great as the original one. Find the number. 
 
 / 
 
 XVII. ON SQUARE ROOT. 
 
 220. In Art. 97 we defined the Square Root, and explained 
 the method of taking the square root of expressions consisting 
 of a single term. 
 
 The square root of a positive qiiantity may be, as we 
 explained in Art. 97, either positive or negative.. 
 
 Thus the square root of Aa^ is 2(t or - 2a, and this ambiguity 
 is expressed thus, 
 
 s^4a^=±2a. 
 
 In our examples in tnis chapter we shall in all cases regard 
 the square root of a single term as i\ positive quantity. 
 
 W 
 
 
/ 
 
 221. The square root of a product may l)e i'ouiid Ly taking 
 the square root of each factor, and multiplying tlio roots, bo 
 taken, together. 
 
 Thus ^/^^^ah, 
 
 222. The square root of a fraction may be found l)y taking 
 the square root of the numerator and the .s(iuare root of the 
 denominator, and making them the numerator and denominator 
 of a new fraction, thus 
 
 «l/>-"96' 
 
 4. 
 4 
 
 
 bxy'^ 
 
 fj ri 
 
 Examples. — Ixxix. 
 
 Find the Square Root of each of the following expressions ; 
 
 4- 
 
 10. 
 
 I. 4xh/. 
 64(i4&i0ca. 
 
 1662- 
 
 256k^ 
 
 28V 
 
 2. 8la%\ 
 
 5- 
 8. 
 
 II. 
 
 7l289a*¥afi. 
 
 1 
 
 4aV 
 G25a2 
 324&2- 
 
 3. l21mi'*?ii'Vi^ 
 6. leQai^feV-*. 
 
 25a^&« 
 ^* 121a;*^7/W' 
 
 
 I 223. "We may now proceed to investigate a Rule fur the 
 J extraction of the square root of a compound algebraical 
 I expression. 
 
 We know that the square oi a + h h a^ + 2ah + h^, and there- 
 fore a + b is the square root of a^ + 2ah + h'. 
 
 If we can devise an operation by which we can derive a + b 
 from a^ + 2ab + b'^, we shall be able to give a rule for the 
 extraction of the S(j[uare root. 
 
 Now the first term of the root is the s([uare root of the iir^t 
 term of the square, i.e. a is the square root of a\ 
 
 Hence our rule begins : 
 
 " Arrange the terms in the order of magnitude of the indices 
 of one of the quantities involved^ then take the sqiiare root of the 
 
 t 
 
 Ijii 
 
tliere- 
 
 ON SQUARE ROOT. 
 
 i6S 
 
 firni term and set down the renult as the first term of the root : 
 suhtruct ili^ square from the given expression, and bring down the 
 
 rpni'iivdrr :^^ tliiis 
 
 a'^ + 2ah + h'i^a 
 
 a- 
 
 '2ah\-b'i 
 
 Now this reinnindor may be reprosentod thus h('2a + h): 
 hence if we divide 2ah + h'^ by 2a + 6 we shall obtain i 6; the 
 f^econd term of the root. 
 
 Hence our rule proceeds : 
 
 " Donhle the fn'd term of the mot and set "Towr the result as tht 
 first term of a divisor:'' thus our process up to this point will 
 stand thus : 
 
 a^ + 2ah\h'^a 
 
 a» 
 
 2a 
 
 2ah + h^ 
 
 Now if we divide 2ah by 2a the •'esult is h, and hence we 
 obtain the second term of the root, and if we add this to 2a 
 we obtain the full divisor 2a + &. 
 
 TIence our rule proceeds thus : 
 
 ^^ Divide the first term of the remainder hij this first term of the 
 
 divisor, and add the result to the first term of the root and also to 
 
 the first term of the divisor :" thus our process up to this point 
 
 ■will stand thus : 
 
 a'^-\-2iJ) + h^{a + h 
 
 a' 
 
 2a + l 
 
 2a6 + 62 
 
 If now we multiply 2a + hhy h we obtain 2((h + b'^, which we 
 subtract from the first remainder. 
 
 Hence our rule proceeds thus : 
 
 " Multiply the divisor by the second term of the root and sub- 
 tract the result from tlie first remainder :'' thus our process will 
 stand thus : 
 
 i-i 
 
 m 
 
 til 
 
 «f 
 
 ki. 
 
 ■ji 
 
h 
 
 i; 
 
 0^ 
 
 l66 ON SQUARE ROOT. 
 
 
 
 
 
 2rt+6 
 
 2a6 + 62 
 2a& + 62 
 
 
 If tliere is now no remainder, the root has been found. 
 
 If there be a remainder, consider the two terma of the root 
 already found as one, and proceed as before. 
 
 lltii 
 
 1 i 
 
 ■ii 
 
 
 ill' 
 ijlii 
 
 
 224. Tlie following examples worked out will make the 
 process more ch ar. 
 
 (1) 
 
 
 la-l 
 
 - 2ah + 62 
 ~2«6 + 62 
 
 Here the second term of the root, and conseqnentl}' tlie 
 second term of the divisor, will iiave a negative sign prefixed, 
 
 I -2ah . 
 
 becai e — ^- — = -h. 
 2a 
 
 (2) 
 
 Gp + 4q 
 
 9if~ + 242)q+mi\2p + 4q 
 9jj2 
 
 24pq + l6f 
 24pq + Wq^ 
 
 25a;2-6(te + 3G(5K-6 
 25a;- 
 
 10a; -6 
 
 - GOx + 36 
 -60a; + 36 
 
 Next take a case in which the root contains three terms. 
 
 a^ + 2ah + b'^-2ac-2hc + c'^{a + b-c 
 
 a- 
 
 2a + h 
 
 2a + 2b-c 
 
 2ab + 62 - 2ac - 26c + c^ 
 
 2a6 + 62 
 
 -2ac-26c + c^ 
 -2ac-26c + c2 
 
le root 
 
 \.e the 
 
 \y the 
 
 e fixed, 
 
 
 s. 
 
 OiV SQUARE ROOT. 
 
 167 
 
 When we obtained the second remainder, we took the double 
 of a-rh, considered as a single term, and set down the result as 
 the first part of the second divisor. We then divided the first 
 term of the remainder, —2ac, by the first term of the new 
 divisor, 2a, and set down the result, - c, attached to the part 
 of the root alread} found and also to the new divisor, and then 
 multiplied the completed divisor by - c. 
 
 Similarly we may proceed when the root contains 4, 5 or 
 more terms. 
 
 Examples.— Ixxx. 
 
 Extract the Square Eoot of the following expressions : 
 
 1. 4rt2+ 12^6 + 962. 6. x^ - 6x^ + Idx- ~ :iOx + 25. 
 
 2. Wc'^-24¥P + 9l^ 7. dx^+l2c^+l0x'^ + 4x+l. 
 
 3. a%'^+l62ab + 656l. 8. 4r^ - 12r'^+ 13r*' -()/•+ 1. 
 
 4. if'~2Sif + :i6l. 9. 4,i^ + 4n^-1ii---in + 4. 
 
 5. 9a2/,-'c2-102a?>c + 289. 10. 1 -6.K+ 13.«''_ 12^3 + 40;^ 
 
 11. x^-4x^+l0x^-Ux^ + 9x:^. 
 
 1 2. 4i/ - 1 2//'Vj + 25?/%2 _ 24yz^ + IGzK 
 
 13. a'^ + 4ah + 4h-±9c' + 6ac + \2bc. 
 
 1 4. a« + 2a% + 3a^/>2 4. 4^353 + 2a^b^ + 2a&'^ + 6". 
 
 15. x^'-4x^ + 6x^ + Sx:^ + 4x+l. 
 
 1 6. 4,r< 4- 8ax^ + 4a^x^ + 1 66 V.2 + 1 6^62,^ + l Qh\ 
 
 17. 9 - 24.7; + 582;2 - 1 1 Qx^ + 1 29x4 - 1 40.r^ 4- 1 00x\ 
 
 1 8. 16a4 - 40«^/) + 25^252 _ maW. + Q4¥x^ + 64(t2te. 
 
 1 9. Oa* - 24(fc3/)3 _ 30a2^ + ] 6(rp6 + 4{)a'pH + 25^2. 
 
 20. 4//^^ - 1 2 //3.k3 + 1 V7/2x* - 1 2i/r' + 4x8. 
 
 21. 2bx^'\f - 'SOxhf + 29x-i/ - 1 2.r?/5 + 4i/. 
 
 22. 16.r4-24x3|/-f 25x2?/2-12.r»/3 + 4i/4. 
 
 2 3. 9a2 -I2ah + 24ac - 1 Otc + 462 + 1 (j^a. 
 
 ?4. a;* + 9x2 + 25 _ Cx^ + 10x2 - 3(\r. ' 
 
 25. 25x2-20x// + 4?/2 + 9«2-12i/2 + 30x«. 
 
 26. 4x2 (x - 1/) + »/' (y - 2) ty^ (!/-'+!). 
 
 I 1' 
 
 ' ;!!»i 
 
 fi 
 
 ii^' ^i 
 
m 
 
 i68 
 
 OJV SQUARE ROOT. 
 
 225. When any fractional terms are in the expression of 
 which we have to find the Square Root, we may proceed as in 
 the Examples just given, taking care to treat the fractional 
 terms in accordance with the rules relatinfj to fractions. 
 
 D 1 /^ 
 
 Thus to find the square root of x^~^x + 5--. 
 
 9 81 
 
 X^ 
 
 x^ 
 
 ■x + 
 
 16/ _4 
 r ' 81 V ^ 9 
 
 2a;- 
 
 9 
 
 8 16 
 
 9''"*" 81 
 
 8 16 
 ~9^^81 
 
 Since 
 
 8_^ 8_^2^8 1^4 
 9 • 9 * 1"~9^2~9' 
 
 8 16 
 
 Or we might reduce x'^---x ■{■ ^y ^^ ^ single fraction, which 
 
 9 81 
 
 would be 
 
 81a;^-72.c+16 
 81 ' 
 
 and then take the square root of each of the terms of the 
 fraction, A\ith the followin result : 
 
 — - - , which is the same as a; - -. 
 9 J 
 
 Examples.— Ixxxi. 
 
 
 
 ^M 
 
 i 
 
 I. 4rt« + -,,. -a*6-^. 
 1() 
 
 9 „ a' 
 3. '»^-2 + ^. 
 
 a' 
 
 W 
 
 ¥ a^ 
 
 4 
 
 6. x* + 2x^-x+ ,. 
 4 
 
 7. 4(t2 - 1 2ah + alfi + 9/>2 _ •'; + i-. 
 ' ;i It) 
 
\r 
 
 ON CUBE ROOT. 
 
 169 
 
 16 32 
 
 8. a;t 4- 8x- + 24 + -,- + -„. 
 
 9. -'-- + 4a^ + -~a^x^ - 7>d?' - 2a^x + -^w>x. 
 lb 9 o 
 
 of the 
 
 1 4 9 4 6 12 
 
 x^ 7/2 a- xy xz y:i 
 
 n^ 25 on 
 
 n 
 
 5 
 
 12. a^6^-6«?,c^4^^^+9c^d^-;^-^. 
 
 •^ z^ X- z' X z^ 
 
 14. 
 
 4m^ 9n^ 
 
 . „.. , 16m 24?t 
 
 , -+ • .,+4- - - + 
 
 n^ m- 
 
 n 
 
 m 
 
 a^ b'^ c^ cC^ ah 2a.c ad he , hd _cd 
 ^5" 9 ■^16"^25^T~ 6 "^ 15 ~ 3 "lO'^^ ~ 5* 
 
 1 6. 49.(;^ - 28x3 -I7x'^ + Qx + ? 
 
 F 7. 9x4 _ 2ax^ + 66x3 + "^tf. _ ahx^ + 62a;2. 
 
 4 
 
 1 8. 9x4 _ 2x3 _ ? 61^2 + 2^ + 9. 
 
 y 
 
 
 % 
 
 1 
 
 X + T. 
 
 4 
 
 XVIII. ON CUBE ROOT. 
 
 226. The Cube Root of any expression is that expression 
 wliuse cnhc or third power gives the proposed expression. 
 
 Thus a is the cuhe root of a-^, 
 36 is the cube root of 276^. 
 
 The cube root of a negative expression will be negative, foj- 
 since 
 
 ( - (()3= —ay - (t y - a = - a', 
 
 the cube root of - u*^ is - a. 
 
 iSJj 
 
I'/O 
 
 ON CUBE ROOT. 
 
 So also 
 
 and 
 
 - 3x is the cube root of - 27a;^, 
 - 4a^'6 is the cube root of — 64a^6'^ 
 
 The symbol IJ is used to denote the operation of extracting 
 the cube root. 
 
 |.>; 
 
 Examples.— Ixxxii. 
 
 Find the Cube Boots of the following expressions : 
 I. 8a3. 2. n^^\f\ 3. - 125?.uV. 
 
 4. -aiea^'-^/A 5. 34361-V8. 6. - 1000a-''6«c-2. 
 
 7. -1728m2iri24, 8. 1331ft06^8. 
 
 227. We now proceed to investigate a Rule for finding the 
 cube root of a compound algebraical expression. 
 
 We know that the cube of a + h is a^ + 3rt-?) '- ZaW-\-W, 
 and therefore a + /; is the cube root of (.i? + 'M-h + 3a6- + W. 
 
 We observe that the first term of the root is the cube root of 
 the tirst term of the cube. 
 
 Hence our rule begins : 
 
 ^^ Arrange the terms in the order of niafjr'.ioh of the indices of 
 one of the quantities involved, then take tlic c-'beroot of the first 
 term and set down the result as the first term of the root; subtract 
 its cube from the given expression, and bring down the remainder:^' 
 thus 
 
 d^ + Za-b^-Ub'^-^Wi^a 
 
 0? 
 
 Za^b + Zah^ + W 
 
 Now this remainder may be represented thus, 
 
 hence if we divide ^a% + ^ah'^ + ¥ hy 3rt2 + 3a6 + 6^^ we shall 
 obtci'ri +?\ the second term "f the root. 
 
 Ilcni.e CTir rule proceeds : 
 
 " MultiiJy th.o r,q>fare of the first term of the root hy 3, and set 
 doicn th'i rt:i,i't as tiie first term of a divisor:" thus our process 
 up to this p' uit \N ill stand thus : 
 
f>.A« 
 
 ON CUBE ROOT. 
 
 171 
 
 
 3a2 
 
 3a26+3a62 + 63 
 
 Now if we divide Za^h by So.^ tlie result is ?), and so we 
 obtain the second term 0^ the root, and if wc add to 3a~ the 
 expression 3a6 + 6"^ we obtain the full divisor 3(t'- + 3c(i + 6''^. 
 
 Hence our rule proceeds thus : 
 
 "Divide the first term of the remainder hy the first term of the 
 divisor, and add the result to the first term of the root. Then take 
 three times the product of the first and secori terms of the root, 
 and also the square of the second term, and add these results to 
 the first term of the divisor." Thus our process up to this point 
 will stand thus : 
 
 a^ + 3a% + 3ab-^ + U'\a + h 
 
 a^ 
 
 3a'^ + 3ah + b'^ 
 
 :^a% + Sab'^ + b'^ 
 
 If we now multijjly the divisor by b, w j obtain 
 
 3«-6 + 3rt6'^ + /)•', 
 which we subtract from the first remainder. 
 
 Hence our rule proceeds thus : 
 
 "Multiply the divisor by the second term of the root, an> mb- 
 tract the result from the first remainder,-'^ thus our process Avill 
 stand thus : 
 
 a^ + 3a'^b + 3ab^ + b\a + b 
 a3 
 
 3a2 + 3a6 + 62 
 
 'Sa:^b-\-3ab'^ + ¥ 
 3a% + 3ab'^ + b^ 
 
 If there is now no remainder, the root has been found. 
 
 If there be a remainder, consider the two teims of t!e root 
 already found as one, and proceed as before. 
 
 228. The following Examxjles may render the pr 
 clear : 
 
 :i ih 
 
 "•IP 
 
 V. (1 
 
 =n 
 
 M 
 
 uil 
 
 
 ;;j:r 
 
 i'^ 
 
172 
 
 ^iV CUBE ROOT. 
 
 Ex. 1. 
 
 a3-12rt''i-l-48a-64(rt-4 
 
 a' 
 
 3(t,''^-12a + 16 
 
 -12a'^ + 48a-64 
 -12ft2 + 48a-64 
 
 Here observe that tlie second term of the divisor is formed 
 thus : 
 
 3 times the product of a and — 4— -3 x a x 4= — 12a. 
 
 Ex. 2. 
 
 a;6 _ o,e'' + I5.r-* - 2().c'^ + \h:(? - 6;c + 1 {t? - 2iC + 1 
 
 fc" 
 
 3,,;4 _ 6,^3 + 4.g2 j _ (|-,;5 + 15.^4 _ 20.x3 + \hx^ - 6x + 1 
 
 3:/;'' - 1 1x^ 3a;4 - ] '±x^ + 1 5.^2 - 6x + 1 
 
 + 15a:'- - 6:c +1 3^* - ISa;^ ^■ l^a;^ - 6u; 4- 1 
 
 Here the formation of the first divisor is similar to that in 
 the preceding Examples. 
 
 Tlie formation of the second divisor may be explained thus : 
 
 Regardinq; a;2 _ 2.x as one term 
 
 3 (.o2 - 2.r)2 = 3 (cc* - 4x3 + 4.^.2) ^ 3,,;4 _ i2x3 + 12,x'2 
 
 3x(a;2-2x-)xl = 3^2 -6a; 
 
 12 ^ 1 
 
 and adding these results we obtain as the second divisor 
 
 3x*-12u;3 + 15a;2-6a; + l. 
 
 £X/,MPLES.- IXXXiii. 
 
 Find the Cube Root, of each of the foUoAving expressions: 
 
 1. ft3-3;.''6 + 3rtfe2 -?,3, 2. 8(t''+12(i2 + 6a4-l. 
 
 3. rt3 ^ 24^2^ ^.. i92«//2 + 5 126-\ 
 
 4. a^ + 3^26 + 3rt7>2 + /,3 + 3j^2c + 6^5c ^. 3^2^. + 3(^^,2 + 36c2 + c\ 
 
 5. ic3 - 3x2?/ .y 2xf - 1/ + 3r2;j - fj./:?/-; + 37/2^ + 3xx;2 - ',iyz^ + z\ 
 
 6. 27x" 54,/;''' 4 63,/;» - 44.r " -V 21x2 (j,^, + j. 
 
 Fi 
 
 ■^/^^. 
 
foTOied 
 
 -2u; + l 
 
 that in 
 ed thus : 
 
 1 
 
 sor 
 
 ssions : 
 1. 
 
 
 
 ON CUBE ROOT. 
 
 ^1% 
 
 7. 1 - 3a + 6a2 - 7a3 + 6«* - 3a'' + a«. 
 
 8. ic3 - 3x2?/ ^ 3^^,j^2 _ ,^3 + 8^23 + 6.^2;^ - 12.ri/;; + G)/^^ + 12.';;^2 _ i2;'^2_ 
 
 9. a« - 12a ' + 54a4 - 1 12^3 + 108a2 - 48a I 8. 
 
 10. Swt" - 36m'^ + 66??i'* - 63??i"' + 33//i2 - %n + 1. 
 
 11. a;3 + 6^2^^ + 1 2a;j/2 + Si/^ - 3x-;; - 1 2^i/a; - 1 ifz + 3.t,-2 + (Jy.-j^ _ ;.;;. 
 
 1 2. 8?/i'^ - 36)/i-'u + 54w?i2 _ 27>i3 _ ] 2m2r + 30/j/ )> /• - Tnih 
 
 3 1 
 
 13. 971^ + 3?/l2 - 5 4- - r, ... 
 
 229. Tlie/oH/'^/i, root of an expression i.s found by taking' 
 tlie square rout of the scj^uare root of tlie expression. 
 
 Thus 
 
 ^/I6a864= J4a^//- = 2a2&. 
 
 The sixDi root of an expression is found l)y takint,' the cube 
 root of the sq[uare root of the expression. 
 
 Thus ;^^\a>W^ ;^8a063 = 2a26. 
 
 Examples. — Ixxxiv. 
 
 Find the fourth roots of 
 
 1. 16a4-96a3x + 216a2a;2_2i6ax3 + 81cc*. 
 
 2. l4-24a2+16rt^-8a-:i2al 
 
 3. 625 + 2000u; + 2400x2 + 1280x-'' + 256a;*, 
 
 Find the sixth roots of 
 
 4. «« - 6a''^6 + 15a''62 - 20(rVr'' + ISa^ftt _ ^ah'' + ^6, 
 
 5. x« + 6r''4-15x- + 20x-'5+15x-' + 6.^'+l. 
 
 6. m» - 1 2?^''' + mm^ - 1 GO ./t^ ^. 240//t2 - 1 92m + 64. 
 
 ,, 
 
 I 
 
 ill' 
 
 ''113 
 
 ::i^ 
 
 
 3i: 
 
 ;■ 
 
 :'H 
 
 \% 
 
\ 
 
 XTX. QUADRATIC EQUATIONS. 
 
 230. A Quadratic Equation, or an equation of tioo dimen- 
 sions, is one into which the square of an unknown symbol 
 enters, without or with the first power of the symboh 
 
 Thus a:2 = 16 
 
 ;md 
 
 a;2 + 6^ = 27 
 
 are Quadratic Equations. 
 
 231. A Pure Quadratic Equation is one into which the 
 square of an unknown symbol ente? s^ the first power of the 
 symbol not appearing. 
 
 Thus, x^ = lij is a. pure Quadratic Equation. 
 
 232. An Adfected Quadratic Equation is one into which 
 the square of an unknown symbol enters, and also the first 
 power of the symbol. 
 
 Thus, x^-\-6x = '27 is an adfected Quadratic Equation, 
 
 Pure Quadratic Equations. 
 
 233. When the terms of an equation involve the square 
 of the unknown symbol onlij, the value of this square is either 
 given or can be found by the processes described in Chapter 
 XVII. If we then extract the square root of each side of tlie 
 e^pu'tion, the value of the unknown symbol will be determined. 
 
 234. The following are examples of the solution of Pure 
 Quadratic Equations, 
 
 I. 
 
 x- 
 
 4. 
 
 X 
 
 7- 
 
 I 
 
 X 
 
 8. 
 
 (' 
 
 
 8 
 
 lo. 5 J 
 
UA DRA TIC EQUA TIONS. 
 
 175 
 
 Ex. 1. a;2=16. 
 
 Taking the square root of each side 
 
 a;=±4. 
 
 We prefix the sign ± to the number on the right-liand side 
 of the e(|uation, for the reason given in Art. 220. 
 
 Every pure quadratic equation will thcirefore have two rootSy 
 equal in magnitude, hut with ditlereut signs 
 
 Ex. 2. 4.x-2 + G = 22. 
 
 Here 4.c2 = 22-6, 
 
 or 4.(;-=16, 
 or a;2 _ 4 . 
 
 That is, the values of x which satisfy the equation are 2 
 and - 2. 
 
 Ex. 3. 
 
 Here 
 
 128 _ 216 
 
 128(5;«2-6) = 216(3a;2-4), 
 or 640x2 - 768 = 648x-- 864, 
 or a;2=12 ; 
 
 :.x=±^l2. 
 
 Examples.— Ixxxv. 
 
 I. a;2 = 64. 2. x^^a-hK 3. :(;2- 10000 = 0. 
 
 4. a;2~3 = 46. 5. 5x2-9 = 2:^2 + 24. 6. ;iax2=192a^c«. 
 
 4""- 
 
 II. inx-' + n=q. 
 
 8. (500 +x) (500 - x) = 233359. 1 2. a;2 - ax + h = ax {x - I ) 
 8112 . .. 45 57 
 
 = 3x. 
 
 1 
 
 10. b^ - ISx + 65 = (3:/j - 3)2 
 
 13- 
 
 14. 
 
 2x^ + 3 4x- - 5 
 42 35 
 
 Q\ 
 
 l^m 
 
 x^-2 x2-3* 
 
 ill 
 
776 
 
 QUADRATIC EQUATIONS. 
 
 ii 
 
 I! ' '\- 
 
 Adjected Quadratic Equations. 
 
 235. Adfected Quadratic Ef[iiatioiis are solved by adding 
 a certain term to both sides of the equutioii so as to make tlie 
 left-hand side a perfect sq^uare. 
 
 Having arranged the equation so that the first term on tlic 
 left-hand side is the square of the unknown symbol, and the 
 second term the one containing the first power of the unknown 
 quantity (the known symbols being on tlie right of tlie equa- 
 tion), we add to both rddes of the equation the square of half the 
 coefficient of the second term. The left-hand side of the equa- 
 tion then becomes a perfect square. If we then take tlie square 
 root of both sides of tlie equation, we shall obtain tivo simple 
 equations, from which the values of the unknown symbol may 
 be determined. 
 
 236. The process in the solution of Adfected Quadratic 
 Equations will be learnt by the examples which we shall giAe 
 in this chapter, but before we proceed to them, it is desirable 
 that the student should be satisfied as to the way in which an 
 expression of the form 
 
 x^ + ax 
 
 is made a perfect square. 
 
 Our rule, as given in the preceding Article, is this : add tlie 
 
 square of half the coefficient of the second term, that is, the 
 
 •> 
 
 square of L that is, -j-. We have to shew then that 
 
 4 
 
 2 
 
 is a perfect square, whatever a may be. 
 
 This we may do by actually performing the operation of 
 
 a^ 
 extracting the square rQot of x'^ + ax + —, and obtaining the 
 
 result x + ^ with no remainder, 
 
 C0( 
 
 111 
 
 I 
 
 coej 
 
 I 
 
 adc 
 
 ■p 
 
 TOO 
 
Q UA DRA TIC EQ I \l TrO.VS. 
 
 177 
 
 237. Let us examinu this process l)y the aid of numeriad 
 coetiicients. 
 
 Take one or two examples from tlm perfect sc^uares given 
 in page 48. 
 
 We there have 
 
 a!2+18x+ 81 v/hich is the square of 0:+ 9, 
 
 a;2 + 34x + 289 .«+17. 
 
 a;3- 8x+ IH x- 4, 
 
 x2-36x^-324 ;(;-18. 
 
 In all these cases the third term la the square of half the 
 coefficient ofx. 
 
 For 
 
 289 = (17y^=^(=^^^/, 
 
 16 = W =(2/, 
 324 = (18)2 = (''^^^)'. 
 
 238. Now put the question in this shape. What must we 
 add to x^ + ax to make it a perfect square / 
 
 Suppose 6 to represent the quantity to be added. 
 
 Then x^ + a.^ + 6 is a perfect square. 
 
 Now if vve perform the operation of extracting the S(|uare 
 root of x^ + ax + 6, our process is 
 
 x^ + ax- -f 6 ( X + ^^- 
 
 a;' 
 
 2x + 
 
 ii 
 
 ax + 6 
 
 a"" 
 ax + -T 
 4 
 
 b- 
 
 a^ 
 
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 M 
 
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 »> > 
 
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 Sciences 
 brporation 
 
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 WnSTIR.N.Y. MSSO 
 
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 ^ 
 
Z78 
 
 QUADRATIC EQUATIONS. 
 
 Hence in order that aj'''+aic+6 may be aperfect square we 
 must have 
 
 J. ^'^ i\ 
 
 or 
 
 or 
 
 a" 
 
 fc=T. 
 
 '=(!)• 
 
 That 13, 6 is equivalent to the square of half the coefficient 
 ofx. 
 
 239. Before completing the square we must he careful 
 
 (1) That the square of the unknown symbol has no coeffi- 
 cient hut unity f 
 
 (2) That the square of the unknown symbol has a positive 
 sign. 
 
 These points will be more fully considered in Arts. 245 and 
 246. 
 
 240. We shall first take the case in whfch the coefficient of 
 the second term is an even number and its sign po^ itive. 
 
 Ex. 
 
 a;2 + 6x=40. 
 
 Here we make the left-hand side of the equation a perfect 
 square by the following process. 
 
 Take the coefficient of the second term, that is, 6. 
 
 Take the half of this coefficient, that is, 3. 
 
 Square the result, which gives 9. 
 
 Add 9 to both sides of the equation, and we get 
 
 x« + 6x + 9 = 49. 
 
 Now taking the square root of Ijoth sides, we get 
 
 a;+3=±7. 
 
QUADRATIC EQUATIONS. 
 
 179 
 
 I'e we 
 
 fflcient 
 
 il 
 
 > coeffi- 
 
 oositive 
 45 and 
 
 lient of 
 
 perfect 
 
 Hence we have two simple equations, 
 
 a; + 3=+7 
 
 and 35 + 3== -7 
 
 (2> 
 
 From these we find the values of oj, thus : 
 from (1) a; = 7-3, that is, a; = 4, 
 
 from (2) a;= - 7 - 3, that is, a;= - 10. 
 
 Thus the roots of the equation are 4 and - 10. 
 
 EXAMPLES.— IXXXVi. 
 
 I. x^ + Gx = *79. 2. .t2+12x = 64. 3. a;2 + 14a; = 15. 
 
 4. x'^ + 46x = m. 5. rt- + 128a; = 393. 6. a;2 + 8a;-65=0 
 7. a;2+ 18a; -243 = 0. 8. it^-; 16a; -420=0. 
 
 241. We next take the case in which the coefficient of the 
 second tenu is an even number and its sign negative. 
 
 Ex. 
 
 x'^-8x = 9. 
 
 The term to he added to both sides i« (8-f2)''^, that is, (4)'-^, 
 that is, 16. 
 
 Completing the square 
 
 x2- 8a; +16 = 25. 
 
 Taking the square root of both sides 
 
 a;-4=±5. 
 
 This gives two simple equations, 
 
 a;-4=+5 (1), 
 
 a;-4= -5 (2), 
 
 From (1) 
 from (2) 
 
 a;=5 f 4, .'. x~9; 
 
 x= -5 4 4, .'. x= — 1. 
 
 t, I 
 
 Thus the roots of the equation are 9 and - 1, 
 
,1 • 
 
 : 
 
 11 ' 
 
 i8o 
 
 QUADRATIC EQUATIONS. 
 
 EXAMPLES. — IXXXVii. 
 
 I. a;2-6a; = 7. 2. rc2-4a;=5. 3. rc2-20x = 21. 
 
 4. a;2-2a; = 63. 5. a;2- 12.7:+ 32 = 0. 6. a;2- 14x + 45 = 0. 
 7. a;2-234x+ 13688 = 0. 8. (a;-3)(x-2)=3(5K+ 14). 
 
 9. a;(3a;-17)-a;(2a; + 5) + 120 = 0. 
 10. (x-5y'' + (a;-V)2=a-(a;-8) + 46. 
 
 242. We now take the case in which the coefficient of thi> 
 second term is an oM number. 
 
 Ex. 1. 
 
 a;2 - V-f = 8. 
 
 The term to be added to both sides is 
 
 .2 49 
 
 4' 
 
 Completing the square 
 
 OH 49 ^ 49 
 
 or, 7? - Ix + 
 
 49_81 
 4"4- 
 
 Takinj? the square root of both sides 
 
 7 .9 
 •^-2=*2- 
 
 This gives two simple equations, 
 
 7_ 9 
 
 '■'~2~'^2 
 
 7__9 
 ^ 2~ 2 
 
 9 7 16 
 
 ^'^o + o' or, a!=— , .. x = 8; 
 
 From (1) 
 
 I'rom (2) 
 
 Thus the roots of the equation are 8 and - 1, 
 
 2 2' 
 9.7 -2 . , 
 
 (1). 
 
 (2). 
 
QUADRA TIC EQUA TIONS. 
 
 z8x 
 
 Ex. 2. a?2-x = 42. 
 
 
 The coeflRcient of the second term is 1 . 
 
 
 The term to be added to both sides ii 
 
 
 (•-^)-Gr=i' 
 
 
 4 4 
 
 
 ., 1 169 
 
 or, x^-a: + ^= --; 
 
 
 1 .13 
 
 
 Hence the roots of the ef|uation are 7 and - 6. 
 
 Examples.— Ixxxviii. 
 
 I. .r2 + 737 = 30. 
 
 2. a;2-lla;=12. 
 
 4. X"- 13a! =140. 5. x^-\-x=--.Y,' 
 
 lo 
 
 3. u;2 + 9x = 437. 
 ^ 4 
 
 6. x^-x = H% 
 
 7. A- + 37a; = 3090. 8. ^2=56 + 0;. 
 
 9. a: (5 - x) + 2x (.v - 7) - 10 (./J - 6) = 0. 
 10. (5x - 21) (7.C - 33) - (I7.(; -t- 15) (2.o - 3) = 448. 
 
 243. Our next case is that :n whicli the coefficient of the 
 second term is a fraction of which the numerator is an even 
 number. 
 
 :i). 
 
 Ex. 
 
 a;2-^x = 21. 
 
 
 The term to be added to both sides is 
 
 ^4 1\2 /2\"^ 
 
 '. ..2_ % 4. -91 4. 
 
 1)1", X 
 
 ■A 
 
 4 4 520 
 5*'' "^25" 25 ' 
 
r83 
 
 QUADRATIC EQUATIONS. 
 
 2_ 23 
 Hence the values of a; are 5 and - 
 
 21 
 6' 
 
 Examples. — Ixxxix. 
 
 I. a;2_ r = 
 
 35 
 9* 
 
 2. iC^ + ^X = 
 
 4 
 
 5^ 
 
 3 
 
 25' 
 
 „ 28a: 1 ^ 
 
 .1 8 3 - 
 
 I'll _ T — II 
 
 ll'^ 11 "• 
 
 ^ 2 26 16 _ 
 7. a;2--x+2=0. 
 
 r 2_. 4 3 , „ 16 16 
 
 8. a;2 — ^a; = 45. 
 
 I 
 
 244. We now take the case in which the coefficient of the 
 second term is a fraction whose numerator is an odd number. 
 
 Ex, 
 
 „ 7 136 
 ^ 3 3 • 
 
 The term to ho added to hotli sides is 
 
 (3-^)==(3^2)=(6)=36> 
 
 ., 7 49 _ 136 49 
 ••''■'~3'^'^36~~3~"^36' 
 
 nr " -^ ^49_1681 
 or a;---x + --— ^-; 
 
 7_.41 
 
 17 
 Hence the values of x are 8 and — — -. 
 
 o 
 
 Examples.-— xc. 
 
 I. x''--^x = 8. 
 
 3 
 
 4. a;2 + -a: = 76. 
 
 15 
 
 2. a;2--ro = 98. 
 o 
 
 5. a:2-?a; = 16. 
 o 
 
 3. a;2 + ^.r = 30. 
 
 6. ..;'-'- 
 
 2 
 11 
 
 7. a;2-~, a;-34 = 0. 
 4 
 
 O " 
 
 o. X- 
 
 23 _ 3 
 < 4 
 
 i 
 
QUADRATIC EQUATIONS. 
 
 183 
 
 245. The square of the unknown symbol must not he pre- 
 ceded hy a negative sign. 
 
 Hence, if we have to solve the equation 
 we change the sign of every term, and we get 
 
 Completing the square 
 
 x2-6x + 9 = 9-9, 
 or x^-Qx + Q = 0. 
 Hence a; -3=0, 
 
 or a- = 3. 
 
 Note. We are not to be surprised at finding only wte 
 value for x. The interpretation to be placed on such a result 
 is, that the two roots of the equation are equal in value and 
 alike in sign. 
 
 246. The square of the unknown symbol must have no 
 coefficient hut unity. 
 
 Hence, if we have to solve the equation 
 
 5x^-3x = 2y 
 we must divide all the terms by 5, and we l.v.[ 
 
 3x 2 
 5* 
 
 x' — — = 
 
 2 
 
 From which we get a:= 1 and a;= — -. 
 
 
 
 247. In solving Quadratic Equations involving literal co- 
 efficients of the unknown symbol, the same rules will apply as 
 in the cases of numerical coefficients. 
 
 Thus, to solve the equation 
 
 X a 
 
 Clearing the equation of fractions, we get 
 
 2a2-x2-2aa; = 0; 
 therefore -x^ — 2ax= -2a\ 
 
 or x^ + 2ax = 2a\ 
 
 // 
 
i 
 
 I 
 
 1 
 
 184 
 
 QUADRATIC EQUATTom. 
 
 Completing the sqiuire 
 
 whence a! + a=±VJi.a; 
 
 therefore x— -a-^ ^3 . a, or cc = - a - ^3 . a. 
 
 The following are Examples of Literal Quadratic Equations. 
 
 EXAMPLES.— XCi. 
 
 I. x^ + 2ax=a^. 2. x^ — 4ax = 7a^. 
 
 3. x^ + 3mx=-. . 
 4 
 
 „ 5n 3n2 
 5. x^ + {a-l)x = a. 
 
 rt%2 2f7;r 52 
 
 8. adx - acx^ = hex - bd. 
 9' cx + -^lf^ = (a + b)x^. 
 
 10. 
 
 b^' 
 
 + -0 = 0. 
 
 II. abx^ + 
 
 3«2x 6a^ + ah-2h^ h^x 
 
 c C c 
 
 1 2. (4a2 - 9cd^) a;2 + (uh^ + 4a6^i2) ^j + (ac^ + &d2)2 = 0. 
 
 248. If both sides of an equation can be divided by the 
 unknown symbol, divide by it, and observe that is in that 
 case one root of the equation. 
 
 Thus in solving the equation 
 
 - a;3-2x2=3x, 
 
 we may divide by x, and reduce the equation to the form 
 
 a52-2ic = 3, 
 
 from which we get 
 
 x = '3 or x= - 1. 
 
 Then the three roots of the original ei^uation are 0, 3 and - 1, 
 
 W(; shall now give some Miscellaneous Examples of Quad- 
 ratic Equations. 
 

 
 ' 
 
 Q UADRA TIC EQUA TIONS. 
 
 
 
 185 
 
 I. 
 
 4. 
 
 7. 
 
 x^- 
 x'- 
 
 -7x + 2 
 -Ux- 
 
 EXAMPLES.— XCii. 
 
 = 10. 2. .':--5x + 3 = 9. 3. 
 
 = 8. 5. »''i + 7x-18 = 0. 6. 
 = 0. 8. 5a;-3-~A 
 
 X?- Yix- 
 
 X - 
 7a; -6 
 
 -7 = 
 
 X 
 
 3 ' 
 
 = 5. 
 = 22. 
 
 9. a;'-^- Go; -14 = 2. 
 
 10. 
 
 a; -3 
 
 4.r a; - 3 
 
 X + 3 " 2a;~+ 6 
 
 = 2. 
 
 II. 
 
 14. 
 
 17. 
 
 19. 
 
 21. 
 
 23- 
 
 4a; a:- 7 
 ^ + 7" 2^ + 3 
 
 1 „ 1 
 
 = 2. 12. a:2-12 = lla;. 
 
 169 
 
 13- 
 
 -14 = 13a;. 
 
 •aX' 
 
 :a;+7Q = 8. 15. 3a; = 26. 16. 2a;2=18a;-40. 
 
 8 
 
 X 
 
 4 + 3x 15-a; 7a;-14 
 
 10 
 
 a;-'i 
 
 20 
 
 3a; -5 
 
 6a; 
 
 1 
 
 ~ yx' 
 
 3x-25 
 
 ^3' 
 
 ix - 10 
 
 7 -3a; 
 
 7 
 
 a; + 5 
 
 X 
 
 "2- 
 
 a;+ll 
 
 ^ 9 + 4a; 
 
 18. 3a;2 = 24.T-36. 
 
 20. 
 
 7 2a; - 5 3a; - 7 
 
 a; + 5 
 
 2x 
 
 52. (a; -3)2 + 4a; = 44. 
 
 a; 
 
 x^ 
 
 26. x2-a; = 210. 
 
 24. 6a;2 + a; = 2. 
 
 6 2 ., 
 27. — ,-T+ =3. 
 ' a;+ 1 X 
 
 r 2 1 _1 
 
 25. X' -q^ — q; 
 
 28. 
 
 4a;2 
 
 -11=; 
 
 29. 
 
 3 x-l 
 
 x-\ 2 
 
 + 
 
 X 
 
 I 2_^3 
 
 ^^' a;-2 a;-f2~5* 
 
 10 14 -2x 22 
 33. — 
 
 a; 
 
 x" 
 
 35. 
 
 12 
 
 8 
 
 32 
 
 5 - X 4-x x + 2 
 
 37. x^+{a + h)x + ab = 0. 
 39. a;2_2ax + rt2„//2^(). 
 
 a 
 
 41. x''4-7a2 
 
 2a^ 
 
 = 0. 
 
 3c. 15x2-705=46. 
 
 32. 
 
 34. 
 
 36. 
 
 4x 20 - 4x 
 
 5-x 
 
 X 
 
 
 X 
 
 7 
 
 
 x+60 
 
 3x- 
 
 5 
 
 X 
 
 _ L 
 
 7-x 
 
 — 
 
 = 15. 
 
 X 
 
 X 
 
 10' 
 
 38. x^ — {h-a)x-ab = 0. 
 40. X- - (a2 _ a3) X — a^ = 0. 
 
 .\2, X^ 
 
 a- 
 
 ' + ^2 
 
 a& 
 
 a; + l=0. 
 
XX. ON SIMULTANEOUS EQUATIONS 
 INVOLVING QUADRATICS. 
 
 ■ i: 
 
 i! i t 
 
 249. For the Rolution of Simultaneous Equations of a de- 
 gree higher tlian the first no fixed rules can be laid down. We 
 shall point out the methods of solution which may be adopted 
 with advantage in particular cases. 
 
 250. If the simple power of one of the unknown symbols 
 can be expressed in terms of the other symbol by means of one 
 of the given equations, the Method of Substitution, explained 
 in Art. -217, may be employed, thus: 
 
 Ex. To solve the equations 
 
 a; 4-1/ = 50 
 xy = 600. 
 
 From the first equation 
 
 cc = .50 - y. 
 
 Substitute this value for x in the second equation, and we 
 get {bO-y).y = 600. 
 
 This gives 50(/ - 1/2 = 600. 
 
 From which we find the values of y to be -iO and 20. 
 
 And we may then find the corresponding values of x to be 
 20 and 30. 
 
 251. But it is better that the student should accustom 
 himself to work such equations symmetrically, thus : 
 
 To solve the equations 
 
 x + y = 60 (1), 
 
 xy = mO (2) 
 
 From (1) x^ + 2xy + y^ = 2500. 
 
 From (2) 4x1/ =2400. 
 
rxvowrxa oiwrn^. t tics. 
 
 Now subtract tliis from (3), and we get 
 
 1 1 
 
 x" 
 
 " ',nj 
 
 
 = "30' 
 
 • • 
 
 1 
 
 X 
 
 1 
 J. 
 
 ■« 
 -6^ 
 
 and from this equation ami (1) we find 
 
 x = 2 or 3 and y = 3 or 2. 
 
 iJ?9 
 
 H i 
 
 Examples. — xcvi. 
 
 ill) 
 x'^ij'^m' 
 
 --> 
 
 1 I 3 
 
 X y 4 
 
 3- 
 
 I I r 
 
 1 1 -41 
 
 
 1 1 ") 
 
 
 -!4+ 2 = 1 '5. 
 
 1 1 1 
 
 
 1 I ^1 
 
 6. 
 
 1 1 *., 
 
 X 7/ 1 2 
 
 3- 
 
 - ^2 • 
 
 •'■ y ^ 
 
 - = 3. 
 
 X y 
 
 1 I 7 
 
 X? " if U I" 
 
 
 1 1 ^3 
 
 X' y 4 
 
 
 
 255. To solve the equations 
 
 a;2 + 3.i7/=7 (1), 
 
 xy + 4y'=lH (2) 
 
 If we add the equations wu ^'(^t 
 
 ft;'-^ + 4.r»/4-47/=25. 
 
 Taking the square root of each side, and taking only the 
 positive root of the right-hand side into account, 
 
 X + 2// = 5 ; 
 .'. X = 5-2*/. 
 
 Substituting tliis vabie for x in (2) wo get 
 
 {o~2ij)y + 4y''=lS, 
 an equation by which y may be determined. 
 
 Note. In some cxami)h;s we mu.-^t suhtract the second 
 equation from the first in oixler to get a perfect square. 
 
190 ON SIMUI. TA NEC I 'S KQUA T/ONS 
 
 2r)0. To solve the C(j[U:itioii.s 
 
 :r-//^ = rn (1), 
 
 x2 + yy + y-'=13 (2). 
 
 Divifling (I) 1»y (2) we pet a;-?/ = 2 (3), 
 
 sqimring, x-' - 2./-J/ + 7/ = 4 (4). 
 
 Sul>tract this I'roni (2 , and we liavc 
 
 AihWu'A tliis to (4), M'f -et x? + 2.r//+7/= 16; 
 
 .'. a; + 2/=^ ±4. 
 
 Then from tliis eiiuation un<l (3) we lind 
 
 a; = 3 or - 1 , aiul ?/ = 1 or - 3. 
 
 257. , To solve the e<inatioiis 
 
 x^ + f^Gb (1), 
 
 3n/ = 28 (2). 
 
 Multiplying (2) by 2, we have 
 
 2a;?/ = 56)* 
 
 .-. a;2 + 2x7/ + ?/2=121) 
 x''-2xy + y-= 9> ' 
 
 ;. x + y=±n (A), 
 
 x-y=± 3 (B). 
 
 Tlie equations A and B furnish four imra of simple 
 equations, 
 
 x + y^n, a; + ?/ = ll, x + y=-U, x + y=-n, 
 
 x-y=-3, ^-2/=~3, JC-?/ = 3, x-y^--3. 
 
 from which v/e find the values of x to he 7, 4, -7 and -4, 
 and the corresponding values of y to he 4, 7, - 4 and - 7. 
 
 258. The artifice, hy which the solution of the equations 
 given in this article is effected, is applical)le to cases in which 
 the e(iiuitionB are homogeneous ami of the same order. 
 
o.v ."^/Arrrr.TAjvF.ors f.q(^at/oxs, cVf. 
 
 187 
 
 Su I itrai- Ling, x- - 2.i// f- //'- - 1 ( )( >, 
 
 Then I'roiu this eiimvtiou and (1) wo find 
 
 a; = ao or 2L) uiid y =^ 20 ur ao. 
 
 I. a3 + i/ = 40 
 a;j/ = 300. 
 
 4. x~y=\\) 
 
 Examples.— xciii, 
 
 2. a; + y=13 
 ay = 30. 
 
 5. x-\j — A'i 
 
 xy = 2.">( ). 
 
 3. a; + y«29 
 a;*/=100. 
 
 6. x-|/ = S)U 
 a;>/=l()0. 
 
 252. To solve tlu; tM[iiations 
 
 X 
 
 J/=12. 
 
 •(1), 
 .(2). 
 
 From (1) 
 
 a;2-2x'//+»/2=144 (3). 
 
 Subtract this Iroui (2), then 
 
 2.7/ = 
 
 «0. 
 
 4x//=-140. 
 
 Add this to (3), then 
 
 oi^ + 2xy + y--Aj 
 :. a;+y=±2. 
 
 Then from thir^ tM|ii;iti(»ii luiil (1) w^ get 
 
 X = 7 or 5 and y= - 5 or — 7. 
 
 I . .T - y = 4 
 .c- + }/-' = 40. 
 
 4. x-^y^S 
 
 Examples.— xciv. 
 
 2. x-y= 10 
 ic'-J + j/'-i-nS. 
 
 5. a;+//=12 
 u;-' + //'=101. 
 
 3. .6-7/= 14 
 
 6. x + y = 40 
 x^ + y''=lOSl, 
 
 KM '„ 
 
 ii 
 
X 
 
 111 
 
 1 88 ON SIMUL TA NEO US EQ UA T/ONS 
 
 253. To solve the eHuatioiis 
 
 a;'' + ?/'' = 35 (1), 
 
 x + y = rt (2). 
 
 Divide (1) In' (2), then >ve get 
 
 .'.•2 -.7-,, + 7/ = 7 (3), 
 
 From (2) f.- f- ±^:ij + if = 2,0 (4), 
 
 Subtracting (3) from (4), 
 
 3:?i/ = 18, 
 
 .-. 4y?/ = 24. 
 
 Then Iroiu this equation and (4) ve get 
 
 :.x-y=±\; 
 anil Iroju tliis equation and (2) we find 
 
 x = 3 or 2 and 2/ = 2 or 3. 
 
 EXAMPLKS.— XCV 
 
 I. i? + y^=-^\ "2. 9;=* + i/ = 341 3. a;3 + 7/=1008 
 
 cc + |/ = 7. rc+// = ll. .'c + 7/=12. 
 
 4. ,,3_^,3^5(j 5. .,..i_y3^txs 6. .v''-7/ = 279 
 
 a;-2/ = 2. a-?/ = 2. a;-?/ = 3. 
 
 254. To solve the equations 
 
 1 1 '» /IN 
 
 i^ro ^^^' 
 
 From (1), by squarini,^ it, \s\t get 
 
 1 9 1 "•"» 
 
 .,- xij y .U. 
 
 From this subtract (d), and w^- hav«' 
 
 2^1:J, 
 jij " 3(J ' 
 
 4 _24 
 
 tht 
 
ON PROBLEMS RESULTING, ^c. 
 
 193 
 
 Examples.- xcviii. 
 
 1. What number i.s that whose hall' iiiulti[)lied ])y its third 
 part 5^'ives 864? '' 
 
 2. What is the iiumher <j1" wh4<;h the seveiitli and eij^lith 
 ])arts bein|j[ multiplied together and the product divided by 
 
 2 
 
 o the quotient is 298.,? 
 
 3. I take a certain number I'rom 91. 1 then add the 
 number to 94. 
 
 I multiply the two result.s together, and the result is 8512. 
 AMiat is the num1)er ? 
 
 4. AVhat are the numbers Mhose product is 750 and the 
 quotient of one bj' the otlier 3., / 
 
 5. The sum of the squares of two numbers is 13001, and 
 tlie difference of tlie same s(j[uare.s ig 1449. Find the numbers. 
 
 6. The product 'f t'.vu juimbers, one of which is as mucli 
 above 21 as the otlier is below 21, is 377. Find the numbers. 
 
 7. The half, the third, the fourth and the fifth parts of a 
 certain number being multiplied together the jnuduct is G750. 
 Find the number. ' 
 
 8. By what number must 11500 l)e divided, so that 
 the (juotient nuiy be the same as the divisor, and the re- 
 mainde*- 51 ? 
 
 9. Find a number to -whicli 20 being added, and from 
 which 10 being subtracted, the square of the first result added 
 to twice the square of the second result gives 17475. 
 
 10. The sum of two numbers is 20, and the sum of their 
 squares is 436. Find the numbers. 
 
 11. The dill'erence between two numbers is 17, and the 
 sum (»f their scpiares is 325. What are tlie numbers ? 
 
 12. Wliat two numbers arc they whose product is 255 and 
 the sum of wliose S([uaros is 514 ? 
 
 13. Divide 16 in^o two parts such that their product 
 addcil to the sum of llieir squares may be 'l^^'':^, 
 
 (>A.] H 
 
 -'\ 
 
 m 
 
 (.!t r 
 
 
 I 
 
1,1 1 
 
 I 
 
 194 
 
 ON PROBLEMS RESULTTNO 
 
 \\ 
 
 14. Wliat miinbcr addi'd to its r,(|iiarG root ^mvcs as a 
 result 1332 ? 
 
 1 5. AVlj;i^t nuinl)er cxc(!L'(la its fequare root l)y 48*. \ 
 
 16. AVliat number exceeds its square root by 2550 ? 
 
 17. The product of two numbers is 24, and tbeir sum 
 multiplied by their difference is 20. Find the numbers. 
 
 18. AVhat two numbers are those whose sum multiplied 
 by the greater is 204, and whose difference multiplied by the 
 less is 35 ? 
 
 19. What two numbers are those whose difference is 5 
 and. their sum nvultiplied by the greater 228 ? 
 
 20. Find three consecutive numbers whose product is 
 equal to 3 times the middle number. 
 
 21. The difference between the s(|uares of two consecutive 
 numbers is 15. Find the numbers. 
 
 22. The sum of the squares of two consecutive numbers is 
 481. Find, the numbers. ' 
 
 23. The sum of the s(iuares of three consecutive numbers 
 is 305. Find the numbers. 
 
 NoTK. If r buy X apples for 1/ pence, 
 
 - will represent the cost of an apple in pence. 
 If I buy X sheep for z pounds, 
 
 X 
 
 will represent the cost of a sheep in pounds. 
 
 Ex. A boy bought a number of oranges for 16(i. ^lad he 
 bought 4 more for the same money, he would have paid 
 one-third of a penny less for each orange. IIow many did 
 he buy ? 
 
 Let X represent the number of oranges. 
 
 Then - - will represent thci cost of an orange in pence. 
 
 Hence 
 
 IG 
 
 X 
 
 K) 1 
 
 or 1(1 (3,r 1-12) = 48.K + 'j\^ + Ax, 
 or x'^^f 4a;=102, 
 from which we find the values of x to be 12 or - 16, 
 
 Therefore he bought 1 2 oranges. 
 
 or 
 
 or 
 
 4. a;- 
 
IN VOL VINO Q i 'ADR A TICS. 
 
 i9i 
 
 To Bolvo the cqiuvtions 
 
 . - a;- + a-?/ = 1 5, 
 
 Suppose y = mx. 
 
 Then ic'-^ + tnx-= 15, from the tii'-st 6(1 nation, 
 and mx'*-m-'.c2 = 2, from the second equation. 
 
 Dividing one of these eqiuitioun by the otlier, ^ 
 
 jT + iiix^ _ 15 
 
 a;-(l +in) _15 
 
 x^ {m-m')~ 2' ' 
 
 l + ?/i, _15 
 
 m — tu'^ '2 ' 
 
 From tliis equation we can determine the values of w. 
 
 2 
 One of these values i.^ ^, and putting,' this for to in the 
 
 2 
 
 equation x'^ + vix^ = 1 5, we get x^ + -x^ = l 5. 
 
 o 
 
 From which we find £c= ±3, 
 and then we can find y from uae of the original equations. 
 
 259. The examples which we shall now give are intended 
 as an exercise on the metliods of solution explained in the 
 four preceding articles. 
 
 i 
 
 Ex A MPLES.— xcvii. 
 
 or 
 
 or 
 
 14 
 
 I. ;r;3-7/ = 37 
 x^ + xy + y' = ^7. 
 
 xy= IG. 
 
 7. x2+.r// + 7/ = 39 
 3i/--5.ri/ = 25. 
 
 2. x^' + 6xy=Ul 
 ()xy + d(hf=^4',i2. 
 
 5. x'' + //•" = 152 
 
 8. .t'- + ///-GO 
 icy-7/ = 5. 
 
 3. »;- + y7/ = 21(> 
 ,/ + .x7/ = 231. 
 
 6. 4x- + n./7/=19(). 
 4.c-5?/=10. 
 
 9. 3.':2 + 4./J/ =- 20. 
 ' ()xy + 2y''^.]':i. 
 
 I o. a;2 - ./•// + 1/-' - 7 II. x' - xy - 35 1 2. 3.t;2 ^ 4xy + 5//-' =-71. 
 
 3.f2 I 1 3.ri/ + 8/y2 = 1 (52. xy + ?/- = 1 S. 
 
 13. x^'f 7/:»-2728 14. .I'-'-i 9,n/- 340 
 
 x-^ - x-2/ + y'^^lli. Ixy - </^ -- 1 7 1 . 
 
 5/ -1-7?/ -29. 
 
 •» 'I T T- 
 
 x- • //•';^220 
 
 x//=^108. 
 
 
 i: 
 

 ! 
 
 XXI. ON PROBLEMS RESULTING IN 
 QUADRATIC EQUATIONS. 
 
 260. The method of stating' piol>lems resulting in Quad- 
 ratic Equations does not require any general explanation. 
 
 Some of the Examples which we shall give involve one 
 unknown symbol, others involve two. 
 
 Ex. I, "What number ifi that whose square exceeds th-':: 
 number by 42 i 
 
 Let a; represent the number. 
 
 Then x'=x + 4'2, 
 
 a;- - a: = 42 ; 
 
 or, 
 
 1 169 
 
 therefore 
 
 whence 
 And we find tlie values of x to be 7 or — 6. 
 
 .X"' iC + — — J ] 
 
 4 4 
 1 ^13 
 
 Ex. 2. The sum of two numbers is 14 and the sum of 
 their squares is K)0. Find the numbers. 
 
 Let X and y represent the niunbers. 
 
 Then x + ij^U, 
 
 and .x--fy-^ = 100. 
 
 Proceeding as in Art. 252, we find 
 
 x^d> or 6, y = G or 8. 
 Hence the numbers arc 8 and 6. 
 
INDETERMINATE EQUATIONS. 
 
 m 
 
 iivolvti one 
 
 Let —-^ = v), tlK'ii 1 - ?/ -^ 3/». ; 
 
 o 
 
 :. i/-l~3/», 
 
 and a- = 3 - 2?/ + wi — 3 - 2 + 6)>i + w = 1 + 7iii ; 
 
 or the general solution of the ecination in whole numbers is 
 
 x—l + 7m and ?/ = 1 - 3»t, 
 
 where m may ])e o. 1, 2 or any integer, positive or 
 
 negative. 
 
 If 'III = 0, x^ 1,7/ =-- 1 ; 
 
 if j/t^^l, .r-^ 8, 2/= -2; 
 
 if m = 2, x=^ io, y= -b; 
 
 and so on, from which it appears that the only positive inte- 
 gral values of x and y which satisfy the ec|uation are 1 and 1. 
 
 2G2. It is next to be observed that it is desirable to divide 
 both sides of the equation by the smaller of the two coefficients 
 of the unknown symbols. 
 
 Ex. To find integral solutions of the equation 
 
 7.(.- + 5y = 31. 
 Here 5?/ = 31-7x"; 
 
 ' . l-2a; 
 
 ••• y = (i-x+~^-. 
 
 l~2r. 
 
 Let — - - = m, an integer. 
 
 D 
 
 Then 1 -2x = 5m, whence 2a; = l -5m; 
 
 .•. x= — ^ 2m. 
 
 ^ ^\—m . , 
 
 Let =n, an integer. 
 
 Then 1 -m = 2n, whence m = l -2?i. . 
 
 Hence 33 = 7? -2m = n-2 + 4n = 5« ~2 ; 
 
 i/=:6 -.x + m = 6-5?i + 2 + l -2?i = 9-7n. 
 
 Now if n = 0. ic=-- -2, ?/« 9; 
 
 if 71 = 1. x= 3, 2/ - 2; 
 
 if n^%x^ «;V"- -5j 
 and so on. 
 
Io8 
 
 INDETERMIiYATE EQUATIONS. 
 
 .^(53. Ill li(»\v iMiiiiy ways can a person pay a Lill of X13 
 with crowns an<l j^'uincas? 
 
 Let X and ij denote tlie niiinl»(.'r of crowns and guineas. 
 
 Then 
 
 .-. r)x = 2(>0-2l2/; 
 
 a; = 52-4i/-^. 
 ^ 5 
 
 11 ^ 
 
 Let ^ = w, an integer. ' ^ 
 
 Then y = 6m, • 
 
 and x = ii2-4y — m='r2-2lm. 
 
 If m = 0, x = 52, 7/= 0; 
 
 m=l, a; = 31, 7/= o; 
 w = 2, a;=10, i/=10; 
 and liiglier values of m will give negative values of x. 
 
 Thus the number of ways is three. 
 
 264. To find a nunihor whicli wlicn divided b}' 7 and 5 
 will give remainders 2 and 3 respectively. 
 
 Let X l)e the number. 
 
 T - 2 . " 
 
 Then '— ^— =an integer, suppose m; 
 
 and 
 
 a; -3 
 
 = an integer, suppose n. 
 
 Then x = 7m + 2 and a; = 5n + 3 ; 
 
 .-. 7w4-2--5». + 3; 
 
 .'. 5u = 7771 - 1, whence ?t = m 
 Let — ;; — =Pf an integer. 
 
 2m -I 
 
 5 
 
 Then 2m = 6p^ Ij whence m—2p -h— • 
 
 jj-'-l 
 
 2 • 
 
 Let ^-- = q, an integer. 
 
 Then 
 
 1^ = 27-1, 
 
 m = 2/> + g = 47-2 + 7 = 5g-2, 
 ?: = 7w + 2 = 35^-12, 
 
 ■ » II iii ".n» i lWlU« \ u 
 
lisr QUADRATIC EQUATIOXS. 
 
 m 
 
 24. T ))iiy a iiumln r of liiindkorcliicfs lor £\\. Iltul I 
 Ijou.i^'lit 3 nioit! lor the sjime inuiii'V, tlicy would have cost one 
 tjliilliiig each less. ITow :iiaiiy did I buy '{ • 
 
 25. A dealer liouglit a number of calves for .£80. Had he 
 bought 4 more for the same money, each calf would liave cost 
 £\ less. How immy did he buy ? 
 
 26. A man bought some itiecea of doth for £33. 15s., 
 wliich he sold again for £2. Sk. the ])iece, and gained as much 
 as one piece cost him. What did he give for each piece ? 
 
 27. A merchant bought sonie pieces of silk for £180. 
 Had he bouglit 3 pieces more, he would have paid £3 less for 
 each piece. How many di«l he buy ? 
 
 28. For a journey of lOM miles (5 hours less would have 
 sulliced had one gone 3 miles un hour faster. How muny 
 miles an hour did one go ? 
 
 29. A grazier bought as many sheep as cost him £60. 
 Out of these he kept 15, and selling thi; remainder for £54, 
 gained 2 shillings a head by them. How many sheep di«l 
 he buy ? 
 
 30. A cistern can be filled by two pipes running together 
 in 2 hours, 55 minutes. The larger pipe by itself will lill it 
 sooner than the smaller by 2 hours. What time will each 
 pipe take separately to till it \ 
 
 31. The length of a rectangular field exceeds its breadth 
 by one yard, and the an^a contains ten thousand and one 
 hundred sc£uare yards. Find the length of the sides. 
 
 32. A cer>ain number consists of two digits. The left- 
 hand digit is double of the right-hand digit, and if the digits 
 be inverted the luoduct of the nunilxr thus f<;iined and the 
 
 A. 
 
 original number is 2208. Find the number. 
 
 33. A ladder, whose foot n.-sts in a given position, just 
 reaches a window on one side of a street, and when turned 
 about its foot, just reaches a w indow on the other side. If the 
 two positions of the ladder be at right angles to each other, 
 and the heights of the windows be 36 and 27 feet respectively, 
 find the width of the street and the length of the ladder. 
 
 f I 
 
 ?, 
 
■( I 
 
 196 
 
 Ox PROBLEMS RESULTIXG, ^^c. 
 
 1 
 
 16 
 
 34. Clutli, beiiij^ wetted, slirinks up in its Icn^^lli aiitl 
 
 o 
 
 . ill its M'idth. It" llu'! siirlaco of .a piece of clotli is di- 
 
 .3 
 
 iiiinished by 5*; sqiuire jards, and the length of tlie 4 sides 
 
 by 4^ yards, what was Che length and width of the cloth ? 
 
 35. A certain number, less than 50, conf^ists of two dibits 
 Avliose difference is 4. If the digits be inverted, the difference 
 between the .s(|uares of the number thus formed and of the 
 original number is 3900. Find the number. 
 
 36. A plantation in rows consists of lOOOO trees. If there 
 liad been 20 less rows, there would have been 25 more trees iu 
 a row. How many rows are there ? 
 
 37. A colonel wished to form a solid S(piare of his men. 
 The first time he had 39 men over: the second time he in- 
 creased the side of the stpiare by one man, and then he found 
 that he wanted 50 men to complete it. How many men were 
 there in the regiment ? 
 
 XXII. INDETERMINATE EQUATIONS. 
 
 261. When the number of unknown symbols exceeds that 
 of the independent equations, the number of simultaneous 
 values of the symbols will be indefinite. AVe jnopose to ex- 
 ]»lain in this Chapter how a certain number of these values 
 may be found in the case of Simultaneous Equations involving 
 two unknown quantities. 
 
 Ex. To find the integral values "f x and y which will 
 batisfy the e(j[uation 
 
 3.'; + 7//= 10. 
 Here 3..;- 10- 77/ ; 
 
 I -If 
 
 N 
 
 .-. x--'^-2u + 
 
 o 
 
 o\v iix ami y are integers, -^ must also be an inU-ger. 
 
If 
 
 XXIII. THE THEORY OF INDICES. 
 
 265. The iiuiuber placed over a 83'iubol to express the 
 power of the symbol is cabled the Index. 
 
 Up to this point our indices have in all cases been Positive 
 Whoh^ Numbers. 
 
 We have now to treat of Fractional and Negative indices ; 
 and to put this part of the subject in a clearer light, we shall 
 commence from the elementary principles laid down in Arts. 
 45, 40. 
 
 2(56. First, we must carefully ob.servo the following results : 
 
 72 = «6 
 
 crxa''=a' 
 
 (a3) 
 
 3\2^^6, 
 
 For 
 and 
 
 a^xa^=a.a.a.a. « =a'', 
 
 (a^y=a^. a^=a.a. a. a. a.a = a^. 
 
 I 
 
 These are examples of the Two llules which govern all 
 combinations of Indices, The general 2>roof of these llules we 
 shall now proceed to give. 
 
 2G7. Def. When m is a positive integer, 
 
 a'" means a. a. a with a written m times as a factor. 
 
 268. There are two rules for the combination of indices. 
 
 Rule I. a" X «"=»"•+". 
 Rule II, ((*')"=»'"". 
 
 269. To prove Eule I. 
 
 a"* — a . (f . a to ?h -factors, 
 
 a"-*a.a.« to w factors. 
 
 t ,. 
 
 
 ■*, 
 
 I 
 
202 
 
 rnr. TirEORy or- indices. 
 
 Thcicfoiv 
 
 o" X a" = (*< .a .a to m lactorH) x (a . a . tt . . . 
 
 ~a.a .a to \rii + 1\) iactors 
 
 = (*"•+", by the Definition. 
 
 , . to 71 factors) 
 
 / 
 
 To prove Rule 1 1, 
 (a")- = rt* . a* . ft" tc n iactors, 
 
 = (ft . ft . ft to w factors) (ft . ^ . a . . . to m factors) . . . 
 
 repeated n times, 
 
 = ft . ft . ft to mn factors, v 
 
 = «""', Ijy the Detinition. 
 
 I 
 
 270. AVe liave (h^duced immediately from the Dcifinition 
 tliat when 7/1 and n are positive intei^ers ft"* x ft" = ft"'+''. When 
 m and n are not ])osilive intej^'ers, the Definition has no mean- 
 ing. We therefore exteud tlie Definition by saying that a'" and 
 a", whatever m and n may be, shall be such that a" x ft'* = a'*+'', 
 and Ave shall now proceed to shew what meanings wc assign to 
 a"*5 in consequence of this definition, in the following cases. 
 
 271. Case I. To find the meaning of ft', p and q being 
 positive integers. 
 
 p p p ,p_ 
 
 aixaf--=a'' % 
 p p p p.p p 
 «' x ft' X ft' = ft' ' X «' = «'' »; 
 
 ':+:+!- 
 
 and by continuing this process, 
 
 '-(--+'-+... toqlena 
 
 a' X a' X to q factors = ft' ' ' 
 
 = «". 
 
 But by the nature of the symbol ^ 
 
 m 
 
 ^c^ K ^a" X to q factor8 = a^; 
 
 ^ f r , — 
 
 ;. a' X ft' X to ly factors = ^a" x Si/a'' x ...to q factors ; 
 
 p \ 
 
 ;. ft' = A/a", 
 
 • 
 
factors) 
 
 INDE TERM FN A TE EQ C'A TIONS. 
 
 lllfUCi; if 
 
 •<==<', -c-= - 
 
 1-; 
 
 if 
 
 i/=l,a;- 
 
 5;i; 
 
 if 
 
 (^=2, a;=3 
 
 i;o ; 
 
 and so en. 
 
 
 
 199 
 
 1 
 
 'tors) ... 
 
 finition 
 
 When 
 
 ) nicun- 
 
 a'" uiid 
 
 ssign to 
 ases. 
 
 q_ being 
 
 )rg; 
 
 • Examples.— xcix, 
 
 ■ J 
 
 Find positive integral .solutions of 
 
 I. 5x + 7// = 20. 2. 7:c+H)// = 92. 
 
 3. 13x+19(/=1170. 4. \U+'mj^'H). 
 
 5. 14./;-ri//^7. 6. llx- Hi5>/-U);)l. 
 
 7. llx + 7// = 3()8. 8. 46' - 11)// -= 23. 
 
 9. 20./;-9// = G83. ic. 3a; + 7// = 383. 
 
 II. 27*- + 4// = 54. 12. 7x- + 1)^ = 053. 
 
 13. Find two fractions with denoniinalors 7 and 9 and 
 
 tlieir sum ,. -. 
 ()3 
 
 14. Find two proper fractions with denoniinatfus 11 and 
 
 82 
 13 and their difl'erence -^-^^. 
 
 i4.i 
 
 15. In how many ways can a deht of £1. 9.?. he paid in 
 florins and half-crowns / 
 
 16. In how many ways can £20 ho paid in half-guineas 
 and half-crowns ? 
 
 17. What number divided l»y 5 gives a remainder 2 an I 
 hy 9 a remainder 3 ? 
 
 18. In how many different ways may £11. \bs, he paid iu 
 guineas and crowns \ 
 
 19. In how many different ways may £4. lis. (Sd. he paid 
 with half-guineas antl half-crowjis / 
 
 20. Shew that 323x -5272/ =1000 <^aiinot he satisfu'd hy 
 integral values of x and y. 
 
 J 
 
 
 'if, 
 ■fii 
 
200 
 
 INDETERMINATE EQUATIONS, 
 
 21. A fanner buys oxen, sheej), and lions. The whole 
 nnniber bought was 100, and the whole price ^100. If the 
 oxen cost .£5, the sheep £\^ and the hens U. each, how many 
 of each hud he? Of how many .solutions does this Problem 
 admit ? 
 
 22. A owes B 4s. \^d.\ if A has only sixpences in his 
 pocket and /' only four[)enny pieces, how can they best settle 
 the matter ? 
 
 23. A person has ^£12. 4s. in half-crowns, florins, and shil- 
 lings ; the number of half-crowns and florins together is four 
 times tlie number of shillings, and the number of coins is the 
 greatest possible. Find the number of coins of each kind. 
 
 24. In how many ways can the sum of £h be paid in 
 exactly 50 coins, consisting of half-crowns, florins, and four- 
 penny pieces ? 
 
 25. A owes B a shilling. A has only sovereigns, and B has 
 only dollars worth 4s. Zd. each. How can A most easily pay Bl 
 
 26. Divide 25 into two parts such that one of them is 
 divisible by 2 and the other by 3. 
 
 27. In how many ways can I pay a debt of £,% 9s. with 
 crowns and florins ? 
 
 28. Divide 100 into two parts such that one is a multiide 
 of 7 and the other of II. 
 
 29. The sum of two numbers is 100. The first divided by 
 5 gives 2 as a remainder, and if we divide the second by 7 the 
 remainder is 4. Find tha nmiibersi 
 
 30. Find a number less than 400 which is a multiple of 7, 
 and which when divided by 2, o, 4, 5. 6, gives as a remainder 
 in each case 1. 
 
THE THEORY OF INDICES, 
 
 ao5 
 
 N ^TE. Wlicn Examples are given of actual iuiniber8 raisc«l 
 to fractional powers, they may often be put in a form more lit 
 for easy bolution, thus : 
 
 (h 144--'=(l44'^)'' = (v^l44y'-12^=172S. 
 (•J) 12r)S.= (l25^)^ = (^125)'' = 52=S>r.. 
 
 279. Since (./;"')"--- .<.•"'", 
 
 (1) {(x")" ('' = (:»;""') '• = :(;"""'. 
 
 (2) }(a-")~"{'' = 0''"'')'' = a'""', 
 
 (3) } (a:-"')" J" = (:»;-"'") '' = x-'*"". 
 
 280. Since rr' = 
 
 1 
 
 ;<;"' 
 
 we may re})lace an expression raised to a negative power by 
 the reciprocal (Art. 199) of the expression raised to the some 
 positive 2^0 vver : thus 
 
 (1) «-i = \ 
 
 (2) «- 
 
 a- 
 
 
 Examples. — c. 
 
 (1) Express with fractional indices : 
 
 (2) Express with negative indices so as to remove all jjowers 
 from the denominators : 
 
 1 rt ?)2 3 
 
 1. -+— + --3-+ ... 
 
 X- 3.C 4 
 
 2. -^+ . + -T- 
 
 yi yi yi 
 
 U? . 5;c''* X / 
 
 xy I z 
 
 4* 3i2"^5:^^«/-'^;ry' 
 
 (3) Express with negative indices so as to remove all powers 
 from the numerators ; 
 
 J: 
 'I 
 
I. -f „ + ... -\- ., 
 
 a if 0- 
 x^ 3x 5 
 
 tj« t* '■' 
 
 (4) Express with root-symbols and positive indices : 
 
 - x-^ 
 
 2 1" ^ J. 'i^ 4- 
 
 5. IC ^ + 2/ * + » ^• 
 
 2/ •* !/ ■* '^y 
 
 1 
 
 n 
 
 a: 
 
 -2 35 3 a; 3 
 
 ,.« 
 
 
 i[*: 
 
 :'i. 
 
 5;:' 
 
 !|i 
 
 281. Since x- -f x" = - ., = x" . x"" - x"-", 
 
 (1) a;8^a;3 = x8-^ = rl 
 
 1 
 
 (2) x^-T-a^ = x'^~^ = x~^ = -^. 
 
 (3) a*" ^ ic*""" = a;"-'"'-"' = rc'"-'"^" = x\ 
 
 (4) a* 
 (5) 
 
 ^ «*-H' =-. a»-<*+''' = a*-*-' = a-' = 
 
 a' 
 
 •t^ 
 
 
 a;* 
 
 (6) 
 
 
 282. Ex. Multiply a"' - a'" + a' - 1 ?>?/ a' + 1 . 
 
 -rt"-M-a'-l 
 
 a" 
 
 a'+l 
 
 a 
 
 ar 
 
 *" _ /iS' A- «2>- _ 
 
 a"' M a' 
 
 tt' 
 
 + a 
 
 n. ^,2r 
 
 + a'-l 
 
 EXAMPLES.— CI. 
 
 Multiply 
 
 1. .r> + r*:''/ + ?/■'' l>y .'■''' -9"''//'' 1 //'''• 
 
 2. rt="" M 3fe-"'//" -I Da'"//-" + 27?/"' l»y a" - 3i/". 
 
 3. x*^ - 2«a;*' + 4tt» by u^ ' + 2ax-" + 4a', 
 
THE THEOR V OF INDICES. 
 
 203 
 
 272. Case II. To find the meaning of a~', s heing a posi- 
 tive number, ichole or fractional. 
 
 We must first find the lueanini' of a". 
 
 AVe have 
 
 (C X «" = »"•■•■*' 
 
 : a" ; 
 
 ,-. iiy— J. 
 
 ■ ? 
 
 Now 
 
 a? X a ' -- ^' ' 
 
 1 
 
 a' 
 
 •t 1 
 
 / 
 
 273. Thus the interim'fation of a"' lias been (hMluced from 
 Rule I. It reinaiiis to he proved th.it this interpretation 
 aj^ree.^ with Rule II. This we shall do hy shewiiifj that Rule 
 II. follows from Rule I., whatever m and n may be, 
 
 274. To shew that {a"")" = a"" for all values of m and n. 
 
 (1) Let n he a positive integer : then, whatever m may he, 
 
 (a"*)" = «"*.«,"*. rt"* to n factois 
 
 i ™f«-|-m-f-»n-|-« •■•oh terms 
 
 (2) Let n he a jKjsitive fraction, and equal to - , /) and q 
 being [)ositive integers ; then, whatever ho the value of m, 
 
 / v'' / v** .. X /' f''^•. .. toy term* 
 
 ((t'")' X (a"')'/ X to 7lactor.s = ((f"')' ' 
 
 -'f'"'', hy (1). 
 
 "^'' ""-' , "'"4-"'" + ... to, torn.. 
 
 Uuttt' xa* X to 7 1 actors ^(t ' ' 
 
 : a'"'' ; 
 
 tliiJt is, 
 
 
 
 
 i !l 
 
i: 
 
 ii, 
 
 1 1'' 
 
 i 
 
 2d4 
 
 THE THEOk Y OE IXDICES. 
 
 (3) Let H— — s, s being a positive number, whole or frac- 
 tional : then, whatever m luav be, 
 
 (a'")--, ^ , bv Art. 272, 
 
 = ~T7y» % (1) '^"'^ (^) ^^ ^^^'^ Article ; 
 
 («"•)»= 
 
 fhat is, 
 
 275. We shall now give some examples of the mode in 
 which the Theorems established in the preceding articles are 
 applied to particular ca.ses. AVe shall commence with exam- 
 ples of the combination of the indices of two single terms. 
 
 276. Since .0'" x-,/'" = ,/;'"+", 
 
 (1) ^■=xx"-'=-x'"+"-^ = ^''. 
 - (2) ^ xx = 'jf^^. • , 
 
 (4) a"'-".&"-^xa"-"'.6'-".c 
 
 :=1.1.C 
 
 2V7. Since (;«"♦)'• = .«""•, 
 
 (1) (,r"f=::»;«>:3=:a;H 
 
 (2) (..')^^.^^^^=^^ 
 
 
 (3) {a'^)^^a"'^=a^. 
 
 278. Since .'•■'- ;/;c-', 
 
 It 
 (1) J= sj^, 
 
 (2) r«^=4/x^ 
 
THE THEOR V OF INDICES. 
 
 2c 
 
 285. Ex. himdc a - h hij ^At - 4/0. 
 Puttiiij^ a* lor ^rt, and 6* for 4^,'>, we iiroceid thus: 
 J-b'^)a-b{J+Jbi + ah^ + J^ 
 
 3 1 
 
 .1 1 
 aV>^~b 
 
 1 1 
 •t;.-:> 
 
 1 :i 
 
 b'^-a^b^ 
 
 a 
 
 h^~ 
 
 b^-b 
 
 a 
 
 ib 
 
 ^-. 
 
 Examples.— civ. 
 
 Divide 
 
 J 1 
 
 1. x-y hy X--IJ-. 
 
 2. a — b 1)V a^ + b-. 
 
 7. x-Hly by a;* - 3// *■. 
 
 8. 8]a - lC>b hy 'Sai -'2b K 
 
 3. x-yhyx'^-y 
 
 4. a 
 
 + b] 
 
 )\ a' 
 
 KbK - 
 
 9. a 
 
 10. Ill 
 
 X I) 
 
 y ;C-+a: 
 
 -:i4:j 1>y ^vt'^-a. 
 
 5. a;-f-2/by ;(;•• + y- 
 
 II. x+17x- + 7i) l)V,';-+7. 
 
 6. ?/* 
 
 — ?i by Vi* 
 
 1 2. x^ + x^ - 12 by u;^ - 3, 
 
 i-v ?r«-3f;- +3/>-/rM.v //» - 1. 
 
 J 
 
 I I I 
 
 14. x + !j-\-z- 'Sx'^y'^y' h 
 
 y .»:•• 4- // ■ 4-;i' 
 
 1 5. X ~ S.r"' - 4G,r* - \K ) by .';='" + 4. 
 
 — in\ : 
 
 ,1 
 
 //1-* )«* f ;< -. 
 
 1 .1. J 
 
 16. m + ^/t^M^ +• H bv //t- 
 
 311 
 
 17. j> - 4/>* + ()^>- -- 4^/* + 1 by ^>- - 2y;i f I. 
 
 
 Jrl i.V 
 
 1 8. %■<• + .c-?/'2 - 3*/ - 4//-;:- - x-~- - ,^ 1 >y 2x- + 3»/- + ;:; 
 
 •24.;^i24...i 
 
 I 
 
 
 
 i^^-,/ 1)V ,/••■' -,>•■•</•■> +,c^'>/'' —:/;•'•)/•'' + // '. 
 
 * 
 
 19. 
 
 
 fs.A.] 
 
 
 
 
 • 
 
 

 
 2fO 
 
 T//£ TIIEOR V OF INDICES. 
 
 Ne[)alivc Indues. 
 28G. Ex. MalU]>hj .,r ^ 4- a;--'//-' + x'hj-'^ + y^ hy x"' - y'K 
 
 X ^ — y~^ 
 
 - x~hj-^ - x~'hj-'^ - x~^y~'^ - y~* 
 
 X 
 
 -4 _ y-i 
 
 Examples.— cv. 
 
 Multii)ly 
 
 8 
 
 lo 
 
 d' 
 
 » + //-! l)y a-^ -h K 
 
 ^ 1)V X 
 -h 
 
 a 
 
 .r' + X 4 
 
 .-1 
 
 + X- 
 
 X' 
 
 2. x-^ + h-'n)X x-'-^-h-^. 
 
 4. ;,*-'-H-rc-'M>V3;=^+l+.'»:-2. 
 
 -t-^r'-^l 
 
 )V a' 
 
 -'2 
 
 -_/,-i + c'-iby a-i + 6-i + 
 
 r.-l 
 
 1 + ,,/,-! +• (('-'^-^ by 1 - «?ri + a'^b-'^. 
 a^l-2 + 24. ,r-/>'^ by a^/>-2 - 2 - a-'-i'-J. 
 4.'C-3 + :U-- + 2x-^ + 1 bv x-^ - rc-i + 1 
 
 >-2 + 3a;-i-;^ by 2.0-2- 
 
 X 
 
 -I 
 
 1 
 
 287. Ex. Dimh x^ + 1 + oj-^ % a; - 1 + X 
 -l+x'^)x'+l+x-'^{x + l + 
 
 -1 
 
 X 
 
 x 
 
 -I 
 
 x^-x+l 
 
 x + x 
 
 2 
 
 
 x-l 
 
 1 .r- 
 
 1 
 
 1 
 
 — ^t;' 
 
 -i+x- 
 
 1 
 
 -x' 
 
 -i+x- 
 
 Note. Tbe onlcr of ibe powern of a is 
 
 ((•'', a^j aS (^^ <^"^ «"^ «"■'. 
 
 a Belied ^vllich iiuiy ]»e written tlius 
 
!! I 
 
 THE THEORY OF INDICES. 
 
 4. (r I Z/' + t;'" l)y (f'" //' |(f. 
 
 5. a- i-//*-2f' l.v 'la'" hA-c: 
 
 6. .';'"" " - //"• l>y :/;" + j/'""-". 
 
 « 
 
 7. x-" - x"ij" + if" by a;-" + x"i/" + y-\ 
 
 8. a'''+'' - h"' + C by a"^-^ +-6^-^' + c^-". 
 
 9. Form the square of x^'' + x'' + 1. 
 10. Form the square of i(;'^-a;''+ 1. 
 
 283. Ex. Dim(hx'''-\hjx^-\. 
 
 X*" 
 
 -x"> 
 
 ■''' + :r''+a;'' + l 
 
 
 x">- 
 
 -1 
 
 3?'>-XF 
 
 
 x^-l 
 
 207 
 
 I 
 
 
 I 
 
 Examples.— cii. 
 
 Divide • 
 
 1 . «*"• - y"* by a"* - 1/"*. 3. x*' - I/"' by x' - 1/'. 
 
 2. x"' + i/"" by a" + 2/". 4. «»''' + 6'"' by a?" + 6^. 
 
 5. aT - 243 by a;'' - 3. 
 
 6. ft*'" + 4fr'"a;-'' + IGa;*" by ft^"* + Sa'-'x" + 4a;'"' 
 
 7. O.r" + ^x*" + Ha;"" + 2 by 1+ 5.t'' + af*". 
 
 8. 14//"'c"' - IS&^V" - 5Z^'"" + 462"'c'"' l)y t"'"* + tV" - 2//"'c'". 
 
 9. Find tbo square root of 
 
 a«« + Grt"" + 15(1*"' + 2()ft"'" + 1 nrt-'" + Oft" H 1. 
 
 10. Find the square root of 
 
208 
 
 
 THE THEORY OF INDICES. 
 
 284. 
 
 Ex. 
 
 Fractional Indices. 
 Ma It IpJij «^ - aW-^ + h^ by a^ + bK 
 
 ■ J\^ah^ + b^' . ' • 
 a'-'-^-b'^ 
 
 2 1. 12 
 
 ft - a^b'-^ -V aMr' 
 + aH-^ -a'-^b-^ + h 
 
 
 a +h 
 
 ilflii 
 
 '^ Examples.— ciii, 
 
 .Multiply 
 
 2 X 1 
 
 1. x^-2x'^^ + lhy x^ -I. 
 
 2. ir+ifi + y^-i-lhy yi -I. 
 
 ^' a^ - a-^ by a'^ -4 a^.r^ + r/'. 
 
 S. 2 2 1 i- 1 1- 1 J 111 
 
 4. «•■* + /r"' + r^ - n'-'h-^ - «:',•« - />-'5fi3 bv a=^ + i^ + c*. 
 
 5. tjx-i + 2.r^//^ + P^.-i?/^ + 7y^ by 2,x^ - SyK 
 
 , 4 i) 1 2 2 I '5 4 , 11 
 
 6. w •' ^ w •'?( •' + 7H -'7/ •' 4- ?// ••;/ " + n^ by m" - n'\ 
 
 7. 7?i^ - lilhn •■• + 4(f^ 1 )y 7//,^ + 2(?^7/i^ + 4f?i 
 
 Q o J 1 '* *^ 4 4- 
 
 Fomi the square of each of the foUowiug expressions : 
 9. a;'*+a'\ 10. :i**-«^. 
 
 12. a + ?/i. 13. a'^-2.r^+3. 
 
 2 2 
 II. a;'* + 2/''''. ,;^ 
 
 9 1 
 
 14. 2a;' H- 32; '+4. 
 
 ^11 ^111 
 
XXIV. ON SURDS. 
 
 289. All nuiubois which we cannot exactly deterininp, 
 because they are not multiples of a Primary or Subordinate 
 Unit, are called Surds. 
 
 290. We shall confine our attention to those Surds which 
 originate in the Extraction of roots where the results cannot 
 be exhibited as whole or fractional numbers. 
 
 For example, if we perform tlie operatiwU of extracting the 
 square root of 2, we oljtaiu r4142..., and though we may 
 carry on the process to any required extent, we shall never be 
 able to stop at any particular point and to say that we have 
 found the exact number which is ec^uivalent to the Square 
 Root of 2. 
 
 4 4. 
 
 291. We can approximate to the real value of a surd by 
 finding two numbers betiocea which it lies, ditfering from each 
 other by a fraction as small as we plea^'^e. 
 
 Thus, since ^/2 =1-4 142 
 
 J'l lies between , - and .'■-, which differ by _ ^; 
 10 10 "^ 10 
 
 also between ^^^^^ and ^^^^^, which diller by ^^^^; 
 
 , , , ' 1414 , 1415 , . , ,.,,. , 1 
 also between -^^ and ^^^,^, which diller by --^^^ . ., 
 
 And, generally, if we find the square root of 2 to n places 
 of decimals, we shall find two numbers betwe(!n wliidi /v^2 li(!S. 
 
 1 
 
 tlilfering from each other by the fraction 
 
 10" 
 
 11- 
 
 ti 
 
214 
 
 ON .SURDS. 
 
 ) ■ m 
 
 
 292. Next, wc can always fiml a i'ractiou <liffering from the 
 iL-al value of a surd by loss than any assigned quantity. 
 
 For exan^ple, snppose it required to find a fraction differ- 
 ing from ^.' by less than r--. 
 
 Now 2(12)'^, tl.-it is 288, lies between (16)^ and (17)2, 
 
 /UW^ /17\2 
 ;. 2 I'es between ( j and (- , ) ; 
 
 .'. J2 lies between , „ and , ^ ; 
 
 12 12 
 
 1 fi 1 
 
 .-. /y/2 'lill'ers from — - by less than . 
 
 203. Sjirds, though they cannot be expressed by whole or 
 fractional numbers, are nevertheless numbers of which we may 
 form an approximate idea, and we may make three assertions 
 respecting them. 
 
 (1) Surds may be compared so fai as asserting that onci is 
 greater or less than another. Thus ^^3 is clearly greater than 
 
 fJ2, and ^d is greater thaii ^S. 
 
 (2) Surds may be multiples of other surds : thus 2 ^2 is 
 the double of /v^2. 
 
 (3) Surds, Avhen nniltiplied together, may produce as a 
 result a whole or fractional number : thus 
 
 and 
 
 
 3 '-A ' ?. 3 
 
 va 
 CI 
 
 n\. 
 
 204. Tlie symbols t^a, li^fa, ^fa, H^/a, in cases where the 
 second, third, fourth, and 71"" roots respectively of a cannot be 
 exhibited as whole or fractional numbers, will represent surds 
 of the second, third, fourth, and ?<*'' order. 
 
 These symltol.s we may, in aeconliiiicc with the i)riiicii)les 
 
 41 + 1 
 laid down in Chapter XXllI,, replace by a-, a^, a*, a». 
 
TflE THEOR Y OF IN DICES. 
 
 211 
 
 EXAMPLES.— cvi. 
 
 Dividt 
 
 I. x^-x~'^ by * t i 
 
 J. M 
 
 > 6-2 by a -6-1. 
 
 3- '"'" 
 
 + ii~'^ by III + 
 
 H 
 
 -1 
 
 4. c^ -Or"" by (;-(/ 
 
 -1 
 
 5. x-y-- + 2 + x-V ^y ^i/~^ + -^ U 
 
 6. a 
 
 -4 + rt--'6-2 + 6-» by a-'-* - «-'/>-! + 6" 
 
 7. :<;•"'//-•' - ;'> "'V'' - 3-^//"^ + 3./;-i )/ by xij-^ - x-hj. 
 
 8. 
 
 f3x~ 
 
 4.<;^ + - 
 
 7 ..-3 
 
 43.0 
 
 2 :)3.c-i 
 
 8 
 
 + 27 
 
 X' 
 
 % 
 
 + 3. 
 
 9. a^/r^ + a-'^W by «6-i + (r^h. 
 
 10. a 
 
 -3 + ^ -3 + ,; -3 _ 3(fc-i6-i(;-i by a-i + &-' + c 
 
 288. Tosbew tbat(«6)" = a". 6". 
 ((tt)" = ah . ah . ah. . .to n factors 
 
 = (a . a . a ... to 71 fuctors) x{h .h.h ...\o n factors' 
 = «".6". ^ 
 
 We shall iu>vv give a series of Examples to introduce the 
 various forms of coml/iaatioii of iiidict'S exi)laiiie(l in this 
 Chapter. 
 
 ■ ■ . Examples.— cvii. 
 
 1 . Divide x* - 4.ci/ 4 4,/'// -f Uf by x^ + ^x^y^ + 2//. . 
 
 2. Simplify iCO-^Wr'. 3- Sinfpliry(,/;'°\.0--^ 
 
 I 1 1 X 4- a X - '( 
 
 ^. Simplify <^,_;^,-;nj^;^, -T^p" 
 
 1 u 
 
 
 1 
 
 f s 
 
 I 
 
 f 
 
 % 
 
 m 
 
 
' I 
 
 ! i 
 
 n< 
 
 ■j!^! 
 
 I M 
 
 !I2 
 
 T//E TrJEOR Y OF liY DICES. 
 
 1 5. Mult iply /^rc-2 + 4^-1 - ? by :j.c--' - ±vr^ - ^. 
 
 l'""^'* ic""* rfX—'ia 
 
 6. Simplii'y ■'- • .-^^ . 7. Divi<l(! .c'^" - ?/»" by .r" + i/. 
 
 8. Multiply {(h + /^-)3 by «^" - ]/\ 
 
 9. Divide a - i by ,;/a - 4/6. 10. Prove that (a^)"* = («"•)'-'. 
 
 11. If «"*" = («'";", liud m iu terms of yi. 
 
 12. Si iiii)lify 05"+*+'. 3;"+*-". x*"'"* "..';*+"-". . , 
 
 V)^(.c--) • ^^' divide ^a'by^-, 
 
 15. Simplify [I (a-"')"" I "J ^ DM" I"'"]- 
 
 16. Multiply (6'" + /;"- 2c'" l)y 2a"'-3&. 
 
 17. Multiply a"* '7/' '^ by a"-"'Z/''-"c. 
 
 19. Multiply X'^+ic^+l by :<;«-»/« + 1 
 and their product by i'/' - v^^ + 1. 
 
 20. Multiply iV" - ha""'^x + ar'-x- by t<" + ha''-^ x - ar-'^x^. 
 
 21. Divide a;2'"»-i'-7/2'/'p-')|,y a;MT-i) + ^9>-i)^ 
 
 22. Simplify I (^r)*" '"^"'i'. 
 
 23. Multiply ic^" 4 a--'?/'' -i ;<;\//-/' + i/p by a;'' - y\ 
 
 24. Write down the values of C25'^ and 12-"i 
 
 25. Multiply ft'"' ^'"-y/"-"'" by .u"- 2/™. *" 
 26^ Mid t i] )ly x-^ -i- 3X" - I by x* - 2x~K 
 
ON SURDS. 
 
 2V 
 
 Examples.— cix. 
 
 Reduce to eqniviiU'nt c.xpressious vvitli a whole or fractional 
 number as one factor : 
 
 /(50). 
 
 s'i^i)-^)- 
 
 I. ^/{:IA). 
 
 4. ^/(12r)a^(/-'). 
 
 7. V(72Uc2). 
 
 10. a.^^. 
 
 12. ^(^~2..:^u + x,f^. 
 
 14. V(63(;-»//-4:i(;^//^ + 7/r). 
 
 16. ;7(16(U-^//"). 
 
 18. ^(137:>a*^//'"). 
 
 20. ^(a^-3(f'^6 + 3a'-V;^-a//0. 
 
 3. s'i^a^). 
 II. x^(a^ + 2a-./;+rDj-). 
 
 13- 
 
 15- 
 
 17- 
 19. 
 
 ^/(5()a2-10()r^/> + r)()/r). 
 
 ;'/(i()8?/i»ui<»). 
 
 302. An expression containing two i'actors, one a surd, the 
 other a whole or fractional nunib(U', as 3 ^^2, a ^f.n, may he 
 ti-nnsformed into a complete surd. 
 
 Thus 3 ^^2 - (3-0"^' . ^2 = ^/<) . ^f-2 ---^ ^/( 18), 
 
 1 
 
 EXAMPLES.—CX. 
 
 Tveduce to complete Surds : 
 
 1. 4v^3. 2. 3v'T. 
 
 '/3 
 4. -i v"« 5- '-'\/ -• 
 
 7. 4a ^/(3.v;). 
 
 . l/m-v\ 
 Q. (ill + 11).^ [ I. 
 
 IT P"-'"^ /^ '''^''^ ^- 
 
 3. •'> N^». 
 
 6. 3 ^^«. 
 
 8. 2,r-' 
 
 
 •L 
 
2l8 
 
 ON SURDS. 
 
 303. Siinln iiiiiy Ix- vnmjKinil l>y transforrniiij^ tlioiii into 
 Hunls of the saiuf oitlt'i'. Thus if it he liMjiiiivcl to <U!tt'niiiiU! 
 whether y,^2 be i^rejiter or less tluui 4^3, we proceed thus : 
 
 And since ^/I) is greater tliun ^s, 
 >>/3 is greater tliau ^'2. 
 
 i 
 
 4: I 
 
 i 
 
 Examples.— cxi. 
 
 Arrange in order of niamiitude the followin'' Surds : 
 
 1. V3jind ^4. 
 
 2. V'U) tiiid ^15. 
 
 3. 2 ^3 and 3 ^^2. 
 
 5. 3 VT and 4 ^A'3. 
 
 6. 2 VH7 and 3 s!'^\\. 
 
 7. 2 ^22, 3 ^7 and 4 V^. 
 
 8. 3^19, r)v'^ 18 and 3^82. 
 
 9. 2 ^14, 5 ^2 and 3 ;V3. 
 
 11 1 
 
 10. „ mJ'2, sJ'^ and , ^4. 
 
 304. The following are examples iu the application of the 
 rules of Addition, Subtraction, Multiplication, and Division to 
 Surds of the same order. 
 
 1. Find the sum of v' 18, -^128, and v^32. 
 
 v^(18) + v/(128) f- V(32)= s'C^ =< 2) + ^/((;4 x 2) -I- ^(16 x 2) 
 
 = 3n/2 i-«s/2 + 4V2 
 = 15V2. 
 
 2. From 3^/(75) take 4^/(12). 
 
 3 ^(75) - 4 V( 1 2) -- 3 ^/(25 X 3) - 4 ^(4 x 3) 
 -3.5.^:^-1.2. V3 
 -I5v'3-8;v^3 
 
 ' -7v^a 
 
ON surws. 
 
 215 
 
 205. Surda of the same order an) thosu for wliieli the mot- 
 symbol or Hunl-iiidex is tliu suiu(!. 
 
 Tlius V"> ^ VW> 4 V(m?i), r- arc surds of the saiiu! order. 
 
 L?7,-f7 Hur<lH art! those iii which tho stimc root-symbol or siud- 
 iiuk'X appears over (lu; safiic tiuantilij. 
 
 Tims 2 ;v^a, 3 ^A', he- are like surds. 
 
 290. A whole (»r fractional number may be expressed in 
 the form of a surd, by raising the number ti» the power drntitcd 
 by the order of the surd, and placinj< the result under the 
 symbol of evolution that corresponds to the surd-index. 
 
 Thus • a= a/<*", 
 
 297. Surds of different orders may be transformed into surds 
 of the same order by reducing the surd-indices to fractions 
 with the same denominator. 
 
 Thus we may transform ^x and ^y into surds of the same 
 order, for 
 
 and *^fy=:y^ = i/^ = -^lfi/^ 
 
 and thus both surds are transformed into surds of the twelfth % 
 
 order. 
 
 Examples.— cviii. 
 
 Transform into Surds of the same order : 
 I. ^x and 4/7/. 2. 4/4 and 4/2. 3. V(l«) nnd 4/(50). 
 4. V2 and ^2. 5- l^/a and V^. 6. ^(a + h) ami ^{a-h). 
 
 298. If a whole or fractional number be multiplied into a 
 surd, the product will be represented by i)lacing the multiplier 
 and the multiplicand side by side with no sign, or Avith a dot 
 (.) between them. 
 
 Thus the product of 3 and ,y^2 is represenlctl l»y 3 ^f-2, 
 
 of 4 and 5 ^/:> by 20^/2, 
 
 . . ..ofrrund ^fc '}ya^Jc. 
 
 .'; t 
 
: 1 
 
 1 
 
 
 1 !: 
 
 
 p 
 
 1 
 
 
 
 
 i 
 
 i ■ 
 
 
 i'> 
 
 i .J 
 
 lli ,■ 
 
 2l6 
 
 ON SC/A'DS. 
 
 299. Like sunla may l)e cmnl)iiie(l 1>y the ordinary pro- 
 cesses ol" {itUlilioii and subtraetioii, that is, by adding the 
 coefficients of the surd and phicing the result as a coefficient 
 of the surd. 
 
 13 m* 
 
 Thuo 
 
 
 3(10. We now proceed to prove a Theorem of great im- 
 portance, whicli may be thus stated. 
 
 The root of any expression is the same as the product of the 
 roots of the separate factors of the expression, that is 
 
 sl{xyz) = i^x . ^ly . :^::, 
 
 We have iu fact to shew from the Theory of Indices that 
 
 111 
 
 {ahy=a".h'\ • 
 
 I ' 
 
 Now \{aby'\" = (ahy = abf 
 
 11 11 " n 
 
 and \a'\h"\" = {a")".{b"y = .<".b" = a.b', 
 
 111 
 
 Ill 
 
 .-. (ab)"=--a".b\ 
 
 301. We can sometimes reduce an expression in the form 
 of a surd to an equivalent expression witli a whole or frac- 
 tional number as one factor. 
 
 rp 
 
 Thus 
 
 
ON SURDS. 
 
 210 
 
 3. Mnltii>lyv^^l)yV(12). 
 
 ■ = s^(16 X 6) 
 
 = 4^/6. 
 
 4. Divide V32 by V18. 
 
 x/(32) _ v/(16 X _2) ^ 4 ^/2 ^ 4 
 V(18)"V(9x2) 3,V2 3* 
 
 I 
 
 Ex AMPLES.— CXi i. 
 
 Simplify 
 
 1. V(27)+ 2^(48) + 3^(108). 11. ^6x^8. 
 
 2. 3 V(1000)+ 4^/(50) + 12^/(288). 12. V(14) x ^(20). ' 
 
 3. a sl{a^x) + 6 VC^''^-'') + <" V(c^-^)- ^ 3- V('">'>) x V(200). 
 
 4. 4/(128)+ 4/((;8G)+ 4/(10). 
 
 5. 7 4/(54) +3^/(10)+ 4/(432)." 
 
 6. V(96) - x/(54). 
 
 7. V(243) - V(48). 
 
 8. 12 V(72)- 3^(128). 
 
 9. 5 4/(16) -2 4/(54). 
 10. 7 4/(81) -3 4/(1020). 
 
 14. 4/(3«'-^/>) x 4/(9a/y'2). 
 
 15. 4'(12r^i)x 4/(8r<'-7>:^). 
 
 16. V(12)-^V3. 
 
 17. x/(18)-^/(5()). 
 
 18. '^I{a'h)~ 4/(u/>2). 
 
 19. 4^0''^)-^ v^(«'>'). 
 
 20. V(.^2 + .,;{_,,) ^ ^/^,. ^ 2.r'-2// + ro^//"). 
 
 305. We now proceed to treat of XXa MuUijdicatioji u\ 
 Compound Surds, an operation which will Ix- IViMjuently re- 
 quired in a later part of tlio subject. 
 
 'V\\v StmK'ut must bear in mind tlie two following Ruled: 
 
 Rule I. ^''ax v^//--- v'(«^^)» 
 Rule II. ^ax ^a = ay 
 which will be true for all values of a and h. 
 
Jl'' 
 > nil 
 
 B 
 
 I i 
 
 ! 
 
 h ' 
 
 220 
 
 ON SURDS. 
 
 Multi])ly 
 I. V-rl)y^/7/. 
 
 Examples.— cxiii. 
 
 lo. V(r-l)by 
 
 J{x-y)hy ^fij. 
 
 /(x-n 
 
 - J(x- 
 
 Jx l)y -4Jx. 
 
 4- >/{:c-~y)hy>J{x + y). 
 5. () V^ by 3 s^x. 
 
 12. -2 y/a by -S^^/rt. 
 
 1 3. VC-^ - 7) by - Jx. 
 
 o. / 
 
 V(x + l)by 8V(x-+l). 14. -2^/(./J^-7)by -3^/.. 
 
 7. 10V^by9V(a;-l). 15. -4V(tt'-^-l)by-2V(*t"^-l). 
 
 8. V(3:(;)byV(4^;). 16. 2 ^/(«'^ - 2« + 3) by - 3 ^(fi^ _ 2a + 3). 
 
 306. The following Examples will illustrate the way of 
 proceeding in forming the products of Oompouucl Surds. 
 
 Ex. 1 . To multiply Jx + 3 by sf^ + 2. 
 
 ^x + 'S 
 Va;+2 
 
 ic + 3V^ 
 + 2^fx + 6 
 
 x + 5i^fx + (i 
 Ex. 2. To multiply 4^x + 3 y/y by 4 ^x - 3 V*/. 
 
 4/v/x- 3V</ 
 
 lfk + ]2V(«?/) 
 __-12V(;i:// )-9y 
 
 i6x-97/ 
 
 Ex. 3. To fo..ii the sipiare of^/Oc- V) - ^/a;. 
 
 V(^ - V) - x/.^ 
 ^/(x-T)- Va; 
 
 ""«• - 7 -- x/(a;2 - 7a-) 
 
 2x-7-2V(x''^-7a;) 
 
ON SURDS. 
 
 221 
 
 } . i 
 
 1). 
 
 !V('''^-1). 
 2 -2a + 3). 
 
 he way of 
 irds. 
 
 !/• 
 
 Ex AMPLES.— CXiV. 
 
 Multiply ' 
 
 I. v^x + 7 by V^; + 2. 2. V-*^ - 5 by V^+ 3. 
 
 3. VO* + •>) + '5 % >v^('t -t- •>) •5- 
 
 5. li sj'>'^ - 7 by ,y/x' + 4. 
 
 6. 2 ^^(.« - 5) + 4 by 3 ^l{x - 5) -i\. 
 
 7. V(G+.';)+xAyby^/((; + .r)- x^.'- 
 
 8. ^(:Xc + 1 ) + x/(2c - 1) by ^3./; - ^(2-^ - !)• 
 
 9. v^^* + s!{^^ - •'■) ^^y V'*^ ~ V('* - ^O- 
 
 10. V('^ + a") + \^-^ ^'.V V('^ + ■'■)• 
 
 11. ^fx-{- y/y+ ^hhy^/x- ^y+ ^::. 
 
 12. sja+ V(rt-a;)+ V-^by v^«- »J{a-:r)-\- ^x. 
 
 Form the squares ot" tlie following expressions: ^ 
 
 13. 21+ ^/(./j2-9). 17. 2v/./--3. 
 
 14. V(^- + 3) + v^(* + 8). 18. x/6c + 2/) - Vv'' - //)• 
 
 1 5. v^c + v/C*^ - 4). 19- V-*'' • v^C'- + 1) - VC'' - 1 )• 
 
 16. ^{x-iJ)-\- s!^' 20. V(*"+ 1)+ \^-'" • VC'^"- !/• 
 
 307. We may now exten*! the Tiieoreni explained in 
 Art. 101. We there sliewetl how to resolv^e expressions ot 
 the form. 
 
 into factors, restricting our observations to the case of iierjcrt 
 squares. 
 
 The Theorem extends to the dilTereiicc between any two 
 quanlities. 
 
 Thus 
 
 a--h = { s!<i \- y!^>) ( v^(t - sfh). 
 
 x"-y-={r+ ^fy) {.>'- ^hj). 
 
 1 -x-(l+ J.r) (I- Jx). 
 
I u. 
 
 I 
 
 i 
 
 222 
 
 OAT SURDS. 
 
 308. Hence we can always find a multiplier wliicli will 
 i'ree from surds an expression of any of tlie/owr forms 
 
 ♦ 
 
 I. a+ s,fb or 2. ^^ft+ ^b, 
 
 3. a- ^fb 
 
 or 4. 
 
 Ja- 
 
 j'b. 
 
 I'or since the Jirst and third of these expressions give 
 as a product a^~h, which is free from surds, and since the 
 second and fourth ^ive as a product a -by which is free from 
 surds, it follows that the recjuired multiplier may be in all 
 cases found. 
 
 Ex. 1. To find the 'multiplier which will free from surds 
 each of the following expressions : 
 
 I. 5+x/3. 2. V6+V5. 3- 2-V5- 
 
 The multipliers will be 
 I. 5- V3. 2. V6- V5. 
 
 The products will be 
 I. 25-3. 2. 6-5. 
 
 That is, 22, 1, - 1, and 5. 
 
 3. 2 + V5. 
 3. 4-5. 
 
 4- V7-x/2. 
 4. ^f7+ V2. 
 4. 7-2. 
 
 a 
 
 Ex. 2. To reduce the fraction fzr% ^^ ^^^ equivalent 
 
 ■fraction with a denominator free from surds. 
 
 Multiply both terms of the fraction by 6+ ^c, and it be- 
 
 comes 
 
 ab + a fjc 
 b^~c ' 
 
 which is in the rec^ui'^ed form. 
 
 Examples,— cxv. 
 
 Express in factors : • 
 
 1. c-d. 2. c^ — d. 
 
 4. l-(/. 5. l-2xi 
 
 7. 4tt''2 - 3x. 8. 9 - 8h. 
 
 10. v'^ -4r. 11. 2^-3(f. 
 
 3. c-d'i 
 
 6. 5w2 - 1. 
 
 9. 11h'-^-16. 
 
 12. a'^'^-b". 
 
 13- 
 
 16. 
 
ON SVRDS. 
 
 ^23 
 
 Reduce the following fractions to equivalent fractions with 
 (lenoniinatora free from surds. 
 
 16. 
 
 r 
 
 a 
 
 Jh' 
 
 2 
 
 14. 
 
 17- 
 
 2- v/:i- 
 
 I 
 
 4 + 3^2 
 
 18. 
 
 19 
 
 •sja 
 
 20. 
 
 21. 
 
 1 
 
 s'x 
 
 ^/{a + x)- s^{a-xy 
 
 23. 
 24. 
 
 o 2 V2' 
 
 2- V2 
 "2 + ;^72" 
 
 v/(m^+l)4- s/(m--l) 
 a + sj{a" - x''*) 
 
 309. The squares of all nnmhers, negative as well as posi- 
 tive, are 'positive. 
 
 Since there is no assignable number the square of which 
 would be a negative quantity, we conclude that an expression 
 which appears under the form x/( - a^) represents an impossible 
 quantity. 
 
 310. All impossible square roots may be reduced to one 
 common form, thus 
 
 V(-a*^)=v/}a'^x(-l)|=^V.V(-0-f^.V(-l) 
 V(-.0=V|-'- x(-l) {=,/,,; .V(-l). 
 
 AVhere, since a and i^fx are possible numbers, the whole 
 impossibility of the expressions is reduced to the appearance of 
 V( - 1) cas a factor. 
 
 311. Bef. By mJ{~ 1) we understand an expression which 
 when multiplied by itself produces - 1. 
 
 Therefore 
 
 IV(-i)P=-i, 
 U^(-i)P=!V(-i)P.V(-i)-(-i).s'(-i)=-V(-i), 
 
 V A so on. 
 
 a: :| 
 
i 
 
 224 
 
 OJV SU/^DS. 
 
 ! Ifi 
 
 EXAMPLES.— CXvi. 
 
 Multiply, observing tluit ' 
 
 yf -ax sj -h=^ - x/rt&. 
 
 I. 4+ V(-3)l>y4- V(-3). 
 
 2. V3-2V(-2)l.y ^/;i + 2 V(-2). 
 
 3. 4V(-2)-2V2l.y!,V(-2)-;W2. 
 
 4. V(-2) + V(-:i)+ v^(-4ji.y v/(-2)- V(--)- V(--i;. 
 
 6. a + V( - a) by rt - V( - c<). 
 
 7. aV(-«) + K^(-^)i>y"v'(-c^)-^V(-^)- 
 
 8. a + ^V(-l)bya-^x/(-l). 
 
 9. 1- V(i-^'Ob^i+ V(i-<^')- 
 
 I o. cp^^-'^ + c-^^ *-'' by Qf'"' (-'> - e-p'^^ -»>. 
 
 312. AVe shall now give a few Miscellaneous Examijlea to 
 illustrate the principles explained in this Chapter. 
 
 Examples.— cxvii. 
 
 1. bimplify-3^^-^-- 3^^-. 
 
 2. Prove that jl+ ,^(-1)1'+ U " V(- 1)I' = 0. 
 
 3. Simplify -2-^-+ ^^y~- 
 
 ■• 4. Prove that {1+ ^f{-l)\'-\l- V(-l){'= ^(-^0- 
 
 5. Dividea* + a*by a;2+ V2a.c + rt^. 
 
 6. Divide m^ + n^ by m2- V2)'i?i + ?i2. 
 
 7. Simplify x/(.<;=' + 2x^7^ + iy'^)+ J(x"- 2x^7^ + a;7/-). 
 
 8. Simplify -^1-:^^,--^^^^^^^^ 
 i — and 6^4. 
 
 r.s.A.j 
 
ON SURDS. 
 
 225 
 
 9. Fin<l tlie S'piar*^ of f( 
 
 \o. Find the p(|ii;ue of aV- 
 
 /('v?). 
 
 ax- 
 
 il. Simiilify 
 
 
 12. Simplify 
 
 Al x) + 
 
 1_ 
 
 1 + 
 
 £3. ,... K- 1 ( a; - 1 1 -r); ) 
 14. Form the R<|uare of 
 
 Va-o-Vc-> 
 
 15. Form tlie sqiiaro of v^(:);-i f^) - ^^{.I'-a), 
 
 16. :Miiltii)]y ';/(«-"•-"?/•'"" ^ !(;•>) l)y ;/((t"6"'^;'" -''■). 
 
 17. Ruise to the 5"" i)owt'r - 1 - a s!{ ~ 1). 
 
 18. Simplify >/(8l)- ^7(- 512)+ ^/(102). 
 
 19. Simplify 
 
 (if'-' 
 ;-l 
 
 4^5 - 8x^_^ 4x 
 
 )• 
 
 20. Simplify ^/_^\ lyOJ^/V'- G;}^A/;2 + 441 j/V- 1020/) (. 
 
 21. Simi)lifv 20i - 1) '/( - .- -—.,^* -—,,_-- \ • 
 
 22. Simplify 2(/t - 1) V(()3) f ? ^(1 12) - ^'^("'^"'^ 
 
 + ./!lTo(«-I)V|x2-2^(;;). 
 
 23. AVhat is the difference hetween 
 
 Vll7-'V(:i:i)l^ N'!l7+V(33)f 
 I'lud 4/JC5+ s'(129)| X ;;/|(>5- V(129){ ? 
 
 '< \ 
 
 . i 
 
 
 
226 
 
 ON SURDS. 
 
 
 
 313. We have now to treat of tin* iiutliod (»f fiiuling the 
 Square Root of a Binomial Surd, that is, of an expression of 
 one of the following forms : 
 
 m+ x/», m- ^j)\, 
 
 ■where m stands for a wholt^ or fiactionul mimhor, and ,^n for 
 a surd of the second order. 
 
 314. AYe have first to prove two Theorems. 
 Theorem I. 7/ >^f(i = m+ ^fn, m must he zero. 
 
 Squaring both sides, 
 
 a = m^ + '2,m ^n+n; 
 ' .: 2m fjn = ci — m'^ — n; 
 
 Jn = 
 
 a — m^ — n 
 
 2m 
 
 that is, /^n, a surd, is equal to a whole or fractional numher, 
 which is impossible. 
 
 Hence the assumed equality can never hold Utrless m=0, in 
 which case sja= fjn. , 
 
 Theorem II. //&+ Ja — m+ ^hi, then must h = m, av^i 
 
 For, if not, let h=m-\-x. 
 
 Then m + x+ ^a = m-\- ,^n, 
 
 or x+ sja= ijn ; 
 
 which, by Theorem I., is impossible unless cc — 0, in which ca^e 
 h = m and ^fa = i.Jn. » 
 
 315. To find the Square Root of rt + ^/h. 
 
 Assume Aj{a+ \^h)— /^/x+ ^kj. 
 
 Then a + V^-' = •'^ + 2 s'{^]i) + V \ 
 
 .'. x + y = a 
 
 2^f(:xii)-- s^h 
 
 from wliicU we have to find x and y. 
 
 Now 
 and froi 
 
 Also, 
 From 
 
 Si mil a 
 
 316. 
 ■seen fron 
 
 Fine tl 
 
 AssuuK 
 
 Then 
 
 Hence 
 
 
 (1). 
 
 ■ also. 
 
 (2), 
 
 1 Hence 
 
 : 
 
 1 That is, 
 
ON SURDS. 
 
 22^ 
 
 in which case 
 
 Now froi 1 (1) X- + 2.1;?/ 4- J/''* = «'•*, , 
 
 <uid from (2) \ ■ 4x1/ ~b; 
 
 :. x^-2xij + ij- = a'^-b', 
 
 .'. x-ij— vX^"^-6). 
 Also, , x-\-y = a. ' . 
 
 Fioiu these ecjuatioiis we find 
 
 %— -| ana ?/ ^ , 
 
 . . sK't + s'h) = y I ^' - I + y I 2 f . 
 
 Siiiiihirly we may show that 
 
 316. The practical use of this method will he more clciiiy 
 seen from the following example. 
 
 Fine the fcjtiuare Root of 18 + 2 V(77). 
 
 Assume v/|18 4-2 V(77)| = V'«+ V^- 
 
 Then 18 + 2V(77)=a; + 2V(if»/) + i/; 
 
 .-.»; + ?/ = 18 ) 
 
 lien 
 
 ce 
 
 al 
 
 so. 
 
 Hence 
 
 .(;- + 2.t;//-t-9/ = 324) _ 
 4x7/ = 308 r 
 
 . u;''^-2a;// + v/'-=16; 
 .-. a;-?/=±4; 
 a; + i/=18. 
 a; = ll or 7, and ?/ = 7 or 11. 
 
 i^li 
 
 : 
 
 It 
 
 That 18, the square root required is ^(11)+ V7« 
 
h i 
 
 izS 
 
 ON SURDS. 
 
 Jdi 
 
 li 
 
 1 1 
 
 'i' 
 
 ^ilV; 
 
 Examples.— cxviii. 
 
 Find the wqiuire roots of the following Binomial Suuls : 
 1. 10 + 2^(21). 2. 16 + 2V(o5). 3. 9-2^/(14). 
 
 4. 94-42V5. 
 7. 14-4^6. 
 
 5. 13-2v'(3()). 6. 38-12x/(l()). 
 
 8. 103-12^(11). 9- Tr)-12x/(:il). 
 
 10. 87-12v/(42). II. 3,^- x/(l()). 12. 57-12^/(1.-)). 
 
 A 
 
 317. It is often easy to dL'termine the s(jiiare roots ol 
 expressions such as tliose given in the preceding bet oi 
 Examples h\j inspection. 
 
 Take lor instance the expression 18 + 2 V(77). 
 
 What we want is ttj lind two nuinheis whose sum is IS and 
 whose product is 77: these are evidently 11 and 7. 
 
 Then 18-f 2 V(7V)-11 + 7 + 2 ^/(ll x 7) 
 
 -■! v^(ii)-i-x^vr-'. 
 
 That is x/(ll)+ V7 is the square root of 18 + 2 V(77). 
 
 tv 
 
 To effect this resolution by inspection it is necessary that the 
 Coefficient of the surd should he 2, and this we can always ensure. 
 
 For example, if the proposed expression be 4+ ^(15), we 
 proceed thus : 
 
 4+ ^(15) = ^^^—'^^-^^^ ^ + ''^ + ^ x^(^x3) 
 
 / x/5 + ^/3y 
 
 \ J2 J' 
 
 .•. ~ — . ^ — is the square root of 4+ \f(lo). 
 
 Again, to lind the S(|uare Root of 28 - 10 ,^f'3. 
 28-10^3 = 28-2^/(75) 
 
 = 25 + 3-2^(25x3) 
 
 ■ =(5- V3)^; 
 
 .•.5- v'3is the square root required. 
 
lire roots ul 
 
 nil irf IS and 
 
 XXV. ON EQUATIONS INVOLVING SURDS. 
 
 318. Any equation may be cleared of a single surd, by 
 transposing all the other tt^rnis to the contrary side of the 
 ecpiation, and then raisiiii,' eacli side to the power correspond- 
 in;,' to the order of the surd. 
 
 The process will be exjdained by the f(dlowing Examples. 
 
 £x. 1. /^fx — 4. 
 
 Raising both sides to the second power, 
 
 ic=16. 
 
 Ex. 2. ^X=:3. 
 
 Raising both sides to the third po\>er, 
 
 r» = 27. 
 
 Ex. 3. ^f{x^ + 7)~x=l. 
 Tiansposing the second term, 
 
 Raising both sides to the second power, 
 
 x'^ + 7 = l-\-2x + x'j 
 
 2. >Jx = 9, 3. a;2 = 5. 
 
 EXAMPLES.— CXix, 
 
 I. s/x = 7. ' . 
 
 4. ^x = 2.[ 5, x^ = 3. 
 
 7. v/C« + 9) = 6. 8. ./(x-7)-7. 9. ^{x~ro) = H. 
 
 10. (a;-'!)^ = 12. II. ^(4:c-16) = 2. 12, 20-3Va; = 2. 
 
 6. ,^'x = 4. 
 
230 
 
 ON EQUATIONS INVOLVING SURDS, 
 
 14. h-\-c sjx *= a. 
 
 15. V(^^-9) + «^ = '>- 
 
 16. ^{x^-U) = x-\. 
 
 17. ^\'4,/:- + &x-2) = 2x'+l. 
 
 1 8. V(9x''^ - 1 2u; - 5 1) 4- 3 = 3.c. 
 
 1 9. ^'{.c^ - ax + h)-a = x. 
 
 20. ^(25^2 _ 3,,j^jj ^ ,^^ _ 5,1; _ 7^^ 
 
 319. "When tivo Biirtls are involved in an o([nation, one at 
 least may be made to disappear l)y disposing the terms in 
 Rucli a way, tliat one of tlie surds stands by itself on one side 
 of the eqnation, and then raising each side to the power cor- 
 responding to the order of the surd. If a surd he still left, it 
 can be made to stand by itself,- and removed by raising each 
 side to a certain power. 
 
 Ex. 1. ^{x~W)+ ^x = 8. 
 
 Ti'ansposing the second term, we get 
 
 ^{x-l6) = S- sfx. 
 
 Then, squaring both sides (Art. 306), 
 
 a;-16 = 64-16V« + a;; 
 therefore 16Vc = 64+16, 
 
 or 16V^ = 80, 
 or x/ic = 5 ; 
 .-. a' = 25. 
 
 Ex.2. V(a'-5)+ V(.r4-V) = 6. 
 
 Transposing the second term, ~ 
 
 ^{x-b) = Q- ^{x + '7). 
 
 Squaring both sides, a;-5 = 36-12 \f(x + 7) 4- a; + 7 i 
 therefore • 12 yf{x + *7) = 36 + x + 7 -x + 6, 
 or 12V(a; + 7) = 48, 
 
 or >v/(a: + 7) = 4. 
 
 Squaring both sides, cc + 7 = 16; 
 
 therefore aj = 9. 
 
ON EQUATIONS IXVOLVING LURDS. 
 
 231 
 
 Examples.— cxx. 
 
 6. 1+ v'(33-+l)- v'(4x- + 4). 
 
 7. 1- V(l-3-'') = 2V(l-.r), 
 
 8. u - ^]{.ll - a) = ^/x. 
 
 V(16 + .r)4- .v^a; = 8. 
 VC(;+ir))+ /c=15. 
 
 V(,o-l)=:3- nV+4). 10. V(-'^--l)+ x/Ci;-4)-3 = 0. 
 
 ?/i 
 
 9. V't''+ \^('' - "0 = 2"- 
 
 IJ20. When siu'ds appear in the denoiiiinator.s of fractional 
 in equations, tlie equations nuiy bo cleared of fractional terms 
 by the process described in Art. 186, care being taken to 
 follow the Laws of Combination of Surd Factors given in 
 Art. 305. 
 
 ' Vtf 
 
 Examples.— cxxi. 
 
 «36 28 
 
 I . Va: + V(a; - 9) = --/(^z y y 3- V(»^ + 7) + ^x = — ^- 1-7 
 
 2. Va:+V(^c-21) = -^|. 
 
 4. sj((X,- 15)+ sjx.= 
 
 _ 105 
 '^{^- 15)- 
 
 5. x/x+V(^-4) = 
 
 8 
 
 V(«^-4)' 
 9a 
 
 6. V^+V(3« + »=)-^^3„~)=0, 
 
 V(aa;) ■\-h _ h — a 
 x + b b- s!\a'j-)' 
 
 8. (1+Va3)(2- V^) = ^/^. 
 
 A/a;+ 16_ Va ; + 32 
 
 10. 
 
 
 321. The following are examples of Surd Equations result- 
 ing in quadratics. 
 
 Ex. 1. 
 
 riearing the equation of fractions, 2x + 2 = b aJ: 
 
 2jx^-^^--6. 
 
 ■A/, 
 
I '^1111 f 
 
 i4 
 
 232 
 
 ON EQUATIONS INVOLVING SURDS. 
 
 S(iuaring both sides, we get 4x- + 8^ + 4 = 25:c ; 
 
 whence we lind £C = 4 or -v. 
 
 4 
 
 Ex.2. ^/(.-; + 9) = 2V-«-3. 
 
 Squaring l)oth sides, iB + = 4x-12 s^x + 1) ; 
 therefore 12x^.'=3x, 
 
 or 4 /vAt; = «. 
 
 S(|uariiig both sides, * in..;^.^^. . v 
 
 Divide b}^ «, ;md we get 16 = cc. 
 
 Hence tlie values of x which satisfy the equation are 16 
 and4) (Art. 248). 
 
 Ex. 
 
 o 
 
 ,/(2x + 1) +2 six = 
 
 _21 
 
 Clearing the equation of fractions 
 
 therefore 
 
 or 
 
 2x + l + 2V(2.'/;2 + a;) = 21 
 2V(2^^ + *)=20-2x, 
 
 ,/(2x2 + aO = 10- 
 
 X, 
 
 Squaring both sides, 2,c- + x- = 1 00 - 20j; + x^, 
 whence x = 4 or -25. 
 
 322. AVe shall now give a set of examples of Surd Equa- 
 tions some of which are reilucible to Simple and others to 
 Quadratic Equations. 
 
 Examples,— cxxii. 
 
 1 . 4x - 1 2 ^/o; = 1 6. 4. V(6:c - 1 1 ) -= ^(249 - 2,r2). 
 
 2. 45-14V-:«= -a:. 5. ^/(6-a') = 2- VC-.«-l). 
 
 3. 3>v/(7 + 2.t;2) = 5;/(4aj-3). 6. k-2 v'vl -3x) + 12-0. 
 
 7. ,s'(2,(,- + 7) 4- V(3x - 18) = ^'(7..; + 1 ). 
 
 8. 11 s^204 - n.c) - 20 - ^{'?>x. - 08). 
 
 A 
 
Oy EQCATIONS /XlV/.y/iVG SUKD^ 
 
 233 
 
 00 
 
 9. v' 
 
 — '1. — 
 
 10. v'.O 
 
 + 11- 
 
 «0S 
 
 14. V(x+i)i- V(2x-1) = 0. 
 
 7^-11* 
 
 ?• V 
 
 /(13.C-1)- v^(2.';-l) = 
 
 -.r» 
 
 11. v'C'^ -♦- ^'0 • -v^C'-^^ 
 
 12) = 12. 16. V(7^;+l)- x^3.o+l)-2. 
 
 1 2. V('<^ + 3) + V('« + ^'^) = '' v' 
 
 17. ^/(4 + x)+ Vu;-3. 
 
 525 
 
 1 3. V(25 + 0.-) + V(25 - ..•) = 6. 1 B. V.0- + V(-'^ + 9975) = -^ 
 
 I 
 
 ^ 
 
 19. 
 
 20. 
 
 V(^^)W(^^)=V(3) 
 
 '(,<:-- i)-t-^'-^ 
 
 IC 
 
 !)• 
 
 24. 
 
 -43. 
 
 21. 
 
 
 -j- 
 
 V(.'.--f4)- sJ^c==JU'^ 
 
 2 
 
 )■ 
 
 = .« + 
 
 V(4 + .t;)- N^3= V^o. 
 
 26. V(^ + 4)+ v/Co + 5) = 9. 
 
 27. x/^<^+ VG^'-^) = -^(^J4)- 
 
 'ilk^ 
 
 i 
 
 '"flv 'Vf 
 
 'S;^' 
 
 ^^M 
 
 33- 
 
 28. x'-^-21-H x'(.(;--i)). 
 
 29. v'(,r)U + .f)- x^(5()-.o)=2. 
 
 (:-m;) 
 
 A2xr4} 
 
 \ 
 
 G 
 
 31. N'^^^+'0+v''^--;/(3+..y 
 
 1 
 
 + 
 
 3x- V(,4a;-u;-) 
 
 !( I 
 

 f -M H 
 
 .•( 
 
 |.i 
 
 t^' 
 
 / 
 . / 
 / 
 
 I 
 
 XXVI. ON THE ROOTS OF EQUATIONS. 
 
 323. We have already proved that a Simple Equation can 
 have only one root (Art. 193) : Ave have now to proA'e that a 
 Quadratic E< [nation can have only tivo roots. 
 
 324. "We nnist first call attention to the following fact: 
 
 U mn = (), either in = i), or 11 = 0. • 
 
 Thus there is an aiul»i<niitv : hut if we know that m cannot 
 be ecpial to 0, then we know'for certain that 7i = (), and if we 
 know that a cannot he f(|ual to 0, then we know for certain 
 that m = 0. 
 
 Further, if hnn = i), then either 1 = 0, or m = 0, or 7^ = 0, and 
 so on for any number of factors. 
 
 Ex.1. Solve the equation (.<;- 3) (./; + 4) = 0. 
 Here we must have 
 
 that is, cc = 3, or x— —4. 
 
 -i ' 
 
 Ex. 2. (x - lia) (Du; - 2b) = 0. 
 
 Here we must have . 
 
 if _ 3(t =- 0, or Tm; - 26 = 0, 
 
 that ii<, aj = 3a, or a; = -~. • 
 
 5 
 
 riu 
 
OM THE ROOTS OF EQUATIONS, 
 
 235 
 
 ONS. 
 
 Loii ctin 
 that a 
 
 act: 
 
 1 cannot 
 d if we 
 certain 
 
 = 0, and 
 
 Examples.— cxxiii. 
 
 I. (x-2)(;7;-5)=0. 2. (.t;-3)(x + 7)=0. 3. (.(;4-9)(;<; + 2).- --0. 
 
 4. (a; -5rt)(u; -()?/)-(). 6. (19x-227)(14x' + 83)=(). 
 
 5. (2a;4-7)(3.T;-5) = 0. 7. (5.>; - 4??i.) (6.c - 11 n) = 0. 
 
 8. (a;2 + 5rt;c+6a'-^)(a;2-7«a;+12a')=0. 
 
 9. (:>;2-4)(.7;2-2aaj + a2) = 0. 
 
 10. cc (a;2 - 5.0) = 0. 
 
 1 1 . {acx - 2a + h) {hex + 3a - h) = 0. 
 
 12. {cx-d){cx-e) = 0. 
 
 325. Tlie general Ibrni of a (jiiadratic ciiuation is 
 
 tJic- + hx + c = 0. 
 
 Hence a\x^ + -x + -) = 0. 
 
 \ a a/ 
 
 Now a cannot =0, 
 
 .-. x- + -x + -=0. 
 a a 
 
 , h . c 
 
 A\'riting ji? for - and q for -, we may take the following 
 
 m the type of a ([uadratic equation of which the coefficient of 
 the first term is unity, 
 
 x^ + 'px-\-q = 0. 
 
 326. To bIicw that a ({Utidratic equation hati onUj two roots. 
 
 Let x'^-\-px + q = i) he the e([uatioii. 
 
 Sujipose it to have three different roots, c, b, c. 
 
 Then a'^ + «^> + f/ = o (1), 
 
 6-'+62? + 2 = () (2), 
 
 c''* + c^ + 2.-=() (3). 
 
 Subtracting ^2) from (1), 
 or, {n-h){a-\-h-^'p)^(\ 
 
 n;i 
 

 
 i 'M 
 
 236 
 
 Oy THE ROOTS OF EQUATIOh'S. 
 
 Now a - h doet' not equal 0, since a and h are not alike, 
 
 .■.a-]h+j) = (4). 
 
 Again, subtracting (3) from *(1), 
 
 a^-c'^ + {a-c)p = 0, ', 
 
 or, (rt-r)(a + c+p)=0. 
 
 Now a — c does not e(|ual 0, f-ince a and c are ..it alike, 
 
 .-. a + c+j) = (5). 
 
 Then subtracting (5) from (4), we get 
 
 i 6 -0 = 0, and therefore & = c". 
 
 Hence there are not more than hvo distinct roots. 
 
 327. We now proceed to sliow the relations existing be- 
 tween the Eoots of a (quadratic equation and the Coelticieuts 
 of the terms of the e< [nation. 
 
 328. 
 
 x'^ +px + q = 
 
 is the general form of a (juadratic equation, in which the co- 
 etficient of the first term is unity. 
 
 H 
 
 ence 
 
 X' 
 
 X'^ + 2)X = —q 
 
 X + 
 
 2 
 
 ;=.-' + 
 
 2--\ 
 
 
 Now if a and jS be the roots of the ec^uation, 
 
 V 
 
 /3= 
 
 2-\ 
 
 
 .(1), 
 
 ,(2). 
 
 Adding (1) and (2), we get 
 
 u 
 
 +/j= -i> 
 
 ■Hi) 
 
 qu{ 
 
 <|IKI 
 
 sibh 
 
;4)' 
 
 .(5). 
 
 ting l»t-- 
 ulUcieiit.- 
 
 the co- 
 
 (1)» 
 
 ON THE ROOTS OF EQUATIONS. 
 
 237 
 
 Multiplying (1) and (2), we get 
 
 or 
 
 or 
 
 f^P^'l 
 
 .(4). 
 
 From (3) we 'learn that tlie, sum of the roots is equal to the 
 coefficient of the second term \clth its sujn chaiHji.d. 
 
 From (4) we learn that the loroduct of the roots is equal to 
 the last term. 
 
 329. The equation cc2 + 2jx + |7 = has its roo^^s real and 
 different, real and equal, or impossible and different, according 
 as ]f^ is > = or < 4g'. 
 
 For the roots are 
 
 2 
 
 or 
 
 or 
 
 ■J7+ ^f{f-^) 
 
 ■V- \W-^) 
 
 First, let p" be greater than 4q, then \^('P^- 4q) is a possible 
 quantity, and the roots are different in value and bntli real. 
 
 Next, let 2^-^4(7, then eacli of the roots is equal to the real 
 
 quantity 
 
 V 
 
 Lastly, let ^/^^ be less than 4q, then \^(p'^-4q) is an impos- 
 sible quantity and the roots are different and both impossible. 
 
 Examples.— cxxiv. 
 
 I. If the equations 
 
 ax'^ + hx + c = 0, and a'x^-\-h'x + <^ = 0, 
 
 have respectively two roots, one of which is the reciprocal of 
 the other, prove that 
 
 (aa'-cc'f = {ah'-hc')(a'h-h'c). 
 
 ■V 
 
 tr. 
 
 ■ f 
 f 
 
 ■J. f 
 
 )■ I ' 
 
 I, 'i J 
 
 i- 
 
 *:! 
 
23S 
 
 ON THE ROOTS OF EQUATIONS. 
 
 li % 
 
 
 tniii 
 
 i 
 
 111! 
 
 2. If a, /3 be the ruois ul the equatiou ax^ + ftx + c = 0, prove 
 that 
 
 3. If a, ^ be the roots of the ecj[uation as? + 6a; + c = 0, prove 
 that 
 
 4. Prove that, if the roots of the equation ax- + 6ic + c =0 be 
 equal, ax2 + &.<; + c is a perfect square with respect to x. 
 
 5. \ia.^ li represent the two roots of the equation 
 
 •2-(l + a)a; + 2(l + a + a2)=0, 
 Bhow that a^ + P'^ = a. 
 
 330. If a and /3 be the roots of the equation x^ +px + 2 = 0, 
 then x'^+px + q={x-a){x- /3). 
 
 For since jr)= - (a + ^) and 2 = aj8, 
 
 ;i'2 +2rx + q = X' — (a + P) x + a/i 
 = (x-a){x- f3). 
 
 Hence we may form a quadratic equation of which the roots 
 are given. 
 
 Ex. 1. Form the equation whose roots are 4 and 5. 
 
 Here a;-a = a; — 4 and X- jS=a;-5; 
 
 /. the equation is (x - 4) (x — 5) = ; 
 
 or. 
 
 x^-9x + 20 = 0. 
 
 Ex. 2. Form the equation whose roots are ^ and - 3. 
 Here x-a = ic--^ and a;-^--x + 3; 
 
 .'. the equation is ^ u; -- j (x + 3) = ; 
 
 or, 
 or, 
 
 (2x-l)(.x + 3)=0; 
 l?j- + 5x-3 = 0. 
 
 n(jt. 
 
 IR »« 
 
 sion 
 JatteJ 
 
 33) 
 
 of til 
 
 expre 
 
 ]>iinci 
 
 ^\^ \\ 
 
 equal 
 
 at wll 
 
 Will II 
 
- 3. 
 
 UN THE ROOTS OF RQCAl lOXS. 
 
 239 
 
 i Examples.— cxxv. 
 
 Form the equations wliose roots^ are 
 I. 5 and 6. 2. 4 and -5. 
 
 5 
 
 1 , 2 
 
 4. 2 and-, 
 
 5. 7 and - 
 
 9 
 
 7. m \-n and m — n, 8. . - and -„. 
 
 a p 
 
 3.-2 and - 7. 
 6. x/3 and - ^/3. 
 
 a 
 
 9. - yj and 
 p a 
 
 331. Any expression contair.ing x is said to be a Function 
 of X. An expression containing any symbol x is said to be a 
 'positive integral function of x when all the powers of x con- 
 tained in it have positive integral indices. 
 
 3 1 
 
 For example, bx^ + 2xP + -;v* + -.-.a-^ + 3 is a positive integral 
 
 function of .r, but QxP + ^x'-^ + l and 5,» " - 2,k-2 j. 3,r2 -j- 1 are 
 
 not, because tlie first contains a;^, of wliich tlie index is not 
 integral, and the second contains a;~-, of which the index is not 
 positive. 
 
 332. The expression Src" + 4,r2 + 2 is said to be the expres- 
 sion corresponding to the equation 5a;^ + 4.^;^ 4- 2 = 0, and tlie 
 latter is the e([uati(Jn corresponding to the former. 
 
 333. If a be a root of an equation, then x-a is a factor 
 of the corresponding expression, provided the e(£uation and 
 expression contain only positive integral powers of x. This 
 principle is useful in resolving such an expression into factors. 
 AVe have already proved it to be true in the case ot" a (quadratic 
 equation. The general proof of it is not suitable for the stage 
 at whicli the learner is now supposed to be arrived, but wtj 
 will illustrate it by some Examples. 
 
 ■I 
 
 \\ 
 
240 
 
 ON THE ROOTS OF EQUATIONS. 
 
 Ex. 1. Resolve 2aj2 - 5x + 3 into factors. 
 
 If we solve the eciuation 2x2 -5a; + 3 = 0, we sliall find that 
 
 its roots are 1 and o- 
 
 Now divide 2^^ _ 5a; + 3 hy x - 1 ; the quotient is 2* - 3 
 that is 2 f re - '^ j ; 
 
 /. the f];iven v.-. a'^f sion = 2 (a3 - 1 ) ( '• - ^y 
 
 * 
 
 Ex. 2. Resolve 2.x^ + x2 - 11a; - 10 into factors. 
 
 By trial we find that this expression vanishes if we luit 
 x^ -\\ that is, - 1 is a root of the e(|Uation 
 
 2x^ + a;2- 11a;- 10 = 0. 
 
 Divide the expression by «: ■+ 1 : the quotient is 23^*^ - a, - 10 ; 
 .-. the expression = (2x2 _ .,; _ 10) (.,; +1) 
 
 =«2(x2-|-5)(x + l). 
 We must now resolve a-- -\-^ into factors, by solving the 
 
 X 
 
 corresponding e(|uation X- - 2 - '^J =^' 
 
 5 
 
 r. 1 ^ 
 
 Tlie roots of this equation are - 2 and „ ; 
 
 2x: 
 
 ■ + x'-'-llx- 10 = 2(1; + 2)(x-2)(x-+l) 
 = (x+2)(2x-5)(x+l). , 
 
 Ex A MPLES. — CXXVi. 
 
 Resolve into simple factors the following expressions : 
 
 I. x'^-llx2 + 36x-3G. 
 
 3. x3-5x2-46x-40. 
 
 5. Gx3+llx--9x-14. 
 
 7. «3-63-c;^-3a?>(;. 
 
 9. 2.t3-5x2-17x+20. 
 
 2. x3-7x2-il4x-8. 
 
 4. 4x^ + 6x2 + x-l. 
 
 6. ii? + y^ + 2^ — 3x//;.'. 
 
 8. 3x^-x2-23x + 2i. 
 
 10. 15x^ + 41x2 + 5x-21. 
 
21. 
 
 OlSr TlIK ROOTS OF EQUATIONS. 
 
 24t 
 
 334. 11' we can tind one root oi' such an etjiuitioii as 
 
 2.o'' + .'j'--lL'J-10 = 0, 
 we can tiiid all the rut^ts. 
 
 One root ol' the eciuation is - 1 ; 
 
 .-. (.r + l)(2,t*'^-.'-l())-(); 
 ."..'(;+ 1=0, or 2;<;---./;- !(>==(); 
 
 .. x= - 1, or — z, or -. 
 
 Siniihirly, if we can find one root of an e(|Uation involvin<^ 
 the 4"* power of a;, we can deriv<; from it an equation involvinj,' 
 the 3^** and lower pitwers of x, from which we may tind the otlier 
 roots. And if again we can find one root of this, the other 
 two roots can be found from a (juadratic e(|uation. 
 
 335. Any e»|uation into wliicli an unknown syml)ol or ex- 
 pression enters in two terms only, having its index in one of 
 the terms doable of its index in the other, may he solved as a 
 (juadratic e(|uation. 
 
 Ex. Solve the ecfaation x*^ - Gx^ = 7. ' 
 
 Regarding x^ as the (|uantity to he obtained by the solution 
 of the equation, we get 
 
 therefore y;^-3^±4; 
 
 x-" 
 
 7, or x''= - 1. 
 
 therefore 
 
 Hence x= H/l ov x= il^f -I, 
 
 and one value of yii/- 1 is — 1. 
 
 33G. In some cases by adding a certain quantity to both 
 sides of an equaticju we can bring it into a form capable of 
 solution, thus, to solve the equation 
 
 a;'^ + 5x + 4 = 5 ^/(u;- + 5x- + 28), 
 add 24 to each side. 
 
 Then .'«^ + 5x+28 = 5 ^/(x'■*4■5x + 2S) + 24; 
 
 or, X- + 5x + 2S-b x/(.'^" + T)./; + 28) - 24. 
 
 This is now in the form of a quadratic ec^uation, the un- 
 known quantity being y,^{x' + bx -^ 28), and conqdeting the 
 square we have 
 
 ■I 
 
 \i 
 
 
 
 m 
 
 m 
 
 
 :tr 
 
 I* ' 
 
7W 
 
 ■ 4 '^l 
 
 
 M 
 
 242 
 
 OiV THE ROOTS OF JiQUATIO^fS. 
 
 XT + 5a; + 28 - 5 sl{->^ + hx + 28) + ^j' - ^-^ ; 
 ,. ^/(^2 + 5,; + 28)-|=±y; 
 
 whenct; 
 
 V(,«''2 + 5a; + 28) = 8 or -3; 
 .-. a;''*+5.« + 28 = 64 or 9; 
 
 from which we iiuiy find four values of a;, viz. 4,-9, and 
 
 2- 2 ■ 
 
 Examples. — cxxvii. 
 
 Find roots of the following et_[iiations: 
 
 I. LO^- 12u;^=13. 
 3. ;(;^-i-r2.<;' + 21-0. 
 
 5. if 
 
 ..*«_^,^'.^25 
 
 3 12 
 
 7. t'C "T 3i'J — ^ J,. 
 
 2. ;'J«+lU'^ + 24 = (). 
 
 4. .-r"M-3;ij"' = 4. 
 . 9^5 
 
 8. a;-'-^"-a;-'' = 20. 
 
 9. a;--2u; + 6(.c'^-2.r + 5)- = ll. 
 I O. %^ — x + o Vv"''^" - y 't; + n) = ^2 ~ • 
 
 ^ II. a;--2V(3:c--2rtx + 4) + 4=^(;^ + ^ + l). 
 12. nx-^-il >J(x' - ax + a-) =^ ic^ + 2a. 
 
 337. Every ii(|uation has as nwmy roots as it has dimen- 
 sions, and no more. Tnis we have proved in the case of 
 simple and (juadratic ecjuations (Arts. 193, 323). The general 
 ]U'oof is not suited to this work, hut we may illustratt? it by 
 the following Examples. 
 
 Ex. 1. To solve the efjuation ;';^- 1=0. 
 
 One root is clearly 1. 
 
 Dividin<^ b}^ x-\, we obtain x- + x + 1 = 0, of which the roots 
 
 -l+V-3 ,-l-x/-3 
 are -^^ and ^ •• 
 
ON RATIO. 
 
 243 
 
 Hence the fhrte, roots are 1, — — *^ — ^ ami :,- — '-, 
 
 2 2 
 
 Ex.2. To solve tht equation .)r* - 1 =0. 
 
 Two of tlie Toots are evidently + 1 inid - 1. 
 
 TTcncf, dividing,' by (./•- l)(a*+ 1), that is hy.r^-l^vo ohtain 
 5'-+ 1=0, of which tlio roots are sj ~\ iiiid - ^'-1. 
 
 Ifonce tlio/rt?n' roots arc 1,-1, sj ~ 1, and - s! - 1. 
 
 Tlu' ('((nation .r" - fi,/" = 7 will in like manner have 5/x 
 roots, for it may he reduced, as in Art. ;53."), to two cubic 
 e(iuations, a;^ _ 7 _ q .^j,,-[ ,,.:! ^ j _ 0^ 
 
 each of which has ilirce roots, which may be found as in 
 Ex. I. 
 
 XXVII. ON RATIO. 
 
 338. If a and B r^tand for two unequal quantities of the 
 f»ame kind, we may consider their inequality in two ways. We 
 may ask 
 
 (1) 1j\j what quantity one is greater than the other ? 
 
 The answer to this is made by stating the difference be- 
 tween the two o^uantities. Now since ([uantities are represerited 
 in Algebra l)y their measures (Art. 33), if a and h be the 
 nieasures of A and B, the difference between A and /i is 
 represented algebraically by a -h. 
 
 (2) ]jij how many times one is greater than the other ? 
 
 The answer to this question is made l>y stating the inimber 
 of times the one contains the other. 
 
 Note. The (quantities must be of the siame hind. Wo can- 
 not compare inches with hours, nor lines Avith surfaces. 
 
 339. The second method of comparing A and 7? is called 
 findinf' the Eatio of A to B, and we give tlie followinji dcfi- 
 nition. 
 
 Def. Ratio is the relati(^n which one quantity l)ears to 
 another of the same kind with res]>ect to the number of timei 
 the one contains the other. 
 
 % 
 
 
 ■ 4 
 
 ;/''•: 
 
 ■^1: 
 
 
 if 
 
 ni 'U 
 
 ,) * 
 
 , I 
 
 r 
 
 Ji 
 
f. 
 
 244 
 
 OA' RA TIO. 
 
 340. Tli;' ratio of yl to 7? is fxpressod tlms, A : /.'. 
 
 A and II arc rallcil \\u'. Tkhms oCtlio ratio. 
 
 A is oallc'd the Antkcedkxt and /> the Coxskquknt. 
 
 
 « 
 
 341. Now sinct' ipiMntilics arc rcpi'esentcd in Al-^cln-a !>>' 
 thoir nipasurcs, avo nnist represent llie i-alio l»et\veeu two 
 quantities by tlie ratio between tlieir measures. Our next 
 step then must he to sliow how to estimate tlie i-atio between 
 two numhers. This ratio is determined by lindin^' liow many 
 times one contains the other, tliat is, by obtainini,' tlie (piotieut 
 resulting from the division of one by tlie otlier. If a and b, 
 
 then, be any two numbers, the fraction j- will express the ratio 
 
 of a to h. (Art, 136.) 
 
 342. Thus if a and h be the measures of A and B respec- 
 tively, the ratio of A to 7> is represented ali^ebraically by the 
 
 fraction ^. 
 
 
 
 343. If a or h or both are surd numbi'rs, the fraction 
 
 a 
 h 
 
 may also be a surd, and its approximate value can l)e found l»y 
 
 rii 
 
 Art. 291. Suppose this value to be , where m and _n are 
 whole numbers : then we should say that the ratio A ; B is 
 approximately represented by 
 
 m 
 
 344. Ratios may be compared with each other, by com- 
 paring the fractions by which they are denoted. 
 
 Thus the ratios 3 : 4 and 4 : 5 may be compared by com- 
 
 parinjj; the fractions ', and .. 
 1 o 4 o 
 
 These are equivalent to ^ ' and -- respectively ; and since 
 
 ~ is greater than ^^., the ratio 4 ; 5 is greater than the 
 ratio 3:4 
 
ON RATIO. 
 
 245 
 
 Examples. — cxxviii. 
 
 1. Pliic(! ill order of iiiagnitiule tliu ratios 2 : 3, G : 7, 7 : 0. 
 
 2. Conipiiru the ratioH x -v 'Sy : x + 2y tiiitl x + 2ij ; x-\- y. 
 
 3. Coniiuir J tl»c ratioH x - 5// : u; - iy and x-fiy: x - 2//. 
 
 4. What uiiiuluT must Lo addtnl to eacli of thu turiud of the 
 ratio a : ^, that it may become the ratio c ; dt 
 
 5. Tlu! sum of the S(juares of the Aut(!Cc(hMit and Cunsr 
 qucnt of a Jvalio is 181, and tlit^ prmljict of tiie Antecedent 
 and Conse<|Uent is DO. What is the ratio i 
 
 345. A rati<j of greater inetimdily is one "whose antecedent 
 is greater than its conse(|uent. 
 
 A ratio of less inequality is one whose antecedent is less than 
 its conse(|uent. 
 
 This is the same as saying a ratio of greater inequality i>4 
 represented l)y an Improper Fraction, and a ratio of less in- 
 equality by a Proper Fraction. 
 
 346. A Ratio of greater inequality is diminished by adding 
 the same number to both its terms. 
 
 Tims if 1 be added to both terms of the ratio 5 : 2 it Lecomea 
 
 G ; 3, which is less than the former ratio, aince «, that is, 2, is 
 5 
 
 less than 
 
 2' 
 
 And, in general, if x be added to both terms of the ratio 
 a : b, where a is greater tlian b, we may compare the two 
 rcvtios thus, 
 
 ratio a + x : h + x is less than ratio a : b, 
 
 if 
 
 if 
 
 if 
 if 
 
 if 
 
 a + x , , . , a 
 -, — be less than -,-» 
 b+x U 
 
 a!) + bx ^ T ,1 ab + ax ' 
 ,., — 7- be less than -,-. — 7-, 
 b^ + hx b~ + bx' 
 
 <ib + bx be less than ab + ax, 
 
 bx be less than ax, 
 
 b be less than a. 
 
 Now b is less than a ; 
 
 .'. a + X ; b + X is less ':han a : h» 
 
 ri 
 
 
 5i I 
 
 |.i| 
 
 I 
 
246 
 
 ON RATIO. 
 
 i!' 
 
 % 
 
 347. We may observe that Art. 34G is merely a repetition 
 of tliat wliicli we proposed as an Exaiii])le at the end of the 
 chapter on Miscellaneous Fractions, There is not indeed any 
 necessity for US to weary tlie reader with examj^es on Ratio: 
 I'or since we express a ratio hy a fraction, nearly all that we 
 might have had to say aliout llatios has been anticipated in 
 our remarks on Fractions. 
 
 348. TIu; student may, however, work the following Theo- 
 rems as Examples. 
 
 (1) If a : h be a ratio of greater inequality, and .'' a positiv*; 
 quantity, the ratio a-s, : b~x is greater than tlie ratio a : h. 
 
 (2) If <t : h be a ratio of less inequality, and x a i)ositive 
 quantity, the ratio a + x : b +x h greater tiian the ratio a : b. 
 
 (3) If (t : h be a ratio of less ine( quality, and x a j)ositive 
 (quantity, the ratio a — x : b — x is less than the ratio a : b. 
 
 341). In some cases we may from a single equation involv- 
 ing two unknown symbols determine the ratio between the 
 two symbols. In other words we may be able to determine the 
 relative values of the two symbols, though we cauuot determine 
 their absolute values. 
 
 Thus from the equation 4a; = 3?/, 
 
 X 3 
 we get - = -7. 
 
 Again, from the equation 3;«- = 2//'-', 
 
 we get o = -oj and thereiore =-\^. 
 
 
 11 
 
 Examples.— cxxix. 
 
 Find th(^ ratio of x to y from the following equations : 
 I. \)x = ^y 2. a-: — by. 3. ax~by = cx-\ihj, 
 
 4. :/j2+2,ry = 5//". 5. ,-;'-^- 12,17/= 13,r. 6. x^ + mxy = nY' 
 
 7. Find two numbers in the ratio of 3: 4, of which the 
 sum is to tlie sum of their squares :: 7 : 50. 
 
 8. Two numbers are in the ratio of 6 : 7, and when 12 is 
 added to each the resulting numbers arc in the ratio of 12 : 13. 
 Find the numbers. 
 
 W 
 
ON RATIO. 
 
 247 
 
 .h the 
 
 9. The sum ol' two nuiulx'is is lUO, aii>l the iiiimbcis are 
 in the ratio of 7 : 13. Find tlicni. 
 
 10. Tlie difference of the S(|iiare8 of two niunhcrs is 48, 
 and the sum of the numhers is to the dilference of the num- 
 bers in the ratio 1:2 ; 1. Find the numbers. 
 
 11. If 5 gohl coins and 4 silver ones are worth as much as 
 15 gold coins and 12 silver ones, find the ratio of the value of a 
 gold coin to that of a silver one. 
 
 11. If 8 gold coins and 9 sih'er ones are worth as much as 
 G gold coins and 19 silver ones, find the ratio of the value of a 
 silver coin to that of a gold one. 
 
 3o(). llatios are comjwundr.d by nmlliplying together the 
 fractions by which they are denoted. 
 
 Thus the ratio compounded of a : h and c : d is ac : bd. 
 
 Examples.— cxxx. 
 
 Write the ratios compounded of the ratios 
 
 1. 2:3 and 4:5. 
 
 2. 3 : 7, 14 : 9 and 4:3. • 
 
 3. x^ — ij- : j:^ + y'^ and x~ - .I'lj + i/'^ : X + If. ^ 
 
 4. «2 _ ],i ^ 2hc - c- : a- - }fi - 2I)C - c^ and a + h-\-c la + b-c. 
 
 5. m^ + n^ ; m^ - n^ and m — n : m + n. 
 
 6. :o^^ + 5./; + 6 : 7/=^ - 1y -t 1 2, and 1/ - 'Pjij : x'^ + 3x, 
 
 351. The ratio a^ : 6^ j,^ called tlie Duplicate Ratio of a : b. 
 Thus 100 : 64 is tlie duplicate ratio of 10 : 8, 
 
 and 
 
 36,i'- : 25//2 is the (biplicate ratio of 6:«; : 5>/. 
 
 Tlie ratio a^ : P is calhd tlie Tiuplicatk Ratio of a : h. 
 
 Tluis G4 : 27 is the triplicate ratio of 4 : 3, 
 
 and 
 
 343.'j'' : 1331 (/•■' is the triiilicate ratio of 7x : 11//. 
 
 T| 
 
 i 
 
 I i<\ 
 
248 
 
 ON PROPORTION, 
 
 352. Tlie definition ol" Ratio <Mven in Euclid is tlie same as 
 in Algebra, and so also is the expression for the ratio that one 
 quantity bears to another, that is, A : B. But Euclid cannot 
 employ fractions, and hence he cannot represent the value of a 
 
 ratio as we do in Algehi'a. 
 
 If ■ 
 
 
 XXYIJI ON PROPORTION. 
 
 353. Proportion consists m the equality of two ratios. 
 
 The algebraic test of I^roportion is thai the two fractions 
 representing the ratios must be equal. 
 
 Thus the ratio a : b will be equal to the ratio c : d, 
 
 .„ a c 
 
 and the /our numbers a, b, c, d are in such a case said to be in 
 proportion. ^ 
 
 354. If the ratios a : b and c : d form a x^roportion, we 
 express the fact thus : 
 
 a : b = c : d. • 
 
 This is the clearest manner of expressing the equality of the 
 ratios a : b and c : d, but there is another way of expressing 
 the same fact, thus 
 
 a : b '.: c : d, 
 
 which is read thus, 
 
 a is to & as c is to d. 
 
 The two terms a and d are called the Extremes. 
 , 7> and c the Means. 
 
 355. J f lien four numbers are in in-oportion, 
 
 product of extremes = product of means. 
 
 Let ay b, c, d be in proportion. 
 
 ihen 1—1' 
 
 <l 
 
 that 
 
 The 
 « and ( 
 
ON PROPORTION. 
 
 249 
 
 Multij^lyiui^' l>otli sides of the eciiuition by M, we get 
 
 \ ad = hc. 
 
 Conversely, if ad = hc we can show that a : b — c : (J. 
 
 For since ad = hc, 
 
 dividing Loth sides by hd, we get 
 
 ad _hc 
 hirhiV 
 
 that is, 
 
 35 G. liad=r.l)Cj 
 
 r= :j, I.e. a : b — c : a. 
 b a 
 
 a b 
 
 Dividing by cd, we get - = ^, i.e. a : c = h : d ; 
 
 d c . 
 Dividing bv ab, we get , =-, i.e. </ : b = c : ai 
 
 7 7 
 Dividing by ac, we get -7 = --, i.e. d : c = b : a. 
 
 C (v 
 
 357. From this it i"olh)ws that if any 4 numbers be so 
 related that the prixbict of Uvo is equal to the jn'oduct of the 
 other two, we can express the 4 numbers in the form of a pro- 
 portion. 
 
 The factors of one of the products must foiiu the extremes. 
 
 The factors of the other product must form the means. 
 
 358. TJiree (piantities are said to be in Continued Piio- 
 PoUTioN when the ratio of the iirst to the second is equal to 
 the ratio of the second to the third. 
 
 Thus a, b, c aw. in continued })roportion if 
 
 (f : b — b : (\ 
 
 Tlie quantity b is called a Mean Phopoutional between 
 a aiid c. 
 
 )i'' 
 
 l! 
 
250 
 
 ON PROPORTION. 
 
 Four quantities are said to he in Continuerl Proportion 
 when the ratios of the first to the second, of the second to 
 the third, and of the third to the fourth are all equah 
 
 Thus «, h, c, d are in continued pro])ortion when 
 
 a : h — l) : c = c ; J. . . 
 
 359. We showed in Art. 205 the process hy which when 
 Iwo or more fractions are known to "be .?qnal, other rehations 
 between the numbers involved in tliem may he determined. 
 That process is of course applicable to Examples in Ratio and 
 Proportion, as we shall now show by particular instances. 
 
 Ex. 1, II' a : b = c : d, prove that 
 
 Since 
 
 , J a c 
 
 a:b-.c: d, j=^ 
 
 Let ,- = X. Then 4 = X ; 
 a 
 
 Now 
 and 
 
 Hence 
 that is, 
 
 .'. a = \h, and c = \d. 
 ff+^_ X26'^_62 _ &y \2jj ) _ X 2 + 1 
 
 C2_+C^2 _ x2fZ2 4.d^_d^ (X2 +_l^ _ \2 +1 
 C2- rf2-X2rf2 "-(/£- ^2 (X2_ l)-xi^_ l* 
 
 a^+]>:_c^J-d"- 
 a^ + ¥ : (f--b'^ = c'^ + J.^ : c^^-di 
 
 Ex. 2. l( a : b '.: c : d, prove that 
 
 a-.c:: ^/(rr* + b') : */(c^ + d*). 
 
 Let? = X. ThenT = X; 
 6 d 
 
 Ex. 
 
 .*. a = \bf and c = \d. 
 
ON PROPORTION. 
 
 25> 
 
 Now 
 
 
 ''"' 4/0- ' + '0 -^^f^Hrf')";/f^v:4/^x4 + i) 
 
 
 
 Hencft 
 that is, 
 
 a:c::^,'{aU-b'): ^{c^ + d^ 
 
 Ex. 3. li a :h = c: d = e :f, prove that each of these ratios 
 is equal to the ratio a + c + e: b + a +/ 
 
 Let 
 Then 
 
 
 .r^ 
 
 7 
 
 r=:X. 
 
 rt = X7>, c —- \d, e — \f. 
 
 XT a + c + e X/j + XJ, fX/ \(b-\-d + f) - 
 
 b + d-rf b + d+f b + d+f 
 
 a + c-\ e _a_c_e 
 h + d+f^'b^d~P 
 
 a-\-c + c -.b-^-d +f— a : b = c : d = e : /. 
 
 Hence 
 
 that is. 
 
 Ex. 4. If a, b, c are in continued proportion, show that 
 
 a2 + 6^&-' + c2 = a:c. 
 
 Let j = \. Then- = X. 
 b c 
 
 Hence rt = X?) and /> = X('. 
 
 Ex. 5. If L^)rt + /j : 15c + fZ=12a + /< : J2c + (/, prove tliat 
 
 a :h = c : d. 
 Since 1 5^^ -f /> : 1 f/c + (Z= 1 2o + ?> : 1 2c + f?, 
 
 and since product of extremes = product of means, 
 
 
 ■ :1^ 
 
252 
 
 ON rROrORTION. 
 
 or, 
 or, 
 or, 
 or. 
 
 {\-m\-])) (12c + f/) = (15c+(/) (12a + />), 
 
 \mac + 126c- + X^Mxl + hi = \SOar + V2ad + 156c 4- hd, 
 
 \2hc+l6ad=l2ad+l5hc, 
 
 2ad = 36c, 
 
 ad — he. 
 
 Whence, by Art. 355, ^ : 6 = c : d. 
 
 Additional Examples will be found in papje 137, to which 
 we may add the following. 
 
 lO. 
 
 II. 
 
 12. 
 
 13. 
 
 14. 
 
 I 
 
 , 
 
 
 i 
 
 
 
 Examples.— cxxxi. 
 
 1 . If rt : 6 = c : ^, show that a -}• 6 : « = c + d : c. 
 
 2. If a : 6 -= c : fZ, show that a"- - h'^ : 62 = c^ - rZ^ ; fl!-. 
 
 3. If a, : 6, = a, : 6„ show that -^^^^^^ ^ ^^^^^^ = j^-. 
 
 4. If a : 6:: c : (f, show that 
 
 3^2 + ah + 262 . 3^2 _ 262 : : 3,2 + ,,/, + 9,72 . 3,2 _ 2d'-, 
 
 5. If a : 6 = c : (Z, show that 
 
 ft2 + 3a6 + 62 : c2 + 3c(^ + d^ = 2a6 + 36^ : 2c(Z + M'-. 
 
 6. If (I : 6 = c : fZ = c : / then a : 6 = vie - ne : md - nf. 
 
 7. If '^-a ^-\ any parts of a, 6, be talo'U fr^in a and 6 
 respectively, show that a, 6, and tlu> reniaiiuh'rs form a propor- 
 
 tion. 
 
 8. li (I :h = c: d=e :f, show that 
 
 ac : 6rZ = Za2 + „H-2 + «r'2 ; W^ + vuF- + vf. 
 
 9. If ((1 : 61 =^(^2 : 62=-«;i : 6,, show that 
 
 ai' + a^ + a'':b-' + h/ + h^::a'':h^. 
 
 360. 
 
 ratios) is 
 Def. 6 ED 
 
 But th 
 quite dift 
 
 The al 
 represent 
 
 EucUd' 
 thus : 
 
 "Theli 
 
 to the se( 
 ecpiimulti 
 and any e 
 
 "If the 
 the multi] 
 or, 
 
 " If the 
 the multij; 
 
OiV PROPORTION. 
 
 253 
 
 ifZ, 
 
 to wliich 
 
 liV'. 
 
 1 a and ?> 
 I a prepor- 
 
 10. Ifaj : fti = rt2 : ^2 = «3 : ^3, sl'ow that 
 
 ttiaa + aortg + agai : ?)i/>a + M3 + W^i = f*i'^ : ^1^. 
 
 11. It »'-■,, ,.,= .J.-' 7— -79, show that either J = jOrT = -. 
 
 1 2. If a^^ + 6ii : a^ - &''^ = t;^ + (P : c' - d', show that 
 
 a : h — c : d. 
 
 13. If (t : fi — c'.d, show that 
 
 14. If ffj : l\ — a„ : b.,, show that 
 
 On the Geometrical Treatment of Proportion. 
 
 360. The definition of Proportion (viz. the equality of 
 ratios) is the same in Euclid as in Algebra. (Eucl. Book v. 
 Def. 6 and 8.) 
 
 * 
 
 But the ways of testing whei,her two ratios are equal are 
 quite different in Euclid and in Algebra. 
 
 The algebraic test is, as we have said, that the two fractions 
 representing the ratios must be equal. 
 
 Euclid's test is given in Book v. Def. 5, where it stands 
 thus : 
 
 " The first of four magnitudes is said to have the same ratio 
 to the second which the third has to the fourth, when any 
 equimultiples whatsoever of the first and third being taKcn 
 and any equimulti j)Ie8 whatsoever of the second and fourth : 
 
 "If the multiple of the first be less than that of the second, 
 tlie multiple of the third is also less than that of the fourth : 
 or, 
 
 " If the multiple of the first be equal to that of the second, 
 the multiple of the third is also equal to that of 'the fourth : 
 or. 
 
 * ■ ii i 
 
 
 i: 
 
 ^i 
 
254 
 
 ON PROPORTION. 
 
 11 i ' 
 
 iM 
 
 i ; 1 
 
 i, 1 
 
 " If tlie multiple of the first be fjreatcr than that uf the 
 Recond, the multiple of the third is also greater than that of 
 the fourth." 
 
 We sliall now show, first, how to deduce Euclid's test of the 
 equality of ratios from the algebraic test, and ' ecoiidly, how to 
 deduce the algebraic test from that employe a by Euclid. 
 
 361. I. "^ sb that if quantities be proportional accord- 
 ing to the ^r-;(;bvi •f'al test they will also be proportional 
 according to tl ,:"uiu*^trical test. 
 
 If a, h, c, d be proporiional according to the algebraical 
 test, 
 
 a _c 
 b~d' 
 
 77? 
 
 Multiply each side by — , and we get 
 
 ma_mc 
 nb nd^ 
 
 Now, from the nature of fractions, 
 if ma be less than nh, mc will also be4ess than nd, and 
 if ma be equal to n6, mc will also be equal to nd, and 
 if ma be greater than 7ib, mc will also be greater than nd. 
 
 Since then of the four quantities a, b, c, d equimultiples have 
 been taken of the first and third, and equimultiples of the 
 second and fourth, and it appears that when the multiple of 
 the first is greater than, ecpuil to, or less than the multiple of 
 the second, the multiple of the third is also greater than, 
 equal to, or less than the multiple of the fourth, it follows that 
 a, b, c, d are proportionals according to the geometrical test. 
 
 332. II. To deduce the al'^ebraic test of proportionality 
 from that given by Euclid. ^ 
 
 Let a, 6, c, d be proportional according to Euclid. 
 
 TaJ 
 
 Then if 
 
 let 
 
 J is not ecpial to -^, 
 
 , be equal to ,. 
 b + x ^ d 
 
 .(1). 
 
 The 
 
 But 
 
 and, 1)' 
 it Ibllo 
 The 
 HeiK 
 
 Ther 
 
 Wes; 
 Examp] 
 
 I. If 
 between 
 
 '> Tf 
 
 and 
 3. If 
 
 4- 
 
 If 
 
 b is 
 
 a me 
 
 5- 
 
 If 
 
 greatest, | 
 
 If 
 
 a + J 
 
 i is great! 
 
of the 
 
 that of 
 
 t of the 
 , how to 
 d. 
 
 . accord - 
 )ortioiial 
 
 ^ebruical 
 
 d, and 
 and 
 than nd. 
 
 k'B have 
 
 es of the 
 
 Liltiple of 
 
 lultiple of 
 
 .iter than, 
 
 lows that 
 
 il test. 
 
 )r 
 
 tionality 
 
 .(1). 
 
 EXAMPLES ON RA TIO. ■ 255 
 
 Take m and n aueh that 
 
 imi is greater than nh, 
 
 but less than 7i(/j + a;) (2). 
 
 Then, by Euclid's definition, 
 
 mo is greater than nd (3). 
 
 But since, by (1), -,, v = — ;, 
 
 and, by (2), mtt is less than n{b + r), 
 
 il follows that 7rtc is less than n^^ ''4). 
 
 The results (3) and (4) therefore contradict each c' her. 
 
 Hence (1) cannot be true. 
 
 Therefore -r is equal to -^. 
 
 We shall conclude this chapter with a mixed collection of 
 Examples on Ratio and Proportion. 
 
 Examples.— cxxxii. 
 
 1. Ti a-b :b-c ::h : c, show that 6 is a mean proportional 
 between a and c. 
 
 2. l^ a: b :: c : d, show that 
 
 a+o c+d 
 
 and a:b'.: i^{ma^ + nc^) : sji^nib^ + m#). 
 
 3. li a'.b w c \ d, prove that 
 
 ma — nb_mc-nd 
 ma + nb mc + nd' 
 
 4. If 5ffc + 36: 7a + 36:: 56 + 3c: 76 + 3c, 
 6 is a mean proportional between a and c. 
 
 5. If 4 quantities be proportional, and the first be tho 
 greatest, the fourth is the least. 
 
 If a + 6, m + 71, m - ?i, ct - 6 be four such {quantities, show that 
 h is greater than n. 
 
 i 
 
 f i 
 
 I, 
 
 t 
 
 !i 
 
 
 
 
 I 
 
 M 
 
 I ti 
 
256 
 
 EXAMPLES ON RATIO. 
 
 6. Solve the eiiuatioii 
 
 x-\ :x--2 = 2x + i :u; + 2. 
 
 7. It'—, — == — r , show that the ratios a : 6 and c ; rf are 
 ' b a ^ 
 
 ulao equal. 
 
 8. In a mile lace 1»etween a hicycle and a tricycle, tht-ir 
 rates were proportional to 5 and 4. TJie tricycle liad hall'-a- 
 minutc .•start, but wa« beaten ])y 170 yards. Find the ratws of 
 each. 
 
 9. li' a : h :: c : (I and a is the f,'reatest of the four (juauti- 
 ties, show that a- + d'^ is greater than ¥ + c^. 
 
 10. Show that it ^TT- — ,= ,L> 1) ^li'^n a : b .: c : a. 
 
 10c + d l'-2c + d 
 
 11. U X : y :: 3 : 2 and a; : 25 : : 24 : ?/, find x and y. 
 
 12. If a, b, c be in continued proportion, then 
 
 (1) a : a + b :: a-b : a-c; 
 
 (2) (a'^ + b'^(b^ + c')-^--{ab + bcyi 
 
 13. If a : & : : c : £?, show that -j— = — 7— i 
 
 and hence solve the e(|uation 
 
 ab — bc — dx_a — b — c 
 bc + dx b + c ' 
 
 14. If a, 6, c are in continued proportion, show that 
 
 a + mb : a - mb :: b + mc : b - mc. 
 
 15. If ct : 6 :: 5 :,4, find the value of the ratio 
 
 13 
 
 16. The sides of a triangle are as 2- : 3- : 4, and the peri- 
 meter is 205 yards : find, the sides. 
 
 17. The sides of a triangle are as 3 : 4 : 5, and the ])eri- 
 meter is 480 yards : find the sides. 
 
 r8. . 
 of tlie g] 
 the sum 
 
 ^9- ^' 
 lie finds 
 time as ." 
 
 20. T 
 
 tains wat 
 If the cor 
 mixture 1 
 liad been 
 water in t 
 
 21. A 
 
 f^ells agair 
 <Jwu'.s ^16 
 Iiave clear 
 How man} 
 cost him ? I 
 
 22. A 
 
 much per 
 tlie horse ? 
 
 23. I b 
 ^iHich per c 
 
 24- A n 
 
 gaining as 
 give for the 
 
 25. A c( 
 at a (listaiic 
 •'mother cre^ 
 tile former 
 '»etween the 
 
 26. The 
 joiirnoy of 
 !^.-14. Th( 
 'IS it would I 
 t'xpress trail 
 Fs.A.] 
 
c ; rf are 
 
 :k', tli«.'ir 
 d halt'-a- 
 } rates of 
 
 r (|uauti- 
 
 * 
 
 d. 
 
 y- 
 
 ,liat 
 
 tlie peri- 
 
 the peri- 
 
 /IXD PROPORTIOh^ 
 
 257 
 
 18. Assimiiii<,' a + h : p + q :: 'j^-q : a- h, prove that the sum 
 of the ^'reatost and least terni.'^ of any proportion is greater tlian 
 the snni of the other two. 
 
 19. A waterman rows 30 miles and hack in 12 hours, and 
 he finds that he can row 5 miles with the stream in the same 
 time as 3 against it. Find the rate of the stream. 
 
 20. There are three equal vessels A, B, G ; the first con- 
 tains water, the second brandy, the third brandy and water. 
 If the contents of Ji and G be put together, it is found that the 
 mixture is nine times as strong as if the contents of A and G 
 liad been put together. Find the raLio of the l)randy to the 
 Avater in the vessel G. 
 
 21. A factor buys a certain quantity of wheat which he 
 sells again so as to gain 5 per cent, on his outlay, and thus 
 clears £IG. Had he sold it at a gain of 5.9. a quarter he would 
 have cleared as many pounds as each (piarter cost shillings. 
 How many quarters did he buy, and what did each quarter 
 cost him ? 
 
 22. A man buys a horse and sells it for £144, gaining as 
 much per cent, as the horse cost him. What was the price of 
 the horse 1 
 
 23. I buy goods and sell them again for £96, gaining as 
 much per cent, as the goods cost. AVhat is the cost price 1 
 
 24. A man bought some sheep and sold them again for £24, 
 gaining as much per cent, as the sheej^ cost him. What did he 
 give lor them ? 
 
 25. A certain crew, who row 40 strokes per minute, start 
 at a distance ecpiivalent to four of their own strokes behind 
 niiother crew, who row 45 strokes to the miiuite. In 8 minutes 
 tlu: ibrmer succeed in bumping the latter. Find the ratid 
 lictween the lemrths of the strokes of the two boats. 
 
 26. The time which an express train -takes to travel a 
 journey of 180 miles is to that taken by an ordinary train as 
 !) : 14. The ordinary train loses as much tin)j from stoppages 
 ii.s it would take to travel 30 miles withovt stoj^ping. The 
 express train only loses half as much time as the other in this 
 
 [s.A.] ^ 
 
 •J 
 
 ^ 
 
 :H 
 
 U 
 
 i :■§ 
 
 .f 
 
258 
 
 ON VARIATION-. 
 
 I.;ll 
 
 I <r 
 
 If 4ij; lii !< 
 
 manner, iiiul it also travels 15 miles an limir ijnickti. Snp- 
 ])(»sin;^' till* rates of travelling' nniforni, wliat are lliey in miles 
 j>er lionr \ 
 
 27. An article is sold at a loss of as innoli per cont. as It 
 is worth in j>oiin<ls. Show that it cuiiiiot he sold for more 
 than i;25. 
 
 XXIX. ON VARIATION. 
 
 363. If a snm of money is put out at interest at 5 ppr rent, 
 the principal is 20 times as great as the annual interest, wintl 
 ever the sum may he. 
 
 Hence if a: he the principal, and ?/ the interest, 
 
 f'; = 2()7/. 
 
 Now if we change x we inust change y in the same 'projmr- 
 tion, for so long as the rate of interest remains the same, x 
 will always he 20 times as great as y, and hence if a be 
 doubled or trehled, y will also be doubled or trebled. 
 
 This is an instance of what is called Direct Variation, 
 of which we may give the following definition. 
 
 Def. One quantity y is said to vary directly as anothei 
 quantity x, when y depends on x in such a manner that any 
 increase or decrease made in the value of x produces a proj na- 
 tional increase or decrease in the value oi y. 
 
 364. If x = 'iay, where m is a constant quantity, that is, ii 
 
 <piantity which is not altered by any change in the values of -. 
 
 and y, 
 
 y will vary directly as x. 
 
 For any increase made in the value of x must produce .1 
 proportional increase in the value of y. Thus if x be doublcil. 
 y must also be doubled, to ja'eserve the equality of x and ///;/. j 
 since m cannot be changed. 
 
 365. 
 
 Then 
 and 
 
 So tlij 
 number < 
 
 This if 
 of which 
 
 Def. 
 
 quantity 
 ■increase 
 tional dec 
 
 Vol' iiny 
 portional 
 
 y must be 
 For 
 
 367. 
 
 5n 
 and 10 h 
 
 Tliat is, 
 product ol 
 
 TJiis is 
 the lollow. 
 
 Def. C 
 
 // and :.', ^v. 
 change in 
 
 368. 0] 
 inversely a 
 
 ^k 
 
 I 
 
Variation, 
 
 ON VAIUA TIOX. 
 
 ^59 
 
 
 3(15. Suititooo a iH.'in can reap an acre of corn iu a clay. 
 Tlit'ii 10 iiK-n can i('a[) (10 acres in (5 <layM, 
 and 20 men can reap GO acres in 3 day.s. 
 
 So that to do the same amount oF work if we doufilc the 
 nuniher of men we must halve the number of days. 
 
 Tins is an instance of what is called Inverse Variation, 
 of which we may give tiie following detinition. 
 
 Def. One ([uantity y is said to V((rji inversely as anollicr 
 ([uantity x, when // depends on x in such a manner that any 
 increase or decrease made in the value of x produces a propor- 
 tional decrease or increase in the value of y. 
 
 m 
 
 306. l(x=- , where m is constant, 
 
 y 
 
 y will vary invei'sely as x. 
 
 For any increase made iu the value of y; must ])roduce a pro- 
 portional DECREASE in the value of y. Thus if .o he doubletl, 
 
 Ti) 
 
 y must be halved, to preserve the equality of x and — . 
 
 For 
 
 _ 2m __ m 
 2 
 
 367. If 1 man can reap 1 acre in 1 day^, 
 5 men can reap 20 acres in 4 days, 
 
 and 10 men can reap 80 acres in 8 days. 
 
 That is, th' number of acres reaped will dej^cnd on the 
 product of the . limber of men into the number of days. 
 
 This is an exanplc oH joint variation, of which we may give 
 the following deiinition. 
 
 Def. One quantity x is said to vary jointly as two others 
 // and 2-, Avhen any change made in x produces a proportional 
 change in the product of y and :.'. 
 
 368. One quantity x is said to vary directly as y and 
 inversely as z when x varies as -. 
 
 ■' !» 
 
 II 
 
 
26o 
 
 ON VARIATION. 
 
 V \' 
 
 300. Theorem, if x varies as \] when z is constant, and 
 as % when 2/ is constant, then wlien ij and ?j are Loth variable, 
 
 X varies as ?/«. 
 
 Let x=^m.y^. 
 
 Then we have to show that in is constant. 
 
 Now when z is constant, 
 
 X varies as ?/ ; 
 .•. mz is constant. 
 
 Now z cannot involve y, since ?j is constant when y changes, 
 and therefore m cannot ixivolve y. 
 
 Similarly it may be shown that m cannot involve z ; 
 
 .'. in is constant, 
 
 and X varies as y:i. 
 
 370. The symbcl oc is used to express variation ; thus xocy 
 stands for the words x varies as y. 
 
 37 i. Variation is only an abbreviated form of expressing 
 proportion. 
 
 Tlius when we say that x varies as y, we mean that x bears 
 to y the same ratio that any gi\'en value of x bears to the 
 corresponding value of y, or 
 
 X : y=a given value of x : the corresponding value of?/. 
 
 And similarly f(»r the oilier kinds of Aariation, as will be 
 seen from our examples. 
 
 Ex. 1. Uxocy and y<xz, to show that x'oc;:. 
 Let x—my, and y = nz. 
 
 Then substituting this value oiy in the tirst ecpiation. 
 
 X = mnz ; 
 and therefore, since mn is constant, 
 
 T OC .:;. 
 
ON VARIATION. 
 
 261 
 
 ' expressing 
 
 Ex. 2. If xazy and xocz, then will xcc s,f{yz). 
 Xjet x = my, and x = nz. 
 
 Then x- = mnyz', 
 
 :. y- sj{mn). sliyz). 
 Now ^{mn) is constant ; 
 
 ••• ^°= V(^2). 
 
 Ex. 3. If 1/ vary as a;, and when ic=l, 2/ = 2, what will he 
 the value of y when x = 2l 
 
 Here ?/ : fc= a given value of y : corresponding value of cc; 
 
 /. y : x — 2 : I ; 
 ;. y — 2x. 
 Hence, whf^n ,T = 2, 7/^4. 
 
 Ex. 4. If A vary inversely ns B, and when yl =2, 7?=12, 
 what will B iK'come when .4=9? 
 
 Here A : -,-,= a given Vidue of A : :p . ^-j^ ; 
 
 B correspondnig value ot Jj 
 
 1 
 
 A- ' -^ 
 
 1 
 12' 
 
 Hence, when A = 9, 
 
 whence 
 
 A 2 
 
 9 _2 
 
 12" 5' 
 
 ^=24 = ^-2^ 
 ^ 9 S-'^S* 
 
 Ex. 5. If A vary jointly as B and C, and when A =^C),B=6, 
 and (7=15, find the value of A when 7>'=1() and 6*= 3. 
 
 IT 
 
 ere 
 
 i4 : BG= a giv:»n value of A : correspoiKling vidue oi BC\ 
 
 :. A : yv'^f) : «x 15; 
 
 /. <>n.i-(;/;c'. 
 
 ' 1 i 
 

 , 
 
 i - - 
 
 
 262 ON VARIATION. 
 
 
 1; 
 
 i Hence, when B= 10 and C=3, 
 
 
 90^ = 6x10x3; 
 
 
 ^ 180 ^ 
 
 
 I;^ 
 
 
 Ex. 6. If z vary as x directly and y inversely, and if when 
 2 = 2, a; = 3 and 7/ = 4, what is the value of z when a; = 15 and 
 i/ = 8? 
 
 TT X . IP corresponding value of x 
 
 Here r: : -=a Laven value ot ^ : ^ ,. ^ — . — : 
 
 y corresponding value 01 y 
 
 X 
 
 :. z'.~ 
 
 y 
 
 ~ 4' 
 
 •• 4 
 
 2cc 
 2/ 
 
 md y = 
 
 8, 
 
 ^z 
 4~ 
 
 30 
 
 8' 
 
 .'. 2 = 
 
 120 ^ 
 24=^- 
 
 m' 
 
 Examples.— cxxxiii. 
 
 1. If yl oc — and Boz — then will ^ ocC. 
 
 2. If ^ oc^ then will poc-^y 
 
 3. U AozB and Co: Z) then will yl Coc 5Z). 
 
 4. If CCOCI/, and when x = 7, y = b, tind the vahie of x when 
 4/ = 12. 
 
 5. If xoc--, and when x= 10, t/ = 2, find the value of y when 
 
 x = 4. 
 
ON variation: 
 
 263 
 
 6. If xoc^/;.', and when ;«= 1, ?/ = 2, rj = 3, find the vahie of y 
 when 03 = 4 and ?, = 2. 
 
 if when 
 = 15 and 
 
 of cc 
 ofl/ ' 
 
 f X when 
 
 f y when 
 
 7. If icoc"^^, and when a; = 6, 2/ = 4, and ^ = 3, lind the value 
 of r/j when ^ = 5 and ;:! = 7. 
 
 8. If Sit + 5?/ oc 5a; + 3?/, and when a; = 2, ;/ = 5, tind the value 
 
 (it -. 
 
 ?/ -► 
 
 9. If AocJt and B^ocC^, express how A varies in respect 
 
 of a 
 
 10. If ^ vary conjointly as x and y, and ,^=4 when x=l 
 and 2/ = 2, what will bo the value of a; when z = 'SO and 2/ = 3? 
 
 11. If xiccB, and when yl is 8, /i is 12 ; express A in 
 terms of B. 
 
 12. If the square of x vary as the cuhe of y, and ,r = 3 when 
 ?/— 4, fiiul the etjuation between x and v/. 
 
 13. If the square of x vaiy iuvcrscly as the cube of //, and 
 x — 2 when y = 'i, lind the ec^uation between x and ij. 
 
 14. If the cube of re vary as tlie s([uare of y and x = ',l when 
 y = 2, lind the equation l)et\veen x and y. 
 
 It;. If xoz:: and ycc ., show that a;oc-. 
 ^ ^ z y 
 
 16. Sliow that in triani,de3 of equal area tlie altitudes vary 
 inversely as the bases. 
 
 17. Show that in parallelogranis of equal area the altitudes 
 vary inversely as the liases. 
 
 18. If 2/ = j? + 1? + r, where j is invariable, q varies as x, and 
 r varies as x'^^ tind the ivlation between y and j\ suj)j)osin^' 
 that when a;=l, ?/ = G; when a; = 2, 7/=ll ; and when x = 3, 
 '//=18. 
 
 19. The volume of a jnramid varies jointly as the ana of 
 its base and its altitude. A j /raiuid, the base of which is D 
 
 'i-1 
 
 m 
 
 
i 
 
 n. 
 
 264 
 
 ON- ARITHMETICAL PROGRESS^O?r, 
 
 ^eet ^;quare and the height of which is 10 feet, is foir..d to con- 
 tain 10 cubic yards. What must be tlie height of a pyranau 
 upon a base 3 feet square in order that it may contain 2 cubic 
 yards ? 
 
 20. The amount of glass in a window, tlie panes of which 
 are in ev(.Ty resj^ect equal, varies as the numljer, length, anil 
 breadth of the panes jointly. Show that if thc^'r number varies 
 as the square of their breadth inversely, and their length varies 
 as their breadth invers(d.>^the whole area of glass varies as the 
 square of the length of the panes. 
 
 n 
 
 XXX. ON ARITHMETICAL PROGRESSION. 
 
 \ 
 
 
 372. An Arithmetical Prc>gression i,; a series of 
 
 numbers which increase or decrease oy a constant dilference. 
 Thus, the following series are ARiTHiiETiCAii Progressions : 
 
 2, 4, (;, 8, 10; 
 
 O "7 ^ ""' 1 
 
 tJj <, <->, >>, i. 
 
 The Constant Difference b( in- t in the first scries and -2 
 in the second. 
 
 373. In Algebra we express an Arithmotiral Progression 
 thus : taking a to represent the first term and d to represent 
 the constant dilference, we shall have as a scries of numbers in 
 Arithmetical Progression 
 
 a,a->rd,a + 2d, a + 3r/, 
 
 and so on. 
 
 "We observe that the terms of the series differ only in the 
 cocjjident of d, and that each coefficient of d is always less b}'- 1 
 ihan the number of the term in which that particular ooefficieut 
 p'.in.is. Thus 
 
 the coefiiclr it of d in the 3rd Icnu is 2, 
 
 in the 4th 3, 
 
 in the 51 h 4. 
 
 pla( 
 
:d to con- 
 
 pyraniid 
 
 n 2 cubic 
 
 of which 
 ngth, an>l 
 l)er varies 
 
 gth varies 
 
 les as 
 
 the 
 
 3SION. 
 
 series ol' 
 ference. 
 
 RESSIONS : 
 
 ;s and - 2 
 
 I'ogressioii 
 
 represent 
 
 uiuhers in 
 
 nly in the 
 
 Si less by 1 
 coefficient 
 
 OAT A RIT/IME 77 C A T. PR OCR ESS ION, 26 5 
 
 Consequently the coefficient of il in the 7i"' term will be 
 n- v. 
 
 Therefore the n"* term of the series will betH- (71 - 1; d. 
 
 374. If the series be 
 
 a J a + c?, ft + 2f/., 
 
 and % the last term, the term next before ;:; will clearly be j; - dl, 
 and the term next before it will be % - 2d, and so on. 
 
 Hence, the series written backwards will be 
 
 z, z - d, z - 2d, a + 2d, a + d, a. 
 
 375. To find the suni of a series of numbers in Arithmetical 
 Progression. 
 
 Let a denote the first term. 
 
 ... d the constant ;lirterence. 
 
 ... z the last term. 
 
 ... n the number of terms. 
 
 ... s the sum of the ?i terms. 
 
 Then s = a+((fc4-^) + (a + 2(/) + +{z-2d) + {z-d)+z. 
 
 Also s=z-\- {z - d) + (z - 2d) + +{a + 2d) + {<t + d) + a, 
 
 the series in the second case being the same as in tlie first, but 
 written in the reverse order. 
 
 Therefore, by adding the two series together, wo get 
 
 2s = (a + 2!) + (rt + s) + (^ + ,-2) + +{,i + z)-\-{a-\-z)-\-{a-\-z)\ 
 
 and since on the right-hand side of this e([uation we have a 
 series of n numbers each equal to a + z, we get 
 
 2s— )i {a-\-z) ; 
 
 This result may W: put in another form, because in the 
 place of z Ave may put a v [n- 1) (/, by Article 373. 
 
 Hence ^~,r,\ « + <* + ('i - 1} « i» 
 
 thati.j, -=^j 2a +(«-!)(/;. 
 
 Jl 
 
 J- 
 
 u* 
 
 
ki 
 
 -,: 
 
 ii 
 
 •JS 
 
 ^ 266 ON AKITILJE/nCAL PKOGRESSIOiX. 
 
 376. We have now oljUiiiicd Ihc lollowing r.'.sults : 
 
 % = a-^{n-\)d (A), 
 
 ^-^-^'^ (B), 
 
 s = |{2a + (7?-l)(/| (C). 
 
 From one or more of these ef[nations we liave in Examples 
 to determine the values of a, rf, n, s or z. "We shall now pro- 
 ceed to give instances of such Examples. 
 
 Ex. 1. Find the last term of the series 
 7, 10, 13, to 20 terms. 
 
 Taking the equation r.' = a+ (n- 1) d, 
 for a put 7 and for n put L'O, and \\a get 
 
 ^-7-f-(20-l)c/, 
 or, z=-.7 + l9d. 
 
 Now d ifi always found hy takimj the first term from the second, 
 and in this case, 
 
 ^^ = 10-7 = 3: 
 .-. K = 74- 19x3 = 7 + 57 = 64. 
 
 Ex. 2. Find the la^^t term of the series 
 12,8,4, to 11 terms. 
 
 In the etjuation z = a+ (u - 1) d, 
 
 put a= 12 and 71=11. 
 
 Then ^=12\-10d. 
 
 Now . d = S - 12= ~ 4. 
 Hence » ^ 12 - 40 = - 28. 
 
 EXAMPi.ES.— CXXXiv. 
 
 Find the last term of cadx of the following series : 
 
 1. 2, 5, 8 to 17 terms. 
 
 2. -l:,S,]-2 tu oO terms. 
 
 N( 
 
iw pro- 
 
 e second. 
 
 ON ARITHMETICAL PROGRESSION. 
 
 K 29 15 , -,,. ^ 
 
 3. /, ^^ , -„- to lb terms. 
 
 1 5 
 
 4, „, -1, - to 23 terms. 
 
 $• -^, 2, (j to 12 terms. 
 
 6. -12, -8, -4 to 14 terms. 
 
 7. —3, 5, 13 to l(j terms. 
 
 o ?^ - 1 71 - 2 w - 3 
 
 8. , , to 71 terms. 
 
 71 n n 
 
 9. (,0 + 7/)-, :'/- f 7/-, (3J - 7/)" to 71 terms. 
 
 a - h 4/t - ?ih la — 56 
 
 10. ,, , , , to 71 terms. 
 
 a+b a+b ^ a+b 
 
 377. Ex. 1. Find the sum oi" the series 
 3,5,7 to 12 terms. 
 
 In the equation s = ~\2a+{n-l) d\ 
 
 put 3 for a and 12 for n, and we get 
 
 Now d — 5 -3 = 2, and so 
 
 i,=.^J^}64-22| =6x28 = 168. 
 
 Ex. 2. Find the >suni of tlie scries 
 
 10, 7, 4 to 10 terms. 
 
 ■. =|}2rt + (/i-l)(i|; 
 
 I'lit 10 for a and 10 for n, then 
 
 « = -^ }20 + 9iZ}. 
 
 2O7 
 
 I 
 
 ' -.1 -. 
 i « 1 ^i 
 
 1^' 
 
 1^ 
 
 < M 
 
 ia 
 
 ii' ' } 
 
 i ml 
 
 "iff 
 
 'I 
 
 Ifitl 
 
 ■^■^f 
 
26S 
 
 ON ARITHMETICAL PROGRESSION. 
 
 i I 'I 
 
 i ;.3 
 
 P-> :IS 
 
 H^ 
 
 I* 
 
 HiHii 
 
 ill 
 
 r 
 
 
 Now d = 7 - 10 = -3, ami therefore 
 
 10, 
 
 s=V--|20-27{=5x(- 7)= -35. 
 
 Examples.— cxxxv. 
 
 Find the sum of the following series : 
 
 I. 1, 2, 3 to 100 terms. 
 
 7. 2, 4, G to 50 terms. 
 
 3. 3, 7, 11 to 20 terms. 
 
 4. -, -, -- to 15 terms, 
 
 4' 2 4 
 
 5. -9,-7,-5 to 12 terms. 
 
 6. ^, 2» ~ to 17 terms. 
 
 7. 1, 2, 3 to n tcms. 
 
 8. 1, 4, 7 to n terms. 
 
 9. 1, 8, 15 to n terms. 
 
 10. , — — , — - to w terms. 
 
 71 U 11 
 
 378. Ex. What is the Constant Difference when the 
 first term is 24 and the tenth term is - 12? 
 
 Taking the equation (A), 
 
 z=a + {n-~l)d, 
 and regarding the tenth as the last term, we get 
 
 -12 = 24 + (10-1)(?, 
 or - 36 = H 
 
 whence we obtain d= -4. 
 
 5. 
 
 and 
 
ON A RITHME TIC A L PRO CRESS TON. 269 
 
 E when the 
 
 Examples, cxxxvi. 
 
 What is tlie Constant Difference in the following cases? 
 I. When the first tenn is 100 and tlie twentictli is - 14. 
 2 .^; fifty-first is -x. 
 
 3 — - forty-ninth is 5-. 
 
 4 — ^ twenty-fifth is -21*^. 
 
 5 -10 sixth is -20. 
 
 6 150 ninety-first is 0. 
 
 379. Ex. What is the First Term vvlien 
 
 the 40tli term is 28 and the 43rd term is 32 ? 
 
 Taking ef^nation (A), 
 
 and regarding the last term to he the 40th, we gcit 
 
 28 = «-f-.39(/ (1). 
 
 Again, regarding the last term to l)e the l:^*rd, we got 
 
 32 = rt + 42J (r2). 
 
 From eqnations (1) and (2) we may lind the value of a to 
 be -24. * 
 
 Examples,— cxxxvii. 
 
 I, What is the first term when 
 (i) The 59th term is 70 and the 66th term is 84; 
 
 (2) The 20th term is 93 - 356 and the 21st is 98 - 37/> ; 
 
 (3) The second term is - and the 55th is 5-8 ; 
 
 (4) The second term is 4 and the 87th is -30 ? 
 
 
 m 
 
 ^.A 
 
 \ 'M 
 
 
r t 
 
 1:/ 1^ f'. 
 
 '^I" 
 
 iri» 
 
 I .'fl 
 
 m 
 
 270 ON ARITHMETICAL PROGRESSION'. 
 
 2. The sum of llie 3r(l and 8th terina of a scries is 31, and 
 the sum of the 5th and lOth terms is 43. Find the sum of 
 10 terms. 
 
 3. The sum of llie 1st and 3rd terms of a series is 0, and 
 the sum of the 2ud and 7th terms is 40. Find the sum of 
 7 terms. 
 
 4. If 24 and 33 he the fourth and fifth ' rms of a series, 
 wlmt is the lootli term ? 
 
 5. Of liow many terms does an Arithmetical Proj^ression 
 consist, wliose difference is 3, first term 5 and hist term 302 ? 
 
 6. Supposing that a body falls through a space of 16=^ feet 
 in the first second of its fall, and in each succeeding second 
 .32- feet more than in the next preceding one, how far will a 
 body fall in 20 seconds ? 
 
 7. Whiit debt can be discharged in a year by weekly pay- 
 ments in arithmetical progression ; tlie first payment being 1 
 shilling and the last £b. 3?. ? , 
 
 8. Find the 41st term and the sum of 41 terms in each of 
 the following series : 
 
 (i) -5,4, 13 
 
 (2) ^a\ 0, -4ft''^ 
 
 (3) l + a, 5 + 3.T, 9 + 5a; 
 
 (4) -% --1'4. 
 
 ^5^ 4' 20 
 
 9. To how many terms do the following series extend, and 
 what is the sum of all the terins ? 
 
 (i) 1002 10,2. 
 
 (2) -6,2 ,186, 
 
 381. 
 
 Tlie 
 are to 
 term h\ 
 
 Takii 
 We hav 
 
 
 J^ M 
 
\l, and 
 sum of 
 
 0, and 
 sum of 
 
 L series, 
 
 crression 
 
 na 
 
 302? 
 
 L6^ feet 
 rr second 
 ar will a 
 
 Bkly pay- 
 Ijcing 1 
 
 eacli of 
 
 xtend, and 
 
 ON- ARITHMETICAL PROGRESS/ON, 271 
 
 (3) 2^.x, -Sic -nzx, 
 
 (4) |, ^.....-24 
 
 (5) w-1 137(1 -w), 13Pi;i-m). 
 
 (6) a;4-254, x + 2,x-2. 
 
 380. To insert 3 arithmetic means between 2 «'>)rZ 10. 
 
 The number of terms will he 5. 
 
 Taking the equation z = a^-{ii- 1) d, 
 
 we have IO = 2 + (5-l)(i 
 
 Whence 8 = 4(^; .-. d=2. 
 
 Hence the series will be 
 
 2. 4, 6, 8, 10. 
 
 EXAMPLKS. — cxxxviii. 
 T. Insert 4 arithmetic means between 3 and 18. 
 
 2. Insert 5 aritlmietic means between 2 and -2. 
 
 3. Insert 3 aritlnnetic means between 3 and .-. 
 
 4. Insert 4 arithmetic means between - and }. 
 
 2i 3 
 
 381. To insert 3 arithmetic means hetiveen a and b. 
 
 The number of terms in the series will be 5, since there 
 are to be 3 terms in addition to the first term a and tlu; last 
 term h. 
 
 Taking the equation 'ri=a-\-{n—l) d, 
 we have to find d, luning given 
 
 a, 2 = 6 and 7i = 5. 
 
 I W'l 
 
 ! 1 ' ; ! 
 
 ■ 1 ".' 
 
 1 ■ 
 
 < ' H 
 
 ti/ 
 
 J 
 
 
 If 
 
 
 !' 'if' 
 
 Pill] 
 

 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 /. 
 
 
 1.0 
 
 I.I 
 
 12.8 
 
 
 ^^ 
 
 
 2.2 
 
 ui 114 
 
 ^ y£ 12.0 
 
 I 
 
 ifi. 
 
 
 '•25ll'-^ 1'-^ 
 
 
 < 
 
 6" 
 
 ► 
 
 Photographic 
 
 Sdences 
 
 Corporation 
 
 33 WEST MAIN STREET 
 
 WEBSTER, N.Y. 145S0 
 
 (716) S73-4S03 
 
o 
 
 ■0 ^° J%i 
 
 i/.. 
 
 
n 
 
 
 i*'i 
 
 I 
 
 'I 
 
 i 
 
 I 
 
 I* 
 
 ^^ 
 
 £72 
 
 OJV ARITHMETICAL PROGRESS/ON 
 
 Hence 
 
 or, 
 
 h = a + ip-\)dy 
 
 4(1 = b- a, .'. d — — r" - 
 ' 4 
 
 Hence the series ^vill be 
 
 h~a b — a Pj(h — a)j 
 
 that 13, «, — 4— — 2", ~ 4"'^- 
 
 Examples.— cxxxix. 
 
 1. Insert 3 arithmetic means between 7??. and n. 
 
 2. Insert 4 aritlimetic means l^etween m + 1 and m - 1. 
 
 3. Insert 4 arithmetic means between n- and ??-+ 1. 
 
 4. Insert 3 arithmetic means between cc- + 1/" '*^^^*^ ^' ~ IJ'- 
 
 382. We shall now f^ive the c^eneral form of the ])rop()Riti(»ii 
 " To inacrt m arithmetic means between a and b." 
 
 series 
 
 The number of terms in the 
 
 will he ?// f 2 
 
 Then tidcinj^' the etpiation :: — a + {n—\)d^ 
 we have in this case b = a-\- {m 4- 2 - 1) c^, 
 
 or. 
 
 ft = rt4-(m + l)fZ. 
 
 Hence 
 
 rf = 
 
 b — a 
 m + V 
 
 and the form of the series will be 
 
 a, a + 
 
 a 
 
 b- 
 m+ I 
 
 a + - 
 
 '2b - 2a 
 
 o;, 9 
 
 » - -jia 
 
 m + 1 
 
 /// 
 
 + 1 
 
 h- 
 
 b-a 
 m+1 
 
 > "> 
 
 that 
 
 IS, 
 
 a. 
 
 am + ft am - a + :2ft 
 
 m+i 
 
 m + 1 
 
 
 bvi + 
 
 a 
 
 i ^< 
 
 3J 
 thus 
 the 
 Ceo 
 
 iiide.i 
 num 
 
 Til 
 
 b 
 
7/1- 
 
 1. 
 
 \-l. 
 
 
 , X- 
 
 -r 
 
 tositioii 
 
 ,^ 
 
 6. 
 
 XXXI. ON GEOMETRICAL PROGRESSION. 
 
 383. A Geometrical Proj^iossion is a series of numbers 
 wliicli increase or decrecase hy a constant factor. 
 
 Thus the following series are Geometrical Puogiiejjcjion.s, 
 
 2, 4, 8, 1(), 32, (J4; 
 
 ' ' i' KJ' 04' 
 
 4, 
 
 111 1 
 
 2' l(i' 128' 1021" 
 
 The Constant Factors being 2 in the first series, . in the 
 
 4 
 
 second, and — - in the third. 
 
 8 
 
 Note. That which we shall call the Constant Factor is 
 usually called the Common Ratio. 
 
 384. In Algeln-a we express a Geometrical Progi*ession 
 thus : taking a to represent the rtrst term and / to represent 
 the Constant Factor, we shall have as a series of numbers in 
 Geometrical Pro.u'ression 
 
 (t, <if, af-j <ip^ and so on. 
 
 'o 
 
 AVe observe that the terms of tlic series dilU-r only in the 
 indcc of/, and that each index of/ is always les.s by 1 tlian the 
 number of the term in which that particuUir index stands. 
 
 Thus the index of/ in the 3rd term is 2, 
 
 in the 4th 3, 
 
 in the nth 4. 
 
 Consequently the index of/ in the 7ith term will Ije n- 1. 
 
 Therefore the 7tth term of the series will be fi/" '. 
 
 [S.A.J 
 
 s 
 
 if 
 
 1 
 
 \ 
 i 
 
 1 1 
 
 Ti 
 
 
 \ 
 
 ■ i 
 
 
 1 
 
 t 
 
 > 
 
 it 
 
u 
 
 274 
 
 OJV GEOMETRICAL rKOGk'ESSIOiV. 
 
 Jleiice if;; be tliu last Uiui, 
 
 385. If tli(i .series contain n tonus, a being the first term 
 and /tile CoJistant Factor, 
 
 the last term will be a/""^', 
 
 the last term bnt one will be aj'''^ 
 
 the last term bnt two \\\\\ be af"-^\ 
 
 Now r//"-» X /= af" ^xf = ((/"- 1+' = r//", 
 
 ('/"-' >^f=of" ' xf = af"--^' = af"-\ 
 
 «/""^ x/=«/"-^ x/' = «/« -^'^ • = a/"- 2. 
 
 386. AVe may now proceed to find the nam of a scries of 
 numbers in Geometrical Progression. 
 
 Let a deuolc the tirst term, 
 
 / the constant factor, 
 
 71 the number of terms, 
 
 » the sum of the n terms. 
 
 Then s = tH- af+ af-\-...+ af'-^ + a/" ^ + af"-\ 
 
 Now multiply both sides of this eijualiun by/, then 
 
 fs = af+ af + af-\-...-\- af*-"- + af" - » + a/". 
 
 Hence, subtracting the first eciuution from the second, 
 
 fs-s = af"~a. ! ri .X" 
 
 ••• «(/-!) =«(/"-!); 
 
 1^ 
 
 b 
 
 air- 
 ■ / 
 
 :-[''■ I 
 
 Note. The proposition just proved presents a difficulty to 
 a beginner, which we shall endeavour to exi)lain. When wo 
 multiply the series of n terms 
 
 a + af+af-\- + af"-^ + af'*-^ + af''-^ 
 
lilty to 
 lieu wc 
 
 OiV GEOMETRICAL PK OCR ESS/OX. 
 
 275 
 
 l>y/, we sliall obtain anotlier series 
 
 afv ap + ap i -1 «/" '^ + <//" " ' h- uJ\ 
 
 which also eon tains n terms. 
 
 Tlion^'li wc cannot lill u]) llie rrap in each scM'ies comphtely, 
 we see tliat the tiivnis in the two seiits must he tlie same, 
 exce])t ihiijiriit term in the former series, aii<l the htst term in 
 the latter. Hence, when we tiubtract, all the terms will ilis- 
 ai>pear except these two. 
 
 387. From the ibrmula) 
 
 / 
 
 ■ t( 
 
 (( 
 
 'f 
 
 II -1 
 
 7-1 
 
 .(A), 
 
 prove 
 
 the followiiif' 
 
 (a) s^l^^. 
 
 (7) «=/v-(/-l).9. 
 
 s-a 
 
 w /=h" 
 
 388. Ex. Find the last term of the series 
 3, ('), 12 to 9 terms. 
 
 The Constant Factor is ;„ that is, 2. 
 
 In the formula 
 
 '-ap-\ 
 
 putting 3 for a, 2 for/, and 9 for n, we get 
 
 - = .}x 2^ = 3 X 250 = 7(!8. 
 
 Examples.— cxl. 
 
 Find the last term of the following series 
 
 1. 1, 2, 4 to 7 terms. 
 
 2. 4, 12, 3() to 10 terms. 
 
 3. n, 20, 80 to I) teruKs. 
 
 : 
 
 I i 
 
 I 
 
 lit # 
 
 I 
 
 I ;i 
 
 t! 
 
 '^ 
 
 lit 
 
 I:' ' 
 
 3» 
 
 !&■ i : \ 
 
 ' ', :,! 
 
 i 
 
 ,« 
 

 ■r 
 
 I st 
 
 iiji 
 'I 
 
 finil 
 
 iii^ 
 
 276 OJV GEOMETRICAL PROGRESS/OJST. 
 
 4. 8, 4, 2 to 15 terms. 
 
 5, 2, fl, IS to 9 terms. 
 
 ^; M- iV 4 to 11 torn.. 
 
 2 1 1 
 7. - , -, - to 7 terms. 
 
 389. Ex. Find thf; SUM of the series 
 
 o 
 
 G, 3, '^ to 8 terms. 
 
 Genercally, .s=^ - /'- 1 
 
 anf'l liere rt = f>,/=2' '"~'^' 
 
 •• '~ 1_ ' ■ _1 
 
 2 2 
 
 25 () _ ' 250 7()5 
 '"1 "" 1 ""64* 
 '2 2 
 
 . EXAMPLES.— CXli. 
 Fin<l the sum of the f()lh-)wiiig series : 
 
 1. 2, 4, 8 to 15 terms. 
 
 2. 1 , 3, i) to () terms. 
 
 3. a, ax^, ax^ to 1.3 termR. 
 
 4. rt, -, -., to 9 terms 
 
 ^ ' 0: x^ 
 
 c. a^-x^, a- X, - to 7 terms, 
 
 -' ' a + x 
 
ON CEOMETR/CAL PROGRESS/OiV. 277 
 
 6. 2, G, 18 to 71 ttTins. 
 
 7. 7, 14, 28 to 71 terms. 
 
 8. 5, -10, 20 to 8 terms. 
 
 2 1 1 
 
 9. -.,, Tj, ~t to 7 tirms. 
 
 390. To tind tlio sum of an lufmite Series in Geometrical 
 Piogression, when the Constiint Factor is a projier traction. 
 
 Ii'/T)e a i)roper fraction an<l n very larj^'e, 
 /" is a very snjall num})er. 
 
 Hence it" the number of terms he infinite,/" is so small that 
 we may neglect it in the expression 
 
 /-I, ' 
 and we get 
 
 -a 
 _ a 
 
 ""1-7 
 
 391. Ex. 1. Find the sum of the series „ + 1 + 7 + to 
 
 infinity. 
 
 Here /=.l-f-.^ = ^; 
 
 4 
 __rt :)_ HJ_^1 
 
 •*-*~i-/~7^"^ 3 ~^V 
 
 3 2 8 
 Ex. 2. Sum to infinity the series o - ■; + 07 ~ 
 
 TT .234 
 
 Here /=-.^^^=-^^; 
 
 3 3 
 a 2 2 27 
 
 
 
 
 . 
 
 M 
 
 >'i ■' 
 
 
 h'i' 
 
278 
 
 ON GEOMETRICAL PROGRESSION. 
 
 
 T 
 
 Examples.— cxlii. 
 
 Find the sum of the followiii'' iiilinite scries 
 
 1 1 
 
 ' 2' -1' 
 
 1 1 
 
 4' l(J 
 
 9. 4^ 2S 
 
 lo. 2x'', - '250;, 
 
 3 
 
 1 1 
 
 ' 3' 27' 
 
 II. (/ 
 
 > "> 
 
 2 1 1 
 3' 3' 6' 
 
 3 1 
 
 12. 
 
 10' 10'" 
 
 » /I' 
 
 4' 4 
 
 13- a^, -2/> 
 
 8 
 
 > QJ 
 
 14. 
 
 15. 
 
 ^6_ ^6^ 
 100' 10000' 
 
 •54444 
 
 8. 1^, -5 
 
 > "J 
 
 16. -83636, 
 
 392. To insert 3 geometric means hchveen 10 and 160. 
 
 Taking the equation ;:;; = ^'/"~S 
 we put 10 for a, 160 for '4, and 5 for n, and we obtain 
 
 160 = 10./'-^; 
 
 .-. 16=/\ 
 
 • 
 
 Now 16 = 2x2x2x2 = 2*; 
 
 .-. 2*=/*. 
 
 Hence /= 2, and the series will be 
 
 10, 20, 40, SO, 160. 
 
 or. 
 
 or. 
 
ON GEOMETRICAL /'A'0(JA*ESS/OA^. 
 
 279 
 
 m 
 
 Examples.— cxliii. 
 
 1. Insert 3 geometric iiieauH bet ween 3 and 243. 
 
 2. Insert 4 geometric metmH Initwcen 1 and 1021. 
 
 3. Insert 3 geometric means between 1 and 10. 
 
 I 243 
 
 4. Insert 4 geometric means between j, and '. 
 
 393. To inncrt m (jcometric means between a uiui )., 
 TI13 number ot" terms in the series will be ?/H-2, 
 In the formula ~ = <'/""*» 
 
 putting 6 for «;, and ?/i + 2 lor n, we get 
 
 or. 
 
 or. 
 
 5 = (;/"'+l; 
 
 Hence the series will be, 
 -i 2 
 
 rt, ax — I , a X .^- > 
 
 ,m+i 
 
 ,«fi 
 
 » 1 
 
 /)-: — — n-i- 7. 
 
 a'"^" a " a'"'* a" 
 
 that is, 
 
 1- _»_ » 1 
 
 a, (a"'. ?))•"+', (a"'-^ ?>2)m+i^ ^ {a- .h'^-y^\ ((6.6'")""', />. 
 
 394. AVe hall now give some mixed Examples on Aritli- 
 metical and Geometrical Progression. 
 
 i"1 
 
 Ex AMPLES.— CXliv. 
 
 I. Sum the foUowing series: 
 
 (i) 8+15 + 22+ tol2termt^. 
 
 (2) 1 10 + 108 + 100 + to 10 terras. 
 
 4 
 
i§o 
 
 ON GEOi'ifETRlCAL rROCRESSIOM. 
 
 (3 
 (4 
 (5 
 
 (6; 
 
 (7 
 (8 
 
 (9: 
 (10: 
 
 (12 
 
 li H- , I- ,1- to iuliuiLv. 
 
 '2, \'l 
 
 to infinity. 
 
 to 13 terms. 
 
 to G terms. 
 
 — to :2J) terms. 
 
 1 211 
 
 2 3 G 
 
 \ \ 2 
 
 :i :3 ' 1) ~ 
 
 
 5 "2 
 
 / 7 
 
 to 8 ttjrms. 
 
 •J + a + o- + *o iiiiinity. 
 
 3 _ 14 51 
 5 10 15 
 
 to 10 terras. 
 
 ^?- s'G + 2V(l.'^)- to 8 
 
 terms. 
 
 -5+2-4 + 
 
 to 5 terms. 
 
 2. If the coTitinued product of 5 terms in Geometrical 
 Progression be 32, show that the middle term is 2. 
 
 3. If rt, h^ c are in aiithmetic progression, and a, b', c are 
 in geometrical progression, show that ,-, = s- ./-r. 
 
 4. Sliow that the arithmetical mean between a and h is 
 greater than Uie geometrical mean. 
 
 5. The sum of the first three terms of an arithmetic series 
 is 12, and the sixth term is 12 also. Find the sum of the first 
 6 terms. 
 
 6. What is necessary that ct, h, c may be in geometric pro- 
 gression? 
 
 8. 
 
 9- 
 1, con 
 
 of an J 
 
 difFere 
 
 gressic 
 
 10. 
 make 
 
 II. 
 
 series ] 
 
 12. 
 
 «ion be 
 
 13- 
 3„, the 
 
 22. R^ 
 
 14. 
 
 progress 
 in the 1 
 subtract 
 Require 
 
 15. 
 and the 
 
 16. 
 arithmct 
 numbers 
 
 17. f 
 
 18. T 
 tbe const 
 number 
 
 19. P 
 
ON GEOMETRICAL FROGRESSION, 
 
 2Sr 
 
 7. If 'In. X and ^^ aie in fjoomotrio pro^Tossion, what is a? 
 
 i:» 
 
 8. ir '2n, 7/ and ,, arc in arillinu'tic iJronrcs.sion, wlial is )i( 
 
 Zll. 
 
 9. Tlie sum of a <,'e()nietric pro^^Tcssion wlioso first term i^ 
 1, constant factor 3, and nunilKT of terms 4, is (-([ual to the sum 
 of an arithmetic progression, whose first term is 4 and constant 
 difference 4 : how many terms are there in the aritlimetic pro- 
 gression ? 
 
 10. The first (7 + ?i) natural nundurs when achletl togetlier 
 make 153. Find n. 
 
 11. Prove that the sum of any number of terms of the 
 series 1, 3, 5, is the square of tlie nund)or of terms. 
 
 12. It the sum of a series of 5 terms in arithmetic prot^res- 
 sion be^05, sliow that the midcUe term is 19. 
 
 13. There is an arithmetical progression whose first term is 
 
 1 . 4 
 
 .3„, the constant difference is 1-, and the sum of tlie terms is 
 
 22. Required the number of terms. 
 
 14. Tlie 3 digits of a certain nundjer are in arithmetical 
 progression ; if the number be divided by the sum of the digits 
 in the units' and tens' place, the quotient is 1()7. If 396 be 
 subtracted from the number, its digits will be inverted. 
 Required the number. 
 
 15. If the (i? + ?)*'* term of a geometric progression be 7n, 
 and the (2^ — <?)**' term be n, show that the p*^ term is ^/(m»). 
 
 16. The difference betw^een two numbers is 48, and the 
 arithmetic mean exceeds the geometric by 18. Find th«* 
 numbers. 
 
 17. Place three arithmetic means between 1 and 11. 
 
 18. The first term of .an increasing arithmetic series is •034, 
 the constant difference •(M.)04, and the sum 2"748. Find the 
 number of terms. 
 
 ^' 
 
 ) ' t 
 
 19. Place nine arithmetic means between 1 and - 1, 
 
t 
 
 282 
 
 OAT HARMONICA!. PROCRESS/ON 
 
 20. I'rove tliiit every term f»r tlie series 1, 2, 4, i^ 
 
 greater l)y unity tliiui tlio suiii of all tl)iit jjrccede il. 
 
 21. Sliow tiiiit if a scries of lapXi-vuxH foriiiiiif^a fjeometrical 
 ]iro<,'ression wiiose coiistiuit factor is r lie divided into sets of jt 
 consecutive terms, the sums of tlie sets will form a geometrical 
 progression whose (M)nstant factor is ?•''. 
 
 22. Find five numhers in arithmetical progression, such 
 that their sum is 55, and the sum of their squares 7G5. 
 
 23. In a geometrical progression of 5 terms the difference 
 of the extremes is to the dili'erence of tlie 2iid and 4th terms 
 as 10 to ?>, and the sum of the 2nd an<l 4th terms ecjuals twice 
 the product of the 1st and 2nd. Find the series. 
 
 24. Show that the amounts of a sum of money put out at 
 Compound Interest form a series in geomt^tiical pn\iycssion. 
 
 25. A certain number consists of three digits in geometrical 
 progression. The sum of the digits is 13, and if 71^)2 he added 
 to the number, the digits will be inverted. Find the number. 
 
 26. The ])opulation of a county increases in 4 ycarsi from 
 lOOOU to 14641 ; Avluit is the rate of increase ] 
 
 refore 
 and ti 
 
 3J)G. 
 Sine 
 
 or 
 
 or 
 
 or 
 
 397. 
 
 First 
 
 Proce 
 
 I n 
 
 XXXII. ON HARMONICAL PROGRESSION. 
 
 .395. A Harmonicai Progression is a series of mmibcis 
 of which the reciprocals form an Arithmetical Progression. 
 
 Thus the r^eries of nundiers a, /*, r, J, is a IIaumontcai, 
 
 . 1 1 • 1 • * •., , 
 
 PiioGUKSHioN, it the series , , , , „ is an Anthmctual 
 ' a h c a 
 
 Progression. 
 
 Jf a, h, c be in ITarmonical Progression, h is called the 
 JIarmonical Mean between a and c. 
 
 Note. There is no way of finding a general expression f^r 
 the sum of a llarinonical Series, but many proble/us with 
 
 or 
 
 Hence 
 
 1 1 
 
 a a 
 
 Theref 
 
i, 
 
 1' 
 
 nnietriciil 
 ( sets (»r i> 
 ^ometrical 
 
 Acm, such 
 15. 
 
 «li(lVrcm'*- 
 
 4tli terms 
 
 [U.'ils twict' 
 
 •j)ut out at 
 itycssion. 
 
 iTt'oinetricnl 
 )2 l)e a<l(U'<l 
 liu nuiii1>^r. 
 
 ycara froia 
 
 :ssiON. 
 
 (of mmilni^ 
 
 'aumonicai. 
 .ritlinictic-il 
 
 called tlie 
 
 ll)leius with 
 
 OAT IIARMOXICAT. PROGRESSION. 283 
 
 H'fereiiee to wirli a series nmybo solved \\\ inverlliiL,' the teiius 
 and trcutiug the reciprocals a.s an A.itbiiietical Series. 
 
 3!)G. 7/ a, b, c he in Harmnuical Pmiression^ to xhoiv that 
 
 <( : (' :: a h -, Ji— ,*. 
 
 Since , ,, - are in Arithiuotical Proj-'ressiou, 
 
 tt be o » 
 
 11 11 
 
 h - /■ a — /) 
 
 or / J » 
 
 he ah 
 
 oh (I - h 
 
 or ; ^ ; . 
 
 fir h~i) 
 
 (( a - h 
 or = f — . 
 
 c b-c 
 
 397. To insert m'harmonic means between a and b. 
 
 First to insert w arithmetic means between and t. 
 
 <t 
 
 Proceeding as in Art. 357, we liave 
 
 a 
 
 or « = /> + (?« -t- \).abd 
 
 ' 1- J*"l„ 
 ah{m + \) 
 
 Hence the arithmetic serii's will be 
 
 1 1 a-h 1 2("-J')_ I ilO^-^) 1 
 
 «' (t f<ft(7/i-Hl)' a ah{iii+\y a ah{mriy 6' 
 
 1 /)>/i + a ////(, + 2a - /;• «//i + h ] 
 
 or — _ 
 
 Therefore the Harmonic Serii^s is 
 
 ah{ m+\) ah{m+l) ah(m + l) , 
 
 ' bill -ill ' bm-r2a-b^ am + b ' * 
 
 ll 
 
 » 
 
 ^1' 
 
 \h 
 
 % 
 
 ! ^ 
 
 H ^ 
 
 II 
 
284 
 
 ON IIARMONICAL PROGRESSION. 
 
 398. riiveii a aii<l h tho fir.^t two terms of a scries in Har- 
 monicul Proj^n-eBsion, to tiud the n"' ti'ini. 
 
 i - are tlie first two terms of an Aritlmetical Series of 
 a' h 
 
 which the common (litrerence is ^-^^. 
 
 The if" term of this Arithmetical Series is 
 
 1 {n - 1 ) (rt - l>) _ h + w^ -n-n h + h 
 
 __ (na-al- ( ?ib - 2 />) ^ (n -J) a -{n -2)6 
 ~ ~ah ah ' 
 
 ;. the ?i"' term of the llannonical Series is 
 
 (^Trrt-(''-2)6' » 
 
 399. Let a ami r l»e any two nnmhers, 
 
 h the Harmonical :^[ean hetween them. 
 
 Then 
 
 or 
 
 i_i_i_i 
 
 h a c 6' 
 
 6" acf' 
 
 .-. 6- 
 
 2a r 
 a 4- (** 
 
 Henc 
 re«2}ecli 
 
 tliat is, ( 
 
 401. 
 ina<,fjiitui 
 
 Since i 
 
 or 
 or 
 
 or 
 
 «.i"iclt IS. A 
 
 Also, si 
 
 400. The following resnlts shonhl ho rememhered. 
 
 a + c 
 
 Arithmetical Mean befwe.-n a and r= ^- 
 
 Geoinetrical Mean hetwicn a :in<l c-^ s^ac. 
 
 2ctc 
 Harmonical Aft^an hotween a and c = -■--' 
 
 III T C 
 
 3- 
 
 4. 
 
ON JIARMONICAL PROGRESSION. 
 
 285 
 
 Hence if Ave denote the Means by the letters li, G, H 
 
 respectively, 
 
 A X II-. 
 
 a + c Z((c 
 
 ^— X 
 
 a + 
 
 etc 
 
 -G"-^ 
 
 that is, is a mean proportional l>etween A an<l 11. 
 
 401. 
 
 niaunitmlc 
 
 o snow 
 
 tliat A, (I, H art' in deseendinu' order of 
 
 '5 ' ■) 
 
 Since ( v^(( - s,fc)'^ must he a positive quantity 
 ( \^" - Vc)'- is <,frcater than 0, 
 
 or 
 
 or 
 
 ur 
 
 (t - 2 ^lac + c greater tlian ' K 
 a + c greater than 2 J<^c 
 
 v<-n-, 
 
 a ^- c 
 
 greater than tjac. 
 
 
 > Ij 
 
 ilctl/ 
 
 is, A is greater than G. 
 
 Also, since a -h <.• is (at-ater than 2 x^ 
 
 V'<'') 
 
 /s/ac {a + c) is greater than 2ac 
 
 ijac is greater than 
 
 a -ft' 
 
 i.(^. G is greater than //. 
 
 
 Examples.— cxlv. 
 
 I. Insert two harmonic nu\uis hetween (5 and 24. 
 
 3 i'oiir 2 and 3. 
 
 3 tiiree and . 
 
 4 lour and ,-„. 
 
 i fin 
 
 I 1 
 
286 
 
 ON ITARMONICAL PROGRESSION. 
 
 5. Insert five harmonic means between — 1 ar i 2~^ 
 6 five 
 
 ^ 1 1 
 
 7- 
 8. 
 
 six 3 and _. . 
 
 n 2.0 and 3//. 
 
 11 
 
 Q. Tlje sum of throe terms of a liarmonical series is ,>,,and 
 ^ 12 
 
 the first term is .^ : find the series, and continue it both ways. 
 
 10. The arithmetical mean between two numbers exceeds 
 the ^'eometrical l)y i:', and the ^geometrical exceeds tlie har- 
 monical by 12. What are tlie numbers \ 
 
 11. There are four numbers (i, />, f, </, the first three in 
 arithmetical, the last three in liarmonical progression ; show 
 that a\h=-c\ d. 
 
 12. If X is the harmonic mean between m and ?«., show that 
 
 1 111 
 
 + = - +-. 
 
 x-tii x-n in, n 
 
 13. The sum of three terms of a harmonic series is 11, and 
 the sum of their s(|uarcs is 49 ; hud tlie numbers. 
 
 14. If X, y, '^ be the y/*", (/"', and f^ terms of a H.P., show 
 that (r - 7) //:; ■\- {}) - r) xz + (q-p) xij = 0. 
 
 15. If the H.M. between each pair of the nund)ers, a, h, c 
 be in A.P., then b-, a', c- will be in h.p. : and if the h.m. be in 
 H.P., b, a, c will be in h.p. 
 
 r>n ^1 ^ f 4- 2(t C 4- 26 , H, ,rK T 
 
 16. Show that - . -i =4, >7, or >10, according as 
 
 t' U C it 
 
 c is the A., G. 01* H. mean between a and b. 
 
 402. 
 cession 
 called J 
 
 • TIiu.s 
 nialce th 
 then on 
 
 If In 
 mutatioi 
 
 I 
 
 403. 
 are taker 
 permutat 
 certain m 
 
 Thus t 
 f\ Q, am 
 
 404. 
 taken r at 
 
 Let a, 
 
 First t( 
 'a ken tioo 
 
 If a 1)0 
 ^vhich the 
 in wliicli 
 
XXXin. PERMUTATIONS. 
 
 (MS 
 
 :ti 
 
 h ways. 
 
 i exceeds 
 the liar- 
 
 tliree in 
 311; sliow 
 
 =liow that 
 
 is 11, and 
 p., show 
 
 ers 
 
 s, (I, h, c 
 
 H.M. hi.' in 
 
 :ortli"g as 
 
 402. The dillereuf. JMTiuij^cfuciits in rrsju'ct of ordiT of siic- 
 CL'Ssion wiiich ciiii he made of u given uuiuber of tilings are 
 
 called Permutations. 
 
 Thus if from a hu.v of letters I .sidect tvo^ J' and Q, I can 
 malce two peiniutations of lliem,phicing P fir.st on the k-ft and 
 then on tlie right of Q, thus: 
 
 r, Q and (,), 1\ 
 
 If I now take three h'ttcr.s, 1\ (^ and li, I can make sic jier- 
 niutationn of tliem, thus : 
 
 F, Q, li ; P, R, Q, two in wliich P stands iirst. 
 
 Q,]\n', Q,P,I', Q 
 
 E,P,Q; E,Q,P, K 
 
 403. hi tlic Examples just given all the things in each case 
 are taken togetlicr ; hut we may be recjuired to iind how many 
 permutations can he made out of a number of things, when a 
 certain nuiiiber onlij of them are taken at a time. 
 
 Thus the permutations that can he formed out of the letters 
 /*, Q, and 11 taken Iko at a tunc are six in numher, thus: 
 P, Q ; P, R ; Q, P ; (J, R ; R, P ; R, Q. 
 
 404. To find the number of jjermutations of n dijf'erent thimjs 
 taken r at a time. 
 
 Let a, h, c, d ... stand for n different things. 
 
 l^'irst to find the numher of permutations of the n things 
 taken txvo at a time. 
 
 If a l)e placed hefore each of tin; otlior things h, r, d ... «jf 
 which the numher is/i-l,we ahull huveri-l perniututionw 
 in which a stands first, thus 
 
 ah, ac, ad, "^ 
 
 I 
 
 1 1 
 
28S 
 
 PERMUTATIONS. 
 
 \ih be placed hel'ore each of tlie other thing's, «, c, f? ... wu 
 shall have n— 1 permutations in which h staucl-j iirst, thus : 
 
 htb, he, bd, 
 
 Similarly there will he u-1 })ermutation8 in which c stands 
 first: and so of the rest. In this way we get every possiblr 
 jiorrnutation of the n things taken two at a time. 
 
 Hence tliere will be u , {n - 1) permutations of n things taki'ii 
 two at a time. 
 
 Next t(j lind tlic number of permutations of the n thini;- 
 taken iJircc at a time. 
 
 Leaving a out, we can form (h- 1) . (/<"2) permutations ol 
 the remaining (n-l) things takm tvo at a time, and if mc 
 place a before each of these permutations we sliall ha\e 
 (71- 1) .(?i- 2) permutations of the n things taken three at a 
 time in which a stands first. 
 
 Similarly there will be (n-l) .(ii — 2) permutations of tlic 
 n tilings taken three at a time in which b stands first : and so 
 for the rest. 
 
 Hence the whole number of permutations of the )i tliini^s 
 taken three at a time will l)e n. {)i-l) . {)i — 2), the factors of 
 the formula decreasing each by \, and thejiyare in the lastfactot 
 being 1 less than the nioiiber taJceii at a time. 
 
 We now assume that the fornmla holds good for the numliur 
 of permutations of n things taken /•- 1 at a time, and we shall 
 proceed to show that it will hold good for the number of per- 
 nuitations of n things taken r at a time. 
 
 The number of permutations of the n things taken r—l at 
 
 a time will Ije 
 
 n.{n-l).(n-2) [n-\{r~l)-l\l 
 
 that is n.{u-l).(ii-2) {n-r+'2). 
 
 Leaving a out we can form (h - 1) . {ii~2) (/i - 1 - r + -) 
 
 permutations of the (h - 1) remaining things taken r - 1 at a 
 time. 
 
 Putting a before each of these, we yhall have 
 
 {)i-i).{n-2) {,i-r+l) 
 
 permutations of the n things taken r at a time in which <t 
 stands lii'st. 
 
 So aga 
 iiiufation:- 
 first; and 
 
 Hence 1 
 taken r at 
 
 If tlien 
 '/■ - 1 at a 
 time. 
 
 But we 1 
 time; henc 
 su on : ther 
 
 405. If 
 
 ^"rniula giv 
 
 that is, 
 
 '1'* ihv. mim 
 fereJit thijjffi 
 
 For l)i{' vit 
 
 ^vJiicli is tlie 
 i^ written | w 
 
 Similarly 
 
 Oh^, 
 
 n 
 
 ill 
 
 l";i<ther when 
 
 '"ft tlie n t 
 '""' suppuse t 
 
 iiid so on. 
 
 * Allot li,. 
 
PERMUTA TIONS. 
 
 289 
 
 
 d ... NVO 
 
 c stiiiitU 
 possible 
 
 igs taken 
 
 1 n lliiii;-;"^ 
 
 tat ions nt 
 unci it' A\ <i 
 hall luivc 
 three at 11 
 
 uiis of the 
 st: ami ^o 
 
 e n thin;^^ 
 factors "f 
 lastfadui 
 
 he niunhcr 
 id wc shall 
 ber of pcr- 
 
 ken r 
 
 - 1 at 
 
 -l-r + 2) 
 lu r - 1 nt a 
 
 which <t 
 
 So again wv shall liavo (// - 1). (?? -2) (»-r ♦ 1) jht- 
 
 iiiutatioiis of tlie )i things taken r at u tinu' in which b standi 
 first ; and so on. 
 
 Hence the whole nnniber of piTiiintatioiis of tlie n things 
 taken r Jit a time will be 
 
 n.{u~]).(u-2) (//-r+1). 
 
 If then the fornnila holds good when the n things are taken 
 r- 1 at a time, it Mill ludd good when tliey are taken r at a 
 time. 
 
 But we have shown it to hold when they are taken 3 at a 
 time; hence it will hold when they are taken 4 at a time, and 
 su on : therefore it is tine for all integral values of r.* 
 
 405. If the n things be taken all together, r = n, and the 
 formula gives 
 
 n.{n-l) .{n -2) (» - 71 4- 1 ) ; 
 
 that is, )^ , (?i - 1) . (n - 2) 1 
 
 as the number of ]iermutjitions that can be formed of ?j. dit- 
 ferciit tilings taken <tll together. 
 
 For brevity the formula 
 
 n.{n-\).{n-2) 1, 
 
 which is the ss ne as 1 . iJ . 3 ??, 
 
 i-* written | n. This symbol is called /ac<on**<^ n. 
 
 Similarly | r is put for 1 . 2 . 3 r ; 
 
 ljr-1 for 1.2.3 (''-I). 
 
 f)h, \n = n.\n - I =n .{71 -I) .11 — 2 = &c. 
 
 400, Tofiml the numher nf 'permutations of n things tahm all 
 li'ijillier when certain of the tliinys are alili'. 
 
 1-et the n things be represented by the htters «, h, f, d 
 
 aiiil suppose tluit a recurs p times, 
 ' h q times, 
 
 c r times, 
 
 iinl so fm. 
 
 * Anotlior proof of Ibis 'riioovnii ni:i y lie seen in Art. 475. 
 fs.A.] , T 
 
 ' 'i 
 
290 
 
 PERMUTATTONS. 
 
 v\ 
 
 
 Let P rejdvsent the whole nunil)cr of pernmtations. 
 
 Then if all the f letters a were clian^'ed into \i other letteis, 
 different from each other and from all the rest of the a httcis. 
 the places of these p letters in uny one permutation could now 
 he iiiterchanj^ed, each interchan.L,^' iciA'ini^' rise to a new juriiin- 
 tation, and thus from each .single permutation we could form 
 
 1 .2 p permutations in all, and the whole number of pei- 
 
 jnutations would be (1 . 2 ... jf)) P, that is ; p . P. 
 
 Similarly if in addition the q letters h were changed into q 
 letters different from each otlu'r and from all the rest of the // 
 letters, the whole number of permutations would be 
 
 an<l if the r letters r were also similarly changed, the whole 
 number of permutations would be 
 
 and so on, if moio wiire alike. 
 
 But when the p, q, and r, Sec, letters have thus been changed, 
 we shall have n letters all dilferent, and the number of permu- 
 tations that can be formed of them is j n (Art. 405). 
 
 Hence P . \v . \q . \r = 1 71 
 
 p= 
 
 Examples.— cxlvi. 
 
 1. How many permutations can be formed out of 12 thinj,'> 
 taken 2 at a time ? 
 
 2. How many permutations can be lormed out of 10 things 
 taken 3 at a time I 
 
 3. How many permutations can be formed out of 20 things! 
 talien 4 at ii time I 
 
 4. How many changes can be rung with 5 bells out (»f S I 
 
 5. How many permutations can be made of il:*^ letters in 
 the woril Kxitmiimtion taken all together? 
 
 6. In how many ways can 8 men be placed side by side? 
 
 7. Ii 
 
 8. T 
 
 signals 1 
 being 4 
 
 9. H 
 
 the lette 
 
 10. 1 
 things tn 
 
 11. 1 
 
 time : th 
 a time = 
 
 12. 1] 
 together, 
 
 13- F 
 
 product a 
 
 14. Fi 
 
 out of th 
 
 Talavera, 
 
 407. T 
 different c 
 a certain 
 wliieh the 
 
 Thus tl 
 i(h, ac, ad, 
 
 Here iix 
 tions: thu 
 Lombinatic 
 
 Similarl 
 nre uhc, ak 
 
 Here frc 
 tions; thut 
 
T 
 
 rOMIUXA TfOXS. 
 
 291 
 
 7. Ill how many ways can l(> lui'ii be jtlaced .side by side ? 
 
 8. Three Hags are reijuired to make a signal. How many 
 signals can l)e given l)y '1^ Hags ul' 5 dillereiit colours, there 
 being 4 of each colour \ 
 
 9. How many dillVrent permutations can be Ibrmed out oi 
 the letters in Ahjehm taken all together i 
 
 10. ihe number ot things ; numiujr of jiormutations ol the 
 things taken li at a time = 1 : 2<). How many things are there i 
 
 11. The number of pernuitations of 7/1 things taken 3 at a 
 time : the luuuber of permutations of iii + 2 things taken 3 at 
 a time = 1:5. Find m. 
 
 12. In the permutatiims of (/, /), c, d, e, f, <j taken all 
 togvther, lirul how many begin with al. 
 
 13. Find the nund)er of ju rmutations of the letters of the 
 product a'-IPc^ written at full length. 
 
 14. Find the number of permutations that can Ite formed 
 out of the letters in each of the f(dlowing words : Conceit^ 
 Talavera, Calcutta, Proposition, Mistiissijijji. 
 
 
 t 
 
 ^ * li^ 
 
 XXXIV. COMBINATIONS. 
 
 407. The Combinations of a number of things are the 
 dillerent collections that can be formed out of them by taking 
 a certain number at a time, without regard to tlie order in 
 whicli the things stand in each collection. 
 
 Thus the comljinations of a, h, c, d taken ti.co at a time are 
 ah, ac, ad, be, hd, cd. 
 
 Here from each combination we could make tu:o pernuita- 
 tions : thus ah, ha ; ac, ca j and so on : for ah, ha are the same 
 combination, and so are ac, ca. 
 
 Similarly the condjinations of a, b, c, d taken tln'ce at a time 
 are ahc, ahd, acd, hcd. 
 
 Here from each combination we could make six permuta 
 tioiis; thus ahc, acb, bac, bca, cab, cba : and so on. 
 
 
 \^ 
 
> A 
 
 4 \: 
 
 292 C0Af/JrV.t7VO.VS. 
 
 And, ^cnenilly, ill acrordance willi Art. 405, any roniMiiiu 
 tion of n tilings may be made iulo 1 .2.'.'j...u piTniutalions. 
 
 ■tos. Tn find the number of comhinations of n different thiiitj.i 
 taken t at a time. 
 
 Let Cr denote the number of coinVn'nations required. 
 
 Since eacli combination contain.s r things it can be made 
 into \r permutationH (Art. 4(>')); 
 
 .*. the Avlude numl)er of permutations = ! r . C^. 
 
 But also (from Art. 404) the whole number of i)ernmtation.s 
 of 71 things taken r at a time 
 
 — n(n—}) (/i-r-t-1); 
 
 .•. I r . C\ =^a{n- V (n — r+1); 
 
 . /^_ '^^(^^-^ ) (n-r+l ) 
 
 4o9. To alioto that the number if combinations of n </<i»f/,s 
 time is the same as the number taken n — r at a 
 
 takcu r at 
 time. 
 
 a= 
 
 n 
 
 and 
 
 a 
 
 n 
 
 n 
 
 (n-1) \n-(n- ) + l \ 
 
 1.2. J...."..(H-r) 
 
 (n-1) (r+1) 
 
 I 
 
 '•> 'I 
 
 {n - r) 
 
 H 
 
 ence 
 
 0, n.(u~l).... ..(n-r+l) 1.2.3 (n-r) 
 
 t-n-r 
 
 1.2.3 r 
 
 X 
 
 n 
 
 ('t-1) {rtl) 
 
 n 
 
 {n - 1) (n -~r+ 1) . {n - r) 
 
 3.2.1 
 
 1.2.3 r.(r + l) {n~l).n 
 
 n 
 
 11 
 
 That 
 
 = 1. 
 
 18, 
 
 a-c. 
 
 f — ^n—r' 
 
 410 
 
 Henc( 
 one of 1 
 
 that wh; 
 
 With 
 
 (1) I^ 
 
 (2) II 
 
 which de 
 
 (3) t: 
 
 successive 
 each of tl 
 
 respondin 
 
'oml'iiiii- 
 
 COMBTXA TIONS. 
 
 293 
 
 410. Makiii},' r=l, 2, 3 r- 1, r, r+1 iii oiiler, 
 
 
 f 
 
 
 be made 
 
 luiitatioiirf 
 
 )f n thimji 
 n — r at a 
 
 n - r) 
 
 (r+1) 
 
 .1 
 
 r _^ ^' (^^ ~^) (.-r-f-2) 
 
 •^'~ r^ (r-i) " 
 
 . , _ ?t . (n - 1 ) (n- r + 2) . (h - r + Tl ) 
 
 ' l72......(r-l).r 
 
 Cy4-i = 
 
 'r+l 
 
 _n . (n-l) (n-r + \) . (n-r) 
 
 1 .2 r.(r+l) 
 
 c^„=i. 
 
 Hence the general expression for the factor connecting C„ 
 one of the set of numhers Cj, C'a CV+i C„, Avitli Cv_x, 
 
 that which stands next before it, is , that is, 
 
 T 
 
 ^_7i-r+l ^ 
 
 r 
 
 /ij J' _|_ ]^ 
 
 "With regard to this factor - , we observe 
 
 (1) It is always positive, because n+1 is greater than r. 
 
 (2) Its value continually decreases, for 
 
 Qi-r+l n+\ 
 
 r r 
 
 which decreases as r increases. 
 
 -1, 
 
 (3) Though continually decreases, yet for several 
 
 successive values of r it is greater than unity, and therefore 
 each of the corresponding terms is greater than the preceding. 
 
 (4) "When r is such that , - i*^ l^ss than unity the cor- 
 responding term is less than the preceding. 
 
 I 
 
 , » 
 
 :i^ ^ ' 
 
294 
 
 COMBINATIONS, 
 
 n-r-^- 1 
 
 \& 
 
 I 
 
 41 
 
 i 
 
 = 1 , C, and Cv_i aro .i 
 
 (5) If ?i and r be Bnch that 
 pair of C'(|ual t*""iM, each greater tluin any preceding or suhHO- 
 
 •i 
 
 iient term. 
 
 Hence \\\* to a coriain term (or ])air c»f terms) tlie terms in- 
 crease, and ufter that decrease: this term (or ])air of terms) is 
 the greatest oi" the series, and it is the ohject of llie next Article 
 to determine what value of r gives this greatejjt term (or ))air 
 of terms). 
 
 411. To find the value r)f-r for vhirh the nvmher of comhina- 
 tions of i\ thiiKjs taken r to(je.ther in tlie yreatesf. 
 
 . ., _n.(n-\) (n-r + 2) 
 
 '-'~ 1.2 (r-1) 
 
 ■ f-i _n. (n-1) (».-?• + 2 ) (n-r+l) 
 
 " lT2T7....(r-l) * r 
 
 ^ _n.{n-l) (n-r+l) n~r 
 
 C.,, ^._^_-__^___ .^^^. 
 
 Hence, if C, denote the number of combinations required, 
 ^ - and j/" must neither of them be less than 1. 
 
 But ^,.- = - -, 
 
 '-V -1 ' 
 
 1 a r+1 
 
 and rr "" ' 
 
 ('r+l ^'-'* 
 
 Hence — is not less than 1 a'^id is not less than 1, 
 
 r n — r 
 
 or, 71 - r + 1 is not loss than r and r+l not less than n - T, 
 
 or, 71 + 1 is not less than 2r and 2r not less than n — \; 
 
 :. 2r is not greater than ?? + 1 and not less than 7i — 1. 
 
 Hence 2r can have only three values, n— 1, n, n + 1. 
 
 Now 2r must be an eveji number, and therefore 
 
 (1) K n be odd, n - 1 and n + l being both even jiumbe^'s^ 
 2r may be equal to 7i - 1 or r* + 1 ; 
 
 (2) 
 
 6. 
 
COMnilVA TIOXS. 
 
 295 
 
 r — 
 
 V 
 
 n + 1 
 or '/•=—--. 
 2 
 
 (2) If n be even, n - 1 and n+ I being both odd numbers, 
 2r Ctan only be ecjual to n ; 
 
 n 
 
 
 I 
 
 U 
 
 9 
 
 Ex. 1. Of ei<,'ht things how many mnst be taken togetlier 
 tliat the number of combinalions may l)e the greatest pos- 
 sible ? 
 
 Here 7?. = 8, an even num))er, therefore the number to be 
 
 taken is 4, which will give -- - — -r — - or 70 combinations. 
 
 1x2x3x4 
 
 Ex. 2. If the number of things be 9, then the number 
 
 9-1 9 t- 1 
 to be taken is — „ - or —9—, that is 4 or 5, which will give 
 
 respectively 
 
 9x8x7x6 - _„ 1 • .• 1 
 
 , or 120 combinations, and 
 
 1x2x3x4 
 
 9^ 8 X 7 X 6 
 Fx 2 X 3 X 4 X 5 
 
 9x8x7x6x5 , ^,, , . 
 
 ^, or 126 combinations. 
 
 Examples.— cxlvii. 
 
 1. Out of 100 soldiers liow many different parties of 4 can 
 be chosen ? 
 
 2. How many combinations can be made of 6 things taken 
 5 at a time ? 
 
 3. Of the combinations of the first 10 letters of the alphabet 
 taken 5 together, in how many will a occur ? 
 
 4. How many words can be formed, consisting of 3 con- 
 sonf.nts and one vowel, in a language containing 19 consonants 
 and 5 vowels ? 
 
 5. The number of conit)i nations of n things taken 4 at ni 
 time : the number taki'ii 2 at a time =15 ; 2. Find ». 
 
 6. The number of combinations of n things, taken 5 at 
 
 
 M'^ 
 
 r 
 
 r 
 
 
 )(•: 
 
 '!'' 
 
 i 
 
 1; 
 
 i 
 
 \ 
 
 i 
 
 If 
 
 !-■ 
 
 
 ii 
 
 
2^6 
 
 COMBINA TlOArs. 
 
 a tim«, in .'j' times tlio number of cumltinatioUiS laluii 3 'it a 
 time, rind n. 
 
 7. Out of 17 coTisoiiuiit?; and 5 vowels, liow numy wonls 
 cnn Ijc loruR'd, cacli ciiiituiiiint; '1 vowels and W consonants \ 
 
 8. Out ol 12 Consonants aud 5 v(»\vt'ls liow inany words can 
 Ijc formed, each containing; G consonants and .'J vowels \ 
 
 9. The numhor of permutations of 7? thint^'s, 3 at a time, i.s 
 V) times the numher of combinations, t at a time. Find )}. 
 
 TO. ITow many dillerent sums may be foinied with a t^Miinea, 
 a half-guinea, a crown, a half-crown, a shilling, and a sixpeni c >. 
 
 - II. At a game of cards, 3 being dealt to each person, any 
 one can have 425 times as many hands as there are cards in 
 the pack. How many cards are there ? 
 
 12. There are 12 sohliers and IG sailors. How jnany dif- 
 ferent parlies of G can bo made, cich party consisting of 3 
 soldiers and 3 sailors ] 
 
 13. On how many nights can a difhirent patrol of 5 men lie 
 draughtctl from a cor])s of 3G ? Ou how many of these would 
 anv one man bo taken? 
 
 I Ml 
 
 XXXV. THE BINOMIAL THEOREM. 
 POSITIVE INTEGRAL INDEX. 
 
 412. The Binomial Theorem, first explained by 
 Newton, is a method of raising a binomial I'Xju'ession to any 
 power without going through the process of actual multipli- 
 cation. 
 
 413. To iitrcsti<ja(e the Binoiiiial Theorem for a Positive 
 Iniegrul Index, 
 
Till-: ni.\oMi.\i n/EONEM. 
 
 297 
 
 \\\ artuiil imilliitlicati'ii we laii ^liow tliut. 
 
 
 + (<'i'<a + '/i^Ta + r<irt^ + "./^i + <f._. 
 
 rr 
 
 + «3«4) 
 
 x^ 
 
 + C"i"ii":i ! '^i'^.'/ . I f< .'f:j"4 -H ('-/'atO »'-^ + ('i''./' 
 
 i"ii'"a'^4 
 
 Tn llic'se rcsiilts we observe tlic lollowiu'' laws 
 
 ■ 'f 
 
 t il 
 
 I. l'];u:li ]>r(»(lu('t is (•(tiiiposiMl nt' a (lescciKlniL; series of 
 )Ht\vrrs ol' ,'•. The indi.'x of .<: iu the lii.-t ti-nu is the same as 
 tile miiiilier ol' laetois, aiul the indices ot" x decn-asv; hy unity 
 in each succeeUing teini. 
 
 II. TJie number ui terms is j^'reuter by 1 lluui the number 
 of factors. 
 
 III. The coellicieiit of the ///•.s/ ti rm is unity. 
 
 of the sccnnd tlie sum of ^i, a.^, a.^ ... 
 of tlie tlLtrJ the sum of the 2)roduct,s of 
 
 «i, a.,, (1.^ 
 
 fdkciL trvo at a time. 
 
 of \]u'f())u(]t the sum of tlie products of 
 
 a,, (?.,, a-, ... tifleii three at a ti 
 
 •1> ' = :!> ^';t 
 
 nie. 
 
 and the last term is tlie product of all the (piantities 
 
 '1' 
 
 M 
 
 w 
 
 i'M 
 
 "l» 'f-2, "3 
 
 Suppose now this law fo In^ld for /(, — 1 factors, so that 
 
 {x + aO (■€ + rt.,) {x + «;,) (7; 4- (?,,_,) 
 
 — X -f- >^ 
 
 + S.,.x"-''^ + S..x''-* + 
 
 «••••• 
 
 + >S„_ 
 
 n-U 
 
 w 
 
 here aV, = « . + a., + a., + . . . -f- a,. 
 
 «-ij 
 
 that is, the sum of «,, a.„ a 
 
 I, i<._,, i?3 
 
 ((■n- 
 
 H-l> 
 
 i 
 
 that i,s, the sum of the product.s of a^, a^, ag ... a^^, 
 taken two at a time. 
 
 fi 
 
 I 
 
^98 
 
 THE BINOMIAL THEOREM. 
 
 that is, tlic sum of the products of (6^, ao...a„_i, 
 taken tlireu at a time, 
 
 
 that is, the pi'uduct of ttj, «„, ^3 ... a^-x- 
 Now multiply both tiidus hy x -f tfc„. 
 Then 
 (x + rtj) (,/; + «.„) . . . (rr + a,. . j) (.r + rt,.) 
 
 + ('^3 + ''^^'j) ^"'^'^ I- . .. + (t„»S'„_i. 
 
 Now »Si + ft,. = (f 1 + «. + (^J -I- . . . i a,, j + «„, 
 
 tliat is, the sum of «j, a^, (^3 ... (f,„ 
 
 >So + a„>S\ = N^ + (Y,, (ff 1 + f '.. + . . . -t- rt„ _ I ), 
 
 that is, the sum of the products of a^, a^.^.a^^ 
 taken two at a time, 
 
 /Sy + rt„No = >S*3 + a„ ('(/^J + fY,«3 +...), 
 
 that is, tlie sum of tlie products of a^^ «.^...r(,„ 
 taken three at a time, 
 
 IS 
 
 A, 
 
 that is, the product of rt], cty, rr., ... nr,,. 
 
 If then the law holds good for n-\ factors, it will hold good 
 for n factors : and as we have shown that it holds good up to 4 
 factors it will hold for 5 factors : and hence for 6 factors : and 
 BO on for any number. 
 
"n-v 
 
 , .. Ct-n 
 
 '2 ••• ''»> 
 
 id good 
 up to 4 
 Id : and 
 
 T//£ BINOMIAL THEOREM. 
 
 igg 
 
 Now let each of the n quantities a^, a.^, (^3 ... a„ Ix' e(|ual to 
 a, and let us write oar result thus : 
 
 (x + ai) (.0 + «.,) . . . (./: + a„) - .;;" + vli . x""' + A„ . .c--^ + . . . + ^4 ^_ . 
 
 The left-hand side hecomes 
 
 {x + a) (x + a) . . . (:c + «) to n factors, that is, {x + a)". 
 
 And on the ri<,dit-hand side 
 
 yli = «4-a + a + ...to 7i terms = ??((, 
 
 ^2 = (fc- + a- + (i'"^+ ...to as many terms as are equal to the 
 luunber of combiuutiona of n things taken two at a time, that 
 
 . n.(n-l) 
 IS — ^ ' - 
 
 12' 
 
 ■ 1 -^'•("-^) „2 
 
 -43 = a^ + rt''4-«'^+ ...to as many terms as are equal to the 
 number of condjinations of n things taken three at a tiuie, that 
 M;/i - 1) . {,1-2) , 
 1.2.3 
 
 •''^■'- 1.2.3 
 
 v1„ = a . rt .rt... to n factors =<*". 
 Hence we obtain as our tinal result 
 
 1.2 
 
 ?? . (11-- 1) . ('h-2) ,, 
 + ^__^ .. ^ ^ . (fi^^u-j 4. . . . + a". 
 
 " 414. Ex. Expand (x' + a)". 
 
 Here the numljer of terms will he sevai, and we have 
 
 0.5.4.3 4 .. (1.5. 4.3.2 , 
 . ^r. 2. 3. 4 ""^17273:^. 5"'^ + ^" 
 
 t «* 
 
 ill 
 
 1- I "i ! 
 
 

 H 
 
 ft,, I 
 
 m 
 
 I 
 
 300 
 
 TT/Ti BINOMIAL THEOREM 
 
 NoTK. The cofffieients of terms e(|UiMistaiit from the enrl 
 and from the he,i:riiiuiiif( are the «iuue. The general ])roof of 
 this will be j^nven in Art. 42U. 
 
 Hence in tlie Exain})le jnst given when the coefRcients of 
 jom- terms had been found thusc of the other three might have 
 been, written iluwn at once. 
 
 Examples.— cxlviii. 
 
 Expand the following expressions : 
 
 I. (rt + x)"*. 2. (/> + f)". 3. ('( + /))^. 
 
 4. (.0 + 7/)^. 5. (5 + 4(0*. 6. {ii^^hcf, 
 
 415. Since 
 
 (.0 + «)" = •'■" + ^?<'.'''"~^ -I- '^ ' 1 ' I . rt-'t;""- + . • . + «", 
 
 J. • A^ 
 
 if we put a:= 1, we shall have 
 
 (1 + (()" = 1 + ^<fi + — : • ^ ^ «-+... + cfc". 
 
 416. Every binomial may be roduce<l to such a form that 
 the part to be expanded may have 1 fur its lirst term. 
 
 Thus since cc + a = x* ( 1 + 1, 
 
 1+ j and multiply each term of 
 the result by .t". 
 
 Ex. Expand (2,/;-!- 3//)'\ 
 (2a; + 3# = (2,.)^.(n-^?^)' 
 
 .n2,.^|i.-5.^^^^t(^n%^^^^.(^n^ 
 
 ( 2x 1 . 2 \2.';/ 1.2.3 \2.c/ 
 
 5.4^.J}^.2 A'/V , /3//y) 
 1.2.3.4'V2,/-/ ^\2a;/ ) 
 
 417. 
 that of 
 powers 
 even te 
 
 Tiius 
 
 for 
 
 Ex. 
 
 ("-ry'=r 
 
 Exjiauf 
 I. 
 4. 
 
 418. A 
 'y (lie Bin 
 
le end 
 :oot' ut' 
 
 ciii^ of 
 it liavc 
 
 bcf. 
 
 orm 
 
 that 
 
 r//£ BINOMIAL THEOREM. 
 
 15// 90//- 27()//3 405//^ 243// 
 
 301 
 
 ^mx>{ 1+ ^^ + 
 
 4- 
 
 8j 
 
 - + 
 
 16. 
 
 + 
 
 32.. 
 
 1 
 
 - 32.*;'^ + 24CU'// + 72(Ki-'^//2 + lOSO.t^^/^ + 810.-;//^ + 243//^ 
 
 417. Tlie expansion of (./• — a)" will be precisely tlic same as 
 that of (.r + rt)", except that the sign of terms in wliich the orhl 
 ])o\vers of a enter, tliat is the second, fourtli, sixth, and other 
 even terms, will he negative. 
 
 Thus (.0 -(()" = .';" -?u<.o"-^ + 
 
 n 
 
 (n-\) 
 
 2,M-'i 
 
 1.2 
 
 a-.i. 
 
 w.(//-1).(h-2) 
 
 aV-'-h 
 
 for (x-a)"=\x + (-a)\' 
 
 = x" + n(-a)x"-'^ + 
 
 yi 
 
 (n-l) 
 1.2 
 
 na:c 
 
 )i-i 
 
 n 
 
 (n - 1) 
 
 1.2 
 
 a\c" 
 
 (-rt)V-2+ &c. 
 
 + &c. 
 
 Ex. Expand {a - rf. 
 
 xr r r 1 5.4^0 5 . 4 . 3 ., n 5.4.3.2 . , , 
 
 = a* - Ott'ic + lOu'c- - lOuV + 5«i< - ./' 
 
 :\ 
 
 H 
 
 
 if 
 
 \' 
 
 T i 
 
 
 
 term of 
 
 3?/ 
 l'2x 
 
 fax 
 
 :y} 
 
 EXAMPLES.—CXlix. 
 
 Expaml the following exjuvssions : 
 
 I. {a-xf. 2. {h-cy. 3. (2.r-.3//)\ ' 
 
 4. (l-2.i-)^ 5. (I -.'•)'". 6. (*/•■' -//-')«. 
 
 418. A trinomial, as rf + 6 + c, may be raised to any ])ower 
 hy the Binomial Theorem, if we regard two terms as one, thus : 
 
 {a + h + cY = (a + by + n . (a + by~' . c 
 
 i 
 
 ft. • 
 

 'If'i 
 
 
 302 
 
 T//^ BINOMIAL THEOREM. 
 
 3.2 
 
 Ex. Expand ( 1 + .r + s?f. 
 
 (1 +.a- + .(;'^)3 = (l +r/-)3 + 3(l ^-aJ)'^.^2 + 'f• "(1 + a;) . .x^ + .^e 
 
 1 . A 
 
 = ( 1 + 3x + Sx'-' + :<?) + 3(1+ 2.C + .^•2) ./^ 
 
 + 3(l+a:)x^ + .v« 
 
 = ] + ZX + 3x2 ^ ^3 ^. 33,2 j^ (j-,.3 4. 3_^4 + 3^4 
 
 + 3r' + .'•" 
 - 1 + 3x' + 6.>:2 + 7^;3 + Gx^ + 3:c^ + a;«. 
 
 Examples.— cl. 
 
 Expand the following expressions : 
 I. (a + 2&-c)3. 2. (l-2a; + 3./;2)-\ 3. {^r?-x^^-£f. 
 
 4. (3.r* + 2.'jHl)l 5. (x+l-iy. 6. (a4 + 6i-t;i)3. 
 419. To jind the 1^^ or general term of the expansion of 
 
 {x + ay. 
 
 We liave to determine three things to enable us to write 
 down the ?*'*• term of the expansion of (x + a)". 
 
 1. The index of x in that term. 
 
 _ 2. The index of a in that term. 
 
 3. The coefficient of thai term. 
 
 Now the index of x, decreasing by 1 in each term, is in the 
 T^ term n -r+l ; and the index of a, increasing by 1 in each 
 term, is in the r"' term ?•- 1. 
 
 , For example, in the third term 
 
 the index of ./■ is ?i - 3 + 1, that is, n - 2 ; 
 the index of a is 3 - 1, that is, 2. 
 
 in assigning its jjioper coeihcient to the r**" term we have to 
 determine the last fictor in the denominator and also in tlio 
 numerator of the fraction 
 
 9i.(n-l).(?t-2).(u-3) 
 
 1.2.3.4...... 
 
 X 
 num 
 tJie I 
 last i 
 
 T], 
 from 
 2 to t 
 
 TI 
 
 H 
 
 nnd f 
 
 RO 
 
 0])S€ 
 
 unity, J 
 that thi 
 
 Coll 
 of (x + , 
 
 e< 
 
 Ohs. 
 Jenoniini 
 
 Find 
 
;-* + x" 
 
 if + .'•'' 
 
 -/)i>3 
 
 ision of 
 
 ■o write 
 
 lis in the 
 
 in each 
 
 |e have to 
 ^0 in the 
 
 T//E BINOMIAL TIT EG REM. 303 
 
 Now the lad factor of the (leiiominator is h'ss bv 1 thiiu the 
 niiinher of the term toAvhich it helongs. Thus in the li"' term 
 tlie lust factor of the doiiominiitor is 2, and in the ?•"* term the 
 h\st factor of the denominator is r~ 1. 
 
 • 
 
 The hint factor of tlie numerator is formed hy suhtracting 
 from n the number of tlie term to which it helonj^'s and adding 
 2 to the result. 
 
 Thus in the 3"' term the last factor of the numerator is 
 
 n- 3 + 2, that is j/ - 1 ; 
 
 in the 4'" '»-4 + 2, that is ?/ - 2 ; 
 
 and so in the /"' '\\ -r \:L. 
 
 Observe also that the factors of the numerator decrease by 
 unity, and the factors of the denominator increase ' _^ unity, so 
 that the coefticient of the 7"' term is 
 
 7i.{n-l).{ n- 2) (n -r + 2) 
 
 1.2.3..7..".(r-T) 
 
 Collecting our results, we write the r"* term of the expansion 
 of (x + ay thus : 
 
 iu{>i -I), {a -2) {n -r -r_2) _, ^,+, 
 
 1.2.3 (r-1) ''' '" ' 
 
 Ohs. The index of a is the same as the last factor in the 
 denominator. The sum of the indices of a and x is n. 
 
 Find 
 
 Examples.— cli. 
 
 1. The8*Hermof (I+.7;)". 
 
 2. The 5"^ term of (a^ - h'-y\ 
 
 3. The 4"^ term of {a - hy^. 
 
 4. The 9*'' term of {2ah - c(iy\ 
 
 5. The middle term of («-?>)'". 
 
 6. The middle term of (u^ + b^)\ 
 
 7. The two middle terms of (a - hy'K 
 
 8. The two middle terms of (a + xy^. 
 
 '4 \ 
 
 ■' i. 
 
 Jt' 
 
 II iy 
 
 \^ I 
 
 
 ■ -^ M 
 
 ' 
 
 1 
 
 \ 
 
 ■ 11 
 
 ) i 
 
 \ 
 
 ■1^ 
 
 
 1 
 
 i 
 
 i 
 
 P 
 
 m 
 
\i 
 
 t 
 
 ■f 
 
 ■i 
 
 304 77/i^ BINOMIAL THEOREM. 
 
 9. Show that the coctHcieiit of the middle term of 
 ,, . „„ 1.3.5 (4»-l) 
 
 lo,* SIhow tliiit the coefficient of tlie middle term of 
 
 ^ ' 1.2 n 
 
 420. To .s/<o?y ;/irt.^ tlie, coefficient of the r"" term from the 
 hecjinnimj of the expansion of (.x + a)" is identical ivith the coeffi- 
 cient of the i"' term from the end. 
 
 Since the nnmher of terms in the expansion is n+l, there 
 are n+l-r terms l)efore the ?-"' term from the end, and there- 
 fore the r*"" term from the end is the {n — r + 2)"* term from the 
 beginning. 
 
 Thus in the expansion of (cc + «)■'', that is, 
 
 x^ + 6ax* + lOa-x^ + lOalc'^ + 5a*x + a^, 
 
 the 3rd term from the end is the (5 — 3 + 2)"", that is the 4"" term 
 from the beginning. 
 
 Now if we denote the coefficient of the r^^ term by 6% 
 and tlie coefficient of the (ii-r + 2y^ term by C„_,^o, 
 we have 
 
 ^, _??.(??,- 1) (n-r + 2) 
 
 ^'"^ 1.2 ;.:::. (7^1)" " ' 
 
 C„_^2 - — j; y 2 _ - ;; ^ -^ _ ,. ^ g - 1) 
 
 _ n.{n-l) r 
 
 ~ l.2....77{n-r+iy 
 
 Hence 
 
 C, ^ n.{n-l ) (7i-r + 2) 1.2 (n-r+1) 
 
 6U+2~ 1 • 2 ...... (r- 1) ^ : ?i .V- 1) J^ 
 
 _?i. (?t-l) ...... (u-r+ 2). (71-r-t-l) 2.1 
 
 lT2......(r-l).r......(?V-l)./t 
 
 In , . , , 
 
 =:i^: = l which proves the proposition, 
 
 \n 
 
THE lUXOMlAl. THEOREM, 303 
 
 421. Ti) fmtl the fjrcatcst term in the cjpamio^i of {xv «)", '^ 
 h iiKj a iiositive intcijcr. 
 
 Tlic r"* term of tho expimi^ioii (.'>}-(«)" is 
 
 r:2 (r-i)~'" "^ ' 
 
 The (/•+ l/** term uf tlie expansion (.v + c)" is 
 n.{n-l) {n-r + 2). {n-r+l ) 
 
 i.2.....7(i^i).7 ''■'"/' ' 
 
 Hence it lollow.s that we obtain the (r+ 1/'' term by niulti- 
 
 „tii 
 
 plying the »•"' term by 
 
 n-r+l (t 
 
 When thi.s nniltipUer is tirst less tlian 1, the r"" term i.-i the 
 reute.st in the expansion. 
 
 low 
 
 7i-r-\-l a 
 
 \ 
 
 
 - is first less than 1 
 
 w 
 
 hen 
 
 na-ra + a is first less than rx. 
 
 or 
 or 
 
 9Jft 
 
 + rt first less than rx-i- nt. 
 
 r (x' -t- 11) tirst greater than a (ii + 1), 
 
 or 
 
 a 
 
 r first "reater than — 
 
 0' + 1) 
 
 It' r be erpial to 
 
 o 
 
 /(. i )i 
 
 a; + a 
 
 + 1) 
 
 + rt 
 
 th 
 
 en 
 
 // -?*-f 1 'r 
 
 1 , and the 
 
 (r+l)"' term is e(nuil to the /•'*", and each is greater tlian any 
 other term. 
 
 Ex. Find the greatest term in the expansion of (4 -1- a)", 
 when a — --. 
 
 IIer( 
 
 a(n + l) 2^' '^^ 12 24 ,„ 
 ^x-i-a " , ;} "ll 11 ^i"i* 
 
 The first whole number greater than 2j\ is 3, therefore the 
 greatest term ot the expansion is tint .3rd. 
 
 [S.A.] u 
 
 il f 
 
 i 
 
 f 
 
 
 'Jk 
 
 i 
 
3o6 
 
 TJJE BIXOMIAL T^KORE^r. 
 
 422. To Ji lul (he sum of (ill the coefficients in the erpansion 
 
 Sill 
 
 n . (n - 1 ) ., 
 
 1.2 
 
 
 pultin;.,' x—l, wo. get 
 
 or. 
 
 2" = 1 + M+--^ ^ ^+ + — ^^~-^+nil; 
 
 2" = tlic auiii urall the cuetiiciciitci. 
 
 423. To slioin (hat the sum of (he coefficient!^ of the odd term 
 in the expansion of (1 +.^)" is ctfud to the $um of the coefficients 
 of (he even terms. 
 
 Since 
 /I N., 1 n.(n i) ., n.{n-l).(n-2) .. 
 
 putting .(;= - 1, -we g(;t 
 
 (i-i)«=i-«i"-<":'>-«-(":'M"~^>+... 
 
 or. 
 
 1.2 
 -1) 
 
 1.2.3 
 
 0=jl-,-^l^). } 
 
 f n.(n-]).{n~2) ) 
 
 n-h 
 
 = suni of coeHicients ot" odd terin.s - sum of cu- 
 ellicieiits of even terms ; 
 
 .•. sum of coeHicients of odd terms = siuu of coetticients of 
 even terms. 
 
 Hence, by the preceding Article, 
 
 2" 
 
 sum of coeflicients of odd terms = „ = 2""^ ; 
 
 2" 
 sum of coefficients of even terms = - = 2"~\ 
 
M\ 
 
 I, I 
 
 
 XXXVI. THE BINOMIAL THEOREM. 
 FRACTIONAL AND NEGATIVE INDICES. 
 
 424. We have sliown that M'lieii in is a 2>ositive integer, 
 
 n™ , m.(iii-\) „ 
 (1 + x)" = 1 + mx + — 1 - o - 't;'^ + 
 
 We have now to show that tliis e(|uation liohls -ood when 
 
 m is a positive fraction, as -, a negative integer, as -3, or a 
 
 negative fraction, as -'. 
 
 4 
 
 "We shall give the proof devised hy Euler. 
 
 425. Tf ?/i be a positive integer Ave know that 
 
 /, \,» 1 . w.(w-l) o 7?/ .(//(.- 1) . (m- 2) ., 
 
 (1 +:.)"'= I +7«rc+ L___^a;^ + __.v -^ ^ AA ^^^a+ 
 
 Let us agree to represent a series of the form 
 
 , m . (rih - 1) ., 
 l+vrx+ j ^^-+ 
 
 by tlie symhol /(ih), irhatever the value o/ni ?»«?/ he. 
 Then we know that when w is a positive integ(;r 
 
 (1 +.•)•" =/(7/0; 
 
 and we have to show that, also, when m is fracti 
 
 onal 
 
 nr 
 
 iieuative 
 
 Since 
 
 (1 +.■)'■' =/(m). 
 
 f{m) = 1 + mx + 
 
 f(v) = I + nx + 
 
 111 
 
 n 
 
 1.2 
 
 (" - 1) 
 1.2 
 
 X- + 
 
 X" + 
 
 . M 
 
 
 r. 
 
 ? ' 
 
 
 i 
 
3''8 
 
 THE IUKOI\rrAL TUP:OKhM. 
 
 If we uinltij^ly to^^'etlicr llir two series, we sluill obtain an 
 exjiression olllie ff)iiu 
 
 \ ^ (ix.-Vhx' + r.r'' \(}x^ -V 
 
 lliiit in, a sci'ii'S of jifcrndiiiL,' ]>(i\vrrs <»t' .c in wlii<'li tlie coelli- 
 
 cicut.s ({^ I), r ai'i' loiined liy varidus combinations of 
 
 m and //, 
 
 To determine the mode in whicli a and h are formed, let us 
 c'onnnonee Ibe multiplication of the two series and continue it 
 as far as terms invcdving .r-, thus 
 
 
 J., . , n. (n - 1) ., 
 f{n)^l + nx + — J- -^ x^+ 
 
 /(w) x/(^/) - 1 +7n.T+ - J ^ ^ ^ .'--2+ 
 
 + nx + «?7?r/;- ^- 
 
 ?< . (u - 1) 2 
 
 Til 
 
 / N (m.(m- 1) 
 
 n .(n—1), „ 
 
 + 77171 + )-2~ ^ ^-^ + 
 
 Comparing this product with the assumed expression 
 l+ax + hx^ + cx^-]-(b:^+ 
 
 we see that 
 and 
 
 , 7H.(7»-1) n. ()! — }) 
 
 _7?r -m + 2)1111 + 7r — 7?. 
 = ^ ^ ._„_. ... 
 
 _(7)l + 7i) . (nt + 7< - 1) 
 — — - , ,^ . 
 
 Hi 
 
 Hen 
 
FRACTIOXAL AND NKCATIVE /XPICES. JO9 
 
 {Similarly we could show hy actual innlt'qdication lliat 
 {m + n) . (m + )i-l). {m + u- 2) 
 
 c = 
 
 1.2.3 
 
 ■thn -1- n) . {iii + n-l) . (m -f n - -2) . {in I- a - 3) 
 
 rt_. i.2.a.T ~ • 
 
 Tlius wt' iiii.Lrlit (IctcniiiiK! tlu^ siiciH'ssivc coetHcieiits to any 
 extent, but we may ascirtaiu the law of their formation l)y tlu^ 
 fullowin<' con^^icU rations. 
 
 Til..' /or///.s' of the coetlicicnts, that is, tlic way in which w, 
 and a are involved in them, do not (h-nend in anv wav on the 
 values of m and ?/, hut will l)e precisely the same whether m 
 and u he positive integers or any mimhers whatsoever. 
 
 If then we can determine the law of their formation when 
 m and )i are positive integers, we shall know the law of their 
 forination for all values of m and n. 
 
 Now when m and n are j^ositivc integers, 
 .•./(m)x/00 = (l+.r)'"x(l-t-aO" 
 
 
 / , (m + n) .(iu + )i 
 
 'A+Cia-^- n)j: + ^ ^> - — 
 
 ^ ^ 1.2 
 
 --/{m + n). 
 
 Hence wo conclude that whatever he the values of m and n 
 f(m)xf{n)=f{m + u). 
 
 Hence / {m + n +}}) =/ O'l) .f{n + 1)) 
 
 =fOn)./in).f{p), 
 
 and so generally 
 
 f{m + n+p + ...)-=f{rn).f{n)./{2))... 
 
 ill! 
 
 1 f, 
 
 h %M 
 
 i 
 
 y- I. a 
 
 ill 
 
 :1 
 
 'I 
 
 r 1! 
 
 'I 
 
 < i 
 
3io 
 
 THE BINOMIAL THEOREM 
 
 u 
 
 Now let in — n = i^= .., = ,,h and k being positive integers, 
 
 then 
 
 /( 
 
 /( /' h , , , 
 
 . -f I. + 7. + ... to k term 
 iC ic Ic 
 
 ) 
 
 ^•^(/■)--^(i)--^(i)-'"'^'"'^'"'^' 
 
 nr. 
 
 or, 
 
 Sfii'W 
 
 •(/.)= j/C.) I 
 
 /'\^ 
 
 (1 +-)*=/(!) 
 
 = 1 + -,:C + 
 
 
 ( 
 
 J .:^ 
 
 VC- + 
 
 wliicli proves the theorem for a positive fractional index. 
 
 Again, since f{w).f{n)=f{m + n) for all values of m and 
 let ?i = - m, then 
 
 }i. 
 
 =/(0). 
 
 Now the series 
 
 1 + 9)ia; + 
 
 ??i 
 
 ("''-I) .2 
 
 1.2 
 
 ./Jf + 
 
 becomes 1 wlien m = 0, that is,/(0)=l 
 
 .•./(m)./(-^/0=i; 
 
 /(->'0 = 
 
 /('") (1 + '^)' 
 
 -(l-f.;)- 
 
 (l+.cO-"'=/(-m) 
 
 = 1 + ( - m) X + Y'2 ' ' -'^^ + • • • 
 
 which proves the theorem for a neyatLve index, integral or 
 fractional. 
 
FRACTION A I. AXD KEG AT/IT. IXD/CES. 311 
 
 426. Ex. Expand {a-¥x)^ to four terniH 
 
 (a + x) 
 
 
 .<; + ' 
 
 1 /I 
 2 
 
 CO 
 
 1.2 
 
 a- 
 
 
 1 /I 
 
 a-')C-) 
 
 I ■' 
 
 d' 
 
 i-« 
 
 • ^v • • • 
 
 w . 'i 
 
 I. I 
 
 2 . r 4. -—. 
 
 X + 
 
 2 
 
 a 
 
 8 
 
 „J 
 
 = «'! + 
 
 .5 Q^o 
 
 2a 2 8a 2 HvC- 
 
 Or we might procutd thus, us is ex[»luinecl in Ait. 410. 
 
 (a-wc)5 = a^(l+^) 
 
 .V 
 
 1 /I 
 
 ( 2a 1.2 
 
 _ i f , . a; :>^ x? ) 
 ""' ( 2a~8a-'^i(Ja^-) 
 
 2(t2 8a'-i 16a- 
 
 c-0 .= K^oa-o 
 
 1.2.3 
 
 a' 
 
 11' 
 i 
 
 Examples.— clii. 
 
 Expand the folloAving expressions : 
 
 1. (1 +«)- to fivo terniB. 
 
 2. (1 +a)'' to four terms. 
 
 3. (rt I f;)^ to five terms. 
 4 (1 + 2x)^ to five terms. 
 
 a4- .^' y^ to four terms, 
 
 1 I I 
 6. (ft* -\-x^)^' to four terms. 
 
 7. (1 -,r-)- to fivo terms. 
 
 8. (1 -a-)-* to four terms. 
 
 .3 
 
 9. (1 -.3.c)-* to four terms. 
 
 10. ('■''- . j'** to four terniiJ. 
 
 11. (1 -.';)** to four terms. 
 
 1 2. ( 7' - 'J' )■' to three terms. 
 
 ! 'I 
 
 ^'■1 
 
 n 
 
 If 
 
31^ 
 
 THE BIXOM/AL THEOREM. 
 
 i'i 
 
 ■ill 
 
 42 
 
 7. J.0 C.iptllK 
 
 l{l+. 
 
 (1 +.'•)"" = ! + (-'0"'^+ ' 
 
 i-n-l) 
 1 . 2 
 
 + 
 
 -n.{'-n-\). {-n-2) 
 
 1, . 2 . 3 
 
 + 
 
 1 - ii.c + 
 
 n(n+\) ,, n{n+\)(ii ■\-2) 
 
 1.2 
 
 1 . 2 
 
 ? 'A 
 
 + 
 
 •«•••• 
 
 tlie terms being alternately positive and negative. 
 Ex. ]':\pand (I +.'•)-'•' \i) live terms. 
 
 (i-t-,'0 
 
 --3_ 
 
 3.4 
 1 ."'2 
 
 4 . 5 
 
 + 
 
 6 
 
 .i_ 
 
 1.2.3 i . 2 
 
 .u 
 
 + (v;--10.';^+15.'j'- 
 
 428. To expand (!-.>■)" 
 
 '1 - .,)-" = 1 - ( - 7i) . o: + — 
 
 -n.(-ii -1) 
 
 1.2 
 
 
 ■71 
 
 (-1^•-l)(- 
 1.2.3 
 
 )( - 2 ) 3 
 
 .^ + 
 
 = 1 + nx + 
 
 n 
 
 (n + 1) ,, ^ H.(/t.+ l)(H + 2) 
 
 1 .2 
 
 + 
 
 1.2.3 
 
 .;;^ + 
 
 ( he terms Ix'ing all positive. 
 
 Ex. Expand (1 -.>■)"•' to five terniH. 
 
 3.4., 3 . 4 . 5 „ 3.4. 5 . « , 
 
 (^-'^)"=i+^^+r:v-+i.,.3-^+r-^73:4'^'+- 
 
 = 1 + 3.(; + Gx- + lUx'^ + I ox* + ... 
 
 Examples.— clili. 
 
 Expand 
 
 1 . (1 + '0 "^ ^^ fi^'c terms. 4. ( 1 - ^ ) ^^ ^ve terms. 
 
 2. (1 -- 3,-;-' to five terms. 5. {a^--2x)--> to five terms. 
 ^ (l-^j Lu lour terms. 6. (d'^ -.';^)"" to lour terms. 
 
THE BINOMIAL THEOREM. 
 
 313 
 
 429. To expamJ (I +,'•)""• 
 
 (1 +a) " = 1 t-( - Vc+' 
 
 II ^ ,1 / ., 
 
 I .-2 
 
 J(„l_,)(_l_2) 
 
 n \ 11 / \ n / 
 
 1.2.3 
 
 • »'-' 1^ • • • 
 
 
 EXAMPLKS. — Cliv. 
 Expand - 
 
 1. (1 +.'•-) ^to five lorms. 4. (1 +2.') - to five terms. 
 
 2. (1 -a"'-') - to five 1 onus. 5. (n-A-.r") ^' to four terms. 
 3 (.'i/* + r/') •'"» to four terms. 6. (d^ + x^) •■■ ti> four terms. 
 
 430. (Jhservationa on ther/cncml exjiressinn for tJie term, involving 
 uf in the expan.uons (1 +.r)" and (1 -x)". 
 
 The <,^;nerul expression for the term involviuj,' re', tluit is the 
 (r+ ly^ term, in the expansion of (1 +ic)" \h 
 
 n.{n- \) ...{n -r-i- \) ^ 
 '.2 r 
 
 Prom this we must (h^thicc the form in all cases. 
 
 'i'hus the (r+1/'' tcrui of the cxjtausidu of (I -./;)" is fnuid 
 Ijy clianu;iiij4 x into ( — ,'), and therefore it is 
 
 1.2 /• '^ ■''' 
 
 U 
 
 
 lill 
 
 4] 
 
 '^ f 1 
 
 ■ 5- 
 
 
 ^1 
 
V' ! 
 
 I 
 
 i 
 
 314 THE BTNOMTAL THEOREM. 
 
 If n l)e iH'.c^'ative and = -m, the (?• 4- 1)"' term of the expan- 
 sion of (I +;r)" is 
 
 (-w)( -m-1) ... (' -'"? -r-t- 1) 
 
 (-ly. )m.(»i4-J)...fM + r-l)|A 
 or \ / I 
 ^' J. 2 r 
 
 If M be ne.^ative and = -?h, the (r + 1)* term of the expan- 
 sion of (1 +;/')" is 
 
 (-l)-.| m.(m+l)...(w + r-l){ 
 
 1.2 r '^ '^" 
 
 w.(//? + 1) ... (7n-j r- 1) , 
 or, -^ .ft^. 
 
 , Examples.— civ. 
 
 Find the r"" terms of the following expansions: 
 
 V 
 
 5. (H-rc)--. 6. (l-:*>.'•v^ 7. (l-.r)'-. 8- ("+4-1 
 
 7 ^ 
 
 9. (l-2.r) -. 10. (a--:'.-) K 
 
 1 1. Find the (r+ 1)*'" term of (1 -p)-\ 
 
 12. Find tlie (r+l)"" term of (1- 4.x)"''. 
 
 13. Find the (r+l)'" term of (1 + ,/)^ 
 
 14. Sliow that the coefficient of ro'^* in (1 +0;)"+^ is the sum 
 of tlie coefficients of a;' and .x't^ in (I +x')". 
 
 1.5. Wliat is the fonrlli term of yn - j - 
 
 16, What 
 
 s the fifth term of (a'-^- //-)•-' \ 
 
 .A 
 
 17. What is the ninlli term of (a''^ + 2.*-)- ^ 
 
 18. Wliat is tlie ti'iith term r.f {a + b)-"' ? 
 
 1 
 
 19. What is tlie seventli term of (a -1-6)'" / 
 
 431. 
 Binomiii 
 
 (1) 1 
 ^/104 = 
 
 
 -I 
 
 
 = 1 
 
 (2) 
 
 T 
 
 »/2 
 
 -( 
 
 = 1 
 
 ._ ^ 
 
 = 1- 
 
 (3) Tc 
 
 Ilerii \v 
 second tei 
 j.»roeeed as 
 
 Approxi 
 
 1. 4/;i 
 
% 
 
 THE B/XOM/AL THEOREM. 
 
 315 
 
 431. Tlie i'Mllowiiii^ are C'\ain])li'.s of tht- application (>t" tlu 
 Jjinomial Tluovem to the approxiiuatioii to rootn of imiubei'ti. 
 
 (1) To aiiim)ximate to the .srpiare root of 104, 
 
 4 \h 
 
 '104= V(|(M)4 4)-10(l^l^^-) 
 
 i 
 
 a-o 
 
 ( .'Z 100 1.2 Vioo/ 
 
 = ioii+ - --- + .- . 
 
 ( 100 lot too 1000000 
 
 4 ) 
 
 VioiV 
 
 . . r 
 
 ) 
 
 = 10-19804 near! V. 
 (2) To approxiuuito Ut the fifth root of 2. 
 
 «'2=(1+1)'^ 
 
 ^i^!-,-^.!(!-i),V!.(!-i)(!-2) 
 
 o 2 ;>V;) ' 5 V5 / VS / 
 
 + 
 
 5 25^250 2500'*' 
 
 3 9^ 
 
 25 "^2500 
 
 nearb 
 
 = 1-1236 nearly. 
 (3) To approximate to the cuhe root of 25 
 
 V N 
 
 .— O — v'-"' ■— S) — •>- i — 
 
 2 ) \ 
 
 i '27\ 
 
 Here we take tlie culie next above 25, so as to make the 
 second term of the binomial as small as possible, and then 
 l>roceed us Itefort;. 
 
 Examples.- clvi. 
 
 Approximate to tlie following roots : 
 I. x^3l. 2. ;/l08. 3. ^/2(;o. 
 
 4. ^-n. 
 
 \ : 
 
 '"• , 
 
 
 ;i 
 
 fi i'r 
 
 j ' 
 
 p 
 
 li^iitiiil 
 
;"'ki* 
 
 ■ ¥ 
 
 V 
 
 m 
 
 f 
 't - 
 
 1 1- 
 
 XXXVII. SCALE:^ OF NOTATION, 
 
 432. The syml)ols emi)loye(l in our coniinon f^yslcm of 
 Arithmetical Notation are the nine di,L,nts and zero. These 
 digits Aviien written consecntively acc^uire local values from 
 their positions with respect to tlie place of units, the value of 
 every digit increasing ten-fold as we advance towards the h-lt 
 hand, and lience tlie number ten is called tlie Radix of the 
 Scale. 
 
 If we agree to represent the nuuihcr ten l)y tlie h'ttor /, a 
 numher, expressed according to the conventions of Arithmetical 
 Notation hy 3245, would assume the form 
 
 3t" i 21- -I- 4/ I- 5 
 
 if expTessed accoi'ding to the convt-iilions of Algehra. 
 
 433. Let ns now suj^pose that some other numl»er, as/r^. 
 i8 the radix of a scale of notation, then a nund)er exjuessed in 
 tliis scale arithmetically hy 2341 will, i( Jive he represented hy 
 /, assume the form 
 
 if expressed algehraically. 
 
 And, genei-ally, if r he the radix of a scale of notation, .". 
 numher expressed arithmetically in that scale by 6789 will, 
 when expressed algebraically, since tin; value of each digit 
 increases r-fold as we advance towards the left hand, be rei>re- 
 sented by 
 
 (;rf7/-48r-f!). 
 
 434. The nunibcr which denotes the radix of any scale will 
 be represented in that scale by 10. 
 
 Thus in the scale whose radix is five, the number five will 
 be represented by 10. ,/ 
 
SCALES OF NOT AT [OX. 
 
 317 
 
 f. 
 
 lllMll of 
 
 3s from 
 ^'alue of 
 the li-ft 
 ; of Uk' 
 
 liiiietical 
 
 r, i\?,five, 
 cssed in 
 euted hv 
 
 it at ion, " 
 
 m will, 
 
 U'li dii^it 
 
 ^(•a1»' will 
 five will 
 
 In tlic snnio scale seven, l>eing eqnal U) live + two, will 
 therefore 1)C represented l)y ['2. 
 
 Hence the scries of nutiiral nuinl)i rs as lar as (iroity-jlvcwWl 
 be represented in tlie st-alu whost; ra<lix is five thus ; 
 
 1, -2, ;], 4, 1(», 11, 12, ]:\, 1;, lio, 2\, 2:1 23, 24, 'M), 31, 
 
 32, ;53, 31, J(i, II, 42, 43, 44, 100. 
 
 43"). In the scale wlnjse radix is clrrr)!. we sliall ivquin^ 
 a new synihol to express the nuinlu r tin, fur in that scale the 
 iiund>er eleven is icprisentcd l)y |(>. If we a,L,a'ec. to express 
 ten in this scale hy IIh; symhol ^, tlie series of natni;;l nnnihers 
 as far as twenty-three will bo represented in this scale thus : 
 
 1, 2, 3, 4, 5, (), 7, 8, !), t, 10, 11, 12, 13, 11, l."), IC, IT, 
 
 18, 1!), It, 20, 21 
 
 436. In the scale whoso radix is twelve we shall re([uii-c 
 another now symbol to express tiie nninhcr ehiven. If we 
 agree to express this nnmber by the symbol e, the natural 
 numbers from nim; to thirteen will be represented in the scale 
 whose radix is twelve tliiis : 
 
 I), f, r, 10, 11. 
 
 Again, the natural numbeirf fioi:i twenty to twenty-live will 
 bo represented thus : 
 
 iw, 11), 1/, 1'', 20, 21. 
 
 t 
 
 437. The scale of notation of which the railix is lico, is 
 called the Binary Scale. 
 
 The names given to the scales, up to that of which the 
 radix is twelve, are Ternary, Quaternary, Quinary, Senary, 
 Septenary, Oetonary, Nonary, Denary, Undenary and Duo- 
 <lenary. 
 
 438. To perform the operations of Addition, Subtraction, 
 Multiplication, and Division in a scale of notation whose index 
 is /•, we ]iroceed in the same way as we do for nuni1»ers ex- 
 pressed in the common scale, with this dillcrence oidv, that r 
 must be nsetl where ten would be used iu the common scale ; 
 wiiicli will bo under.stood better by the following; examples. 
 
 ( I 
 
 iff . i 
 
 ; H 
 
 iJ tf.K 
 
 r. 
 
 m 
 
 / ill 
 
3'8 
 
 .S c •. / /. A .S- OF NO TA'I 'JO/v 
 
 I 
 
 U; 
 
 
 Kx. 1. i'iml the sum ot 4325 uud 02;3 i in the scnarv scale. 
 
 4325 
 5234 
 
 the sum -rl4(H):^ 
 
 which is nhtainctl l>y a«l«h'ii^f tlm iinnihois iu vcitiral line-, 
 carijiiiL!; I ior every six contained in the several results, aii<l 
 setting down the excesses ahovi- it. 
 
 Thus 4 units and 5 units make nine units, that is, six iinit> 
 together witli 3 units, so we set down 3 and cany 1 to the 
 next column. 
 
 Ex. 2. Find tiie difference between 62345 and 534(36 in 
 
 the septenary scale. 
 
 62345 
 
 5346(5 
 
 the dili'erence 
 
 5546 
 
 wliich is obtained by the following process. We cannot take 
 six units from iivc unils, we therefore add seven units to the 
 five units, making 12 units, and take six units from tw<'lv(' 
 units, and then we add 1 to the lower figure in the second 
 column, and so ou. • 
 
 Ex. 3. Mul'iply 2471 by 358 in the duodenary scale. 
 
 24 7 1 
 3 5 8 
 
 • 17 8 8 
 
 e t e b 
 
 7 193 
 
 8 3 3 318 
 
 Ex. 4. Divide 367286 by 8 in the nonarv scale. 
 
 8 ) 367286 
 42033 
 
 Tlic Following is the process, ^^'e ask how often 8 is containt <1 
 iu 36, wdiich in the nonary scale re[)resents thirty-three nnils; 
 the answer is 1 and 1 over. AVe then ask how often 8 is con- 
 tained in 17, A -iich m \Af^ nonary scale represents n{.rfcen units; 
 the answer is 2 and no reu'uinder. And so for the other digit-. 
 
 I. 
 
 Ad 
 
 2. 
 
 A.l 
 
 3- 
 
 ISu 
 
 4- 
 
 Su 
 
 5- 
 
 Mu 
 
 6. 
 
 Mu 
 
 7. 
 
 ]>iA 
 
 8. 
 
 Div 
 
 9- 
 
 Ext 
 
 lO. 
 
 Ext 
 
 stale. 
 
 
ry seal I 
 
 ;ll line-. 
 Its, ;m4 
 
 ix niiil> 
 
 I to till' 
 
 34G6 in 
 
 not take 
 ;s to the 
 I twelve 
 J secoiiil 
 
 ale. 
 
 )ntain(<l 
 e units ; 
 is con- 
 H iiiiit!«; 
 ■r digit-:. 
 
 SCALES Of NOT A TIOjV. 
 
 519 
 
 Ex. 5. Divide 1184323 by 589 in the duodenary scale. 
 589; 1184323 (2483 
 
 22^3 
 
 3e32 
 39H) 
 
 1523 
 1523 
 
 Ex. 6. Extract the square root of 10534521 in the senary 
 scale. 
 
 
 10534521 ( 
 
 ^ 2345 
 
 
 4 
 
 
 43 
 
 253 
 213 
 
 
 504 
 
 40 45 
 3224 
 
 
 5125 
 
 42121 
 42121 
 
 
 3 
 
 4 
 
 5 
 6 
 
 7 
 8 
 
 9 
 10 
 
 scale. 
 
 Examples.- clvii. 
 
 Add 23561, 42513, 645325 in the septenary scale. 
 Add 3074852, 4635628, 1247653 in the nonary scale. 
 ISuhtract 267862 from 358423 in the nonary scale. 
 Subtract 124321 from 211010 in the quinary scale. 
 Multiply 57264 by 675 in the octonary scale. 
 Multiply 1456 by 6541 in the septenarj-" scale. 
 Divide 243012 by 5 in the senary scale. 
 Divide 3756025 by 6 in the octonary scale. 
 Extract the .S([iuuv root of 25400544 in the senary scale. 
 Extract the square root of 56898 H in the duodenary 
 
 *i^ 
 
 r if 
 
 in 
 
 
 i; 
 
 ^inij 
 
 fi 
 
 II 
 
320 
 
 SCALES O/^ NOrATIO.Y. 
 
 430. To irauii/oriii a given inlnjral number from one scale ( ■ 
 another. 
 
 Let iV be the ^'ivcn iiitp,t;er expressed in llie lirst rfcalo, 
 
 r the radix ot" tlie new Hcule in wliidi the iiuiiiIilt is Id 
 he expressed, 
 
 (I, h, c ""'j?') 7 tli<^ <l'o'^ n+ 1 ill iiumher, expressing 
 
 the iiunihi'r in tin; new scalr ; 
 BO that the niunhcr in tiie new scale will 1m> cxjirc-sed thus : 
 
 (W 
 
 h)"^ + cr" -+ + mr- -\- 2"' + 'I' 
 
 We liave now IVoni the e(j nation 
 
 N=ar" + hr"'^ + cr" -+ + uir- \ pr 17 
 
 to determine the values of a, h, c m, p, q. 
 
 Divide N ])y r, the remainder is q. IaI A Ije the quotient : 
 
 then 
 
 A=ar"-^ + J>r''-- + cr"~^+ +riu- \-p. 
 
 Divide A hy r, the remaimler is p. Let B he the quotient ; 
 
 then 
 
 B = ar"-'^ + br"'''-[-fr"'^ \- +m. 
 
 Hence the 
 first digit to the right of the number expressed in tlio 
 
 new sc'de is ry, tlie tirst remainder ; 
 
 second /', the second renuiinder ; 
 
 third //', the third lemainder ; 
 
 and thus all the digits may be determined. 
 
 Ex. 1. Transform 235791 from the common scale to the 
 scale whose radix is (i. 
 
 6 ! 2.35791 
 
 (3 
 6 
 (3 
 6 
 C 
 G 
 
 39298 remainder 3 
 G549 remainder 4 
 
 1091 remainder 3 
 
 181 renuiinder 5 
 
 30 remainder I 
 
 5 remainder 
 remainder 5 
 
 The number required is therefore 5015343. 
 
 Tlie digit: 
 whose radix 
 are the onl; 
 which ihe il 
 
 Ex. 2. 
 
 440. The 
 scale to anot 
 scales are otl 
 lie careful to 
 witli the prii 
 
 Ex. Tia 
 
 tlie scale win 
 
 The requir 
 
 Express 
 
 1. 1828 
 
 2. 1820 
 
 3. 4375 
 
 [S.A.] 
 
\ie scale T- 
 ale, 
 
 llltLT is 111 
 
 'xpre.ssiii;^' 
 I thu.s : 
 
 quotient : 
 
 (|Uotient • 
 
 iiidcr ; 
 iiiiiiii(k'r ; 
 laiiKlc'i" ; 
 
 ;ale to the 
 
 SCALES OF NOTATION. 
 
 \2\. 
 
 Tlie (li 
 
 he (limits by m iiicli a niuiiuer can be expressLMi in a scale 
 
 whose ratlix is /• Avill l»e 1, 2, :? r - 1, because these, 'nth 0, 
 
 are the only ri;niain<leis which can arise I'roni a clivMion in 
 which ihe divisor is r. 
 
 Ex. 2. Express 359S in the scale., whose radix is 12. 
 12 I :j:)!)S 
 
 \'l 'I'M remainder / 
 
 12 24 remainder e 
 
 12 2 remainder 
 
 I remainder 2 
 
 .•. the nnniber re(|uired is 206^ 
 
 440. The metliod of transforming- a given integer from one 
 scale to another is of course a})plicable tc ca.-^es in which both 
 scales are otheV than the common scale. We must, however, 
 lie careful to perform lint operation of division in accordance 
 with the principles explainetl in Art. 438, Ex. 4. 
 
 Ex. Transform 1425:52 fiom the scale whose radix is G to 
 
 tlie scale A\hi'S(' ladix is o. 
 
 n ! i42r)32 
 
 o 20330 remainder 2 
 
 2303 remainder 3 
 
 300 reir inder 3 
 
 33 reiiainder 3 
 
 5 I 4 remainder 1 
 
 j remainder 4 
 
 The required number is thereh^re 413332. 
 
 Examples.— clviii. 
 
 Ex[)ress 
 
 1. 1828 in the septenary scale. 
 
 2. 1820 in the senary scale. 
 
 3. 43751 in tlie duodenary scale, 
 Ts.A.] 
 
 i 
 
 \ 
 
 
 ■ 
 
 Ti: 
 
 N 
 
322 
 
 SCALES OF NOTATIOX. 
 
 4. 3700 in the qiiiiiarv scale. 
 
 5. 7(3)31 ill tlie biliary scale. 
 
 6. 215^55 in the duodeiiarv scale. 
 
 7. 790158 in the septenary scale. 
 
 Translcjiiii 
 
 8. 34002 from the quinary to tho quaternary scale. 
 
 9. 8978 from the iindenary to tlie duodenary scale. 
 
 10. 3256 from the .septenary to the duodenary scale. 
 
 1 1. 37704 from the nonaiy to the octonary scale. 
 \2. 5056 from the septenary to the qiiate'riiary scale. 
 
 13. 654321 from the duodenary to the septenary scale. 
 
 14, 2304 from the quinary to the undenary'. scale. 
 
 441. In any scale the jutsitive inte.^ral powers of the iiuin- 
 1)61 which denotes the radix of the scale are expressed ly 
 10,100,1000 
 
 Thus twenty-five, which is the square of five, is expressed in 
 ihe scale whose ladix is five by 100 : one hundred and tweiity- 
 ftve will be expressed by 1000, and so on. 
 
 Generally, the 7t"' power of the number denoting the radix 
 in any scale is expressed by 1 followed by n ciphers. 
 
 The highest number that can be expressed by ^j digits in a 
 scale whose radix is r is expressed by y* - 1. 
 
 Thus the highest number that can be expressed by 4 di^ils 
 in the scale whose radix is five is 
 
 10^ - 1, or 10000 - 1, that is 444-}. 
 
 The least number that can be expressed by v digits in a 
 scale whose radix is r is expressed by ?■'' \ 
 
 Thus the least number that can lie expressed by 4 digit- in 
 the scale whose radix is five is 
 
 10*-' or 10'', that is 1000. 
 
cale. 
 
 cale. 
 
 e. 
 
 scale. 
 ^ scale. 
 
 le. 
 
 the iiuiii- 
 )re.ssetl \\\ 
 
 pressetl in 
 1(1 tweiitv- 
 
 liyits ill a 
 
 ty 4 i\vj^\{- 
 
 lii^its in a 
 4 iliuit^^ ill 
 
 SCA/.ES or X0TA770X. 
 
 32J 
 
 442. ill a scale uliose radix is r, llie sum (>f' tlic digits of 
 an integer dividrd l)y (r - 1) will leave the same remainder as 
 the integer leaves when divided Ity ;*- 1. 
 
 Let N be the nuiiiher, and suppose 
 
 JV=rtr"-H?)?-"""^ + rr"~''^4- + mr^ + iir ■\- q. 
 
 Then 
 
 iV = a(r"-l)4-/>(^"''^-l) + K^''"^-l)+ ... +^ii{r"-\)+p{i-\) 
 + }(? + 6 + c+ -V iii-¥p\-(i\. 
 
 N(;w all the expres.sions ?•" - 1, r"'^— 1 ?•-- 1, r—\ arc 
 
 divisible by r - 1 ; 
 
 iV . , rt + /) + r + m + p + q 
 
 ... — -= an mtoger H :j ^ — ~ ; 
 
 9* — 1 ^ '/• - 1 
 
 which proves the proposition, lor sinct; the (jnotients ditl'er by 
 all integer, their fractional parts must bo the same, that is, the 
 remainders after division are the same. 
 
 NoTK. From thi^i proposition is derived the test of the 
 accuracy of the result of Multiplication in Arithmetic by casU 
 iiuj out the nines. 
 
 For let 
 
 A = 9m + a, 
 
 JUK 
 
 I 
 
 i?=9 
 
 )i 
 
 then 
 
 ,1 ;,' = 9{9mn + an + hm) -i- ah 
 
 the radix I that is, AB and ah when divided by 9 will leave the same 
 
 del 
 
 remaimier 
 
 liacUral Fmction-^. 
 
 443. As tht} local value of each digit in a scale wdir>se radix 
 is r increases 7'-fold as we advance from right to left, so does 
 the local value of each decrease in the same proportion as we 
 advance from left to right. 
 
 If then we affix a line of digits to the right of the units' 
 place, eacli one of these having from its position a A'aluo 
 one-r"" part of the value it would have if it were one place 
 further to the left, we shall have on the right hand of the 
 units' place a series of Fractions of which the denominators 
 
 l( 
 
 I j^'M 
 
 i '■■ '!',■' 
 
 A 
 
 u 
 
 ll 
 
 1 
 
 !ll| 
 
ZU 
 
 SCALF.S OF A'07\'t77oy. 
 
 J > 
 
 I f 
 
 
 ^1 
 
 I 
 
 
 are successively r, ;•-', r'\ , wliilc tlic immoriitors may lie 
 
 any nunilKTs Letwien r-1 and zero. These are called 
 Radical Fractions. 
 
 In our cnninion system of notation flic word RiiiHrnl is 
 replaced by Decimal, because ten is the radix ol' the scale. 
 
 Now adopting the ordinary system of notation, and markiii.; 
 the place of units by putting a dot * to the right of it, we lun c 
 the following result j : 
 
 S 
 
 In the denary scale 
 
 in the (quinary scale 
 
 324-42i:i = 3xlO'^ + 2,.<l(, + 4.,-;;,,.jf,,+ , ;,,+ ;;„, 
 
 remembering that in this scale 10 stands I'or Jive and not for ^ /( 
 (Art. 434). 
 
 444. To show that in any scale a radical fraction is a jiroprr 
 fraction. 
 
 Suppose the fraction to contain n digits, n, h, c 
 
 Then, since r- I is the highest value that each of the digits 
 can have, 
 
 (t h . , . . 1 / . / 1 1 
 
 -+-.,+ ... 18 not greater than (r- \.)l i- ., 
 
 ... to n ter 
 
 iriiiri j 
 
 Hence- 
 
 proper fi 
 
 44.'). 
 
 a radical 
 
 Let F 
 
 r 
 
 t<, 
 
 from wh 
 mined. 
 
 Muliip 
 
 Now - 
 r 
 
 Hence 
 new fract 
 
 not greater tium i^r- 1) 
 
 
 not greater than (r - 1) .' 
 
 not greater than — ; 
 
 C-V-l))* 
 
 not greater than I - — . 
 
 Giving 
 
 Multii 
 
 Hence 
 fraction, 
 new frac 
 
•II 
 
 S milV lie 
 
 re calKfl 
 
 iniHrnl is 
 calc. 
 
 i inurkiii;^' 
 , we have 
 
 lot lor ten 
 
 ; a jiro/irr 
 
 tlie (li;;it> 
 
 tcnii;- j 
 
 SCALES OF XOTAT/OJ^, 
 
 325 
 
 Hence tlie {,'iveii fraction is luss tluin 1, and i.s therefore a 
 ]>roper fraction. 
 
 44.'). To transform a fmrtlnn exprr.sniul in a rjivcn saile into 
 a radical fraction in any other Hcalc. 
 
 Let F be tile jLjiven fraetwm ('Xjtrcssed in the first scahf, 
 
 r tlie radix of the ne^r scale in which the fraction in to 
 be ex[>r('s.s('d, 
 
 ((, ?>, c.the diu'lts exprcssin;^' the fraction in the new 
 scale, so that 
 
 ]'~ 4- I- 
 
 J -I I^ •» I • • • 
 
 r /- /•' 
 
 from which equation the values of a, 6, c.are to be deter- 
 mined. 
 
 Multiplying both sides of the equation by r, 
 
 Jf ,• = ((+ +■.,+ ... 
 r /- 
 
 h c 
 Now - + ., + ... is a proper fraction by Art. 444. 
 
 Hence the intej^'ral part of Fr will =a, tJte first ditjit of the 
 new fraction, and the fractional part of Fr will 
 
 h c 
 
 = - + -,+ ... 
 r r- 
 
 Giving to this fractional part of Fr the symbol Fi we have 
 
 /'i^ + „ )- ... 
 r r- 
 
 Multiplying both sides of the equation by r, 
 
 ]i\,'^h + -+ ... 
 r 
 
 Hence the integral part oi^ Fir=^b, the second digit of the new 
 fraction, and thus, by a similar process, all the digits of the 
 new fraction may be found. 
 
 ■I 
 
 i . 
 
 !i if 
 
 \ \ 
 
 -:\ 
 
 ^ 
 
 Kt, 
 
 !i;!'L; 
 
 I ! 1 
 
 ' i; 
 
 I ^ ''' 
 
'( ■ 
 
 I 
 
 iVi 
 
 H 
 
 326 
 
 SCALES OF NOTAlVOAr. 
 
 Ex. 1, Express = us a ratlical fraction in the quinary 
 scale : 
 
 3 , 15 1 
 
 i / / 
 
 1 r ^ r. ^ 
 
 -X5= ^ =0 + ^, 
 
 5 
 
 X D 
 
 n — 
 
 25 
 
 = 3 + 3, 
 
 7 
 
 .4 _ 20 ^ 6 
 
 7 / / 
 
 6 ^ 30 , 2 
 
 2 . 10 , 3 
 
 therefore fraction is '203241 recumng. 
 
 Ex. 2. Express -84375 in the octonary scale : 
 
 •84375 
 8 
 
 6-75000 
 8 
 
 6-00000 
 The fraction required is -66. 
 
 Ex. 3. Transform -42765 from the nonary to the senary 
 scale. 
 
 •42765 
 6 
 
 2-78133 
 6 
 
 5-23820 
 6 
 
 1-55430 
 6 
 
 3-65800 
 '^^he fraction required is -2513 ... 
 
SCALES OF NOTATIOX 
 
 327 
 
 Ex. 4. Trannfonu t\'lA:t'n'o liuiu the duodenarv to tlu> 
 quiiteriiaiy scale : 
 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 
 «il24 
 
 2937 - reniaiiidcr 
 83^ — ivnuiiiider 3 
 %)<i - it'iiiaiiuler 2 
 
 (!2 - remaiiidoi' 3 
 
 1(! - roniaiiidcr 2 
 
 4 - ix-maindci- 2 
 
 1 - reiiuiiuiler 
 
 - remuinder I 
 
 •/275 
 
 4 
 
 3-4/58 
 4 
 
 1-75 < 8 
 4 
 
 2-5fc'08 
 4 
 
 Number required is 10223230-312 1 ... 
 
 Examples.— clix. 
 
 25 
 
 1. Express ^^ in the senary scale. 
 
 3 
 
 2. Express ' in the septenary scale. 
 
 3. Express 23-125 in the nonary scale. 
 
 4. Express l820-3:i75 in the st-naiy scale. 
 
 5. Tn what scale is 17486 Avritten 212542 ] 
 
 6. In what scale is 511173 writUn 1740305 ? 
 
 7. Show that a nnndx r in \\w C'oiniuon scale is divisible 
 (i) hv .'5 if the snni of its di<_nts Is divisilih; l»s 3. 
 
 (2) hv 4 if tilt' last two di«'its he divisible bv 4. 
 
 (3) ^'y "^ itlht' last three <li^i(s be divisible by 8, 
 
 (4) by 5 if the number ends with 5 or 0. 
 
 : f ■ 
 
 I* 
 
 Mil 
 41 
 
 !ii'i|! 
 
328 
 
 OM LOGARITHMS. 
 
 (5) by 11 il" the difference boiween the sum of the digits 
 in the odd places and tlie sum of those in the even 
 places he divisible by 11. 
 
 8. If JVbe a number in the scale whose radix is r, and a 
 be the number resulting \vhen the digits of N are reversed, 
 sliow that iV- ?i is divisible by r - 1. 
 
 XXXVIIT. ON LOGARITHMS. 
 
 446. Dkf. Tlie Logarithm of a number to a given base 
 is the index of the power to which the base must be raised to 
 give the number. 
 
 Thus if 1)1 = a', x is called the logarithm of m to the base a. 
 
 For instance, if the base of a system of Logarithms be 2, 
 3 is the logarithm of the nund)er 8, 
 because 8 = 2'': 
 and if the base be 5, then 
 
 3 is the logarithm of the number 125, 
 because 125 — 5^ 
 
 447. The htgarillim of a inimber ni to the base a is written 
 thus, log„7>j. ; and so, i[iii = a'', 
 
 ft* = l(»g„7R, 
 
 Hence it follows that 7/? --r< '"''''"•. 
 
 448. Since l=a", the logarithm of unity to any base is 
 
 zero. 
 
 Since a ~ a^, the logarithm of the base of any system 
 is imitv. 
 
 449. We now proceed to describe that which is called the 
 Oommon System of logarithms. 
 
 The base of the system is 10, 
 
ox LOG A A' /Tf IMS. 
 
 329 
 
 Bv a sufitem ono<Mritlmis to lliu l>a.<i', 1(», wc mcari. a succes- 
 sion of values of x wliicli satisfy the efiuatiuii 
 
 for all po^<itivo values of in, iutrmal ov fractional. 
 
 Such a system is fonuod Ity the series of logarithms of 
 the natural numl)ers from 1 to 1 00000, which constitute the 
 Ingarithiiis reL,nsttireil in our ordinary tables, and which are 
 therefore called tabular loyaritJinis. 
 
 450. N 
 
 o\v 
 
 1 - 10", 
 
 J0-10\ 
 
 loo 10-, 
 
 1000 r^KT' 
 
 and f^o on. 
 
 !: '! 
 
 
 Hence he lo'Mritiini (4' 1 is o 
 
 of 10 is 1, 
 
 of 100 is -2, 
 
 of iooo is :'>, 
 and so on. 
 
 Hence for all numbers between 1 and 10 tjie lo;.jarithm is a 
 decimal less than 1, 
 
 between 10 and 100 the h)garilhni is a decimal between 1 
 and 2, 
 
 between 100 and 1000 a decimal between 2 
 
 and 3, and so on. 
 
 401. Tile logarithms of tlic natural numbers from 1 to 12 
 stand thus in the tables : 
 
 No. L 
 
 O-OOO )0 
 
 o-:ioio;]oo 
 0-4771213 
 0-()020(100 
 0(){)S!)70() 
 
 (1 , o-77sir)i 
 
 N(i. I Log 
 
 7 , O-S l."')OI)KO 
 
 S 0-!)O30l)0O 
 
 10 I l-oooo(t()() 
 
 u i ro4i:5<)27 
 
 12 I-O701SI2 
 
 The logarithms arc calculated to seven places of decimal: 
 
 I 
 
 it 
 
 i 
 
 U' 
 
'1 
 
 m 
 
 am 
 
 330 ON LOG A KITH MS. 
 
 452. I'lic inU',m';il ]t;ii'ls nf ilie lo;4ai'it1)iiis of inimbers 
 liii^'lier tli;iii lOavc called llu; rharnrfcrisflcs nl" [Lose logarithms, 
 and the decimal parts of the logarithms are called the mantissa;. 
 
 Thus 1 is the cliaracteristic, 
 
 •0701812 the mantissa, 
 of the logarithm of 12. 
 
 453. The logarithms for lOO and the niimbors that .>uc(;eed 
 it (and in some tables those 1 lat precede 1()<») have no charac- 
 teristic prefixed, Ijecause it can be supplii-d by the reader, l)t'ing 
 2 for all numbers between 100 and 1000, 3 for all between 
 1000 and 10000, and so on. Thus in the Tables Ave shall 
 tind 
 
 i No. j Log 
 
 I 100 0000000 
 
 101 ' 0043214 
 
 ' 102 i 0086002 
 
 , 103 0128372 
 
 J 04 0170333 
 
 I 105 1 0211893 
 
 which we read thus : ' 
 
 the logarithm of 100 is 2, 
 
 of 101 is 2"0043214. 
 
 of 102 is 2 •0086002; and so on. 
 
 454. Logarithms are of great use in making arithmetical 
 fomjnitatioiis more easy, for by means of a Table of Loj^^arithms 
 the operation 
 
 of >\[uUiplication is changed into tiiat of .Vddition, 
 
 ... Division Subtraction, 
 
 . . . Involution .Multiplication, 
 
 . . . Evolution Division, 
 
 as we shall show in the next four Articles. 
 
 455. The Jncjarifhm of a j^roduct is (qual to the sum of the 
 logarithms of its factors. 
 
or LOGARFTHMS. 
 
 33» 
 
 and 
 Tlieu 
 
 m^(i 
 
 n -— a*. 
 
 )}ni ^a""'''; 
 
 
 oo' 
 
 IT once it To] lows that 
 
 log,??i??,j) = lo,i,v» + log,*/ 4- log^), 
 .111(1 similarly it may he shown that the Theorem holds good 
 for any nnnibci- of factors. 
 
 Thus the operation of Multiplication is changed into that of 
 Addition. 
 
 Suj)i»ost', for instance, \vq want to lind the product of 24() 
 and 1357, we add the loifarithms uf tlie factors, and the sum is 
 the logaritlnu of the product : thus 
 
 log 246 = 2 lilJOQliOi 
 log357 = 2-r)52G682 
 
 their sum = 4-9436033 
 which is the logarithm of 87822, the product reipiired. 
 
 NoTK. We do not write hjgi„24G, lor so long as we are 
 treating of logarithms to the particular base 10, we m;iy omit 
 the sulHx. 
 
 4.')(j. The lof/arifJiin of a ([iioticnt is equnl to thr. lorjarithm of 
 thi' (licidciid diminished hij fhe hxjarithm of the divisor. 
 
 Let 
 
 and 
 
 Then 
 
 m — a, 
 11 = a''. 
 m 
 
 n 
 
 = (1''": 
 
 •■• If^g.. 
 
 rii> 
 
 -'n-'-^f 
 
 = log„?7?-log„n.. 
 
 Thus the operation of Division is changed into that of Sub- 
 traction. 
 
 in 
 
 1 *i 
 
 11 
 
 
332 
 
 Oy LOGARITHMS. 
 
 If, for example, we fire required to divide 37r49 by 52376, 
 we proceed thus, 
 
 lo^'37l-4n--:2-5600t71 
 log 52 37<)- 1-7191323 
 
 their dilferei ice- -8508148 
 
 which is the logarithm of 7-092752, the (|Uolieiit required. 
 
 457. 77'e logarithni of <inii pnwe)' of a nnmher is equal to th>-. 
 livoduct of the logarithni of thx niiiabcr and the index denoting the 
 
 ■power. 
 
 Let 
 Then 
 
 m = a'. 
 
 m" :.^ a"' : 
 
 .-. logjifc"" =-)•.<; 
 
 — r . log„/». 
 
 Thus the operation of Involution is changed into Multi[>li- 
 cation. 
 
 Suppose, for instance, we have to Knd the fourth power of 
 13, we may proceed thus, 
 
 log 13-1-1130134 
 
 4-4; 
 
 )0/ I. 'A 
 
 wliich is the logarithni of 285G1, the number re<|uired. 
 
 458. The logarithm of ((ng runt of a nnuiher is equal to the 
 quotient ((rising from the division of the logarithm of the numha 
 bg the number denoting the root. 
 
 Let 
 Then 
 
 m — o. 
 
 \ r. 
 
 lou„?)/'" = 
 
 X 
 
 — -.log,,))?. 
 
 Thus the operation of Ea^oIu ion is changed into Division. 
 
» \ 
 
 ox LOGARITHMS. 
 
 333 
 
 If, for example, we have to fiixl the lifth root of 10807, we 
 proceed thus, 
 
 5 I 4-2254902, the log of 10807 
 
 •84r)O980 
 which is the logarilliiu of 7, the root retpiircd. 
 
 459. The coininon system of Loi^'urithms has tliis advantage 
 overall others fen- numerical calculations, that its hase is the 
 same as the radix of llie ('omnion scale of notation. 
 
 ircnce it is that the same mantissa sei'vcs for all numhers 
 which have the same siijuiticant dit/its and diliVr (Jiilv in the 
 position of the phice of units relatively to those digits. 
 
 For, since h)g 00 = log 10 + log 6 — 1 ; log 0, 
 log 0( )()-]. )g 1 00 ■;- log = 2 i log (), 
 log 0000 = log 1000 -flog 6 = 3 + log 0, 
 
 it is clear that if we know the logarithm of any numher, as 0, 
 we also know the logarithms of the numbers resulting from 
 multiplying that number by the powers of 10, 
 
 So again, if we know that 
 
 log 1-7092 is -247783, 
 we also know that • 
 
 log 17-092 is 1-247783, 
 log 170-92 is 2-24778.3, 
 h)g 1709-2 is 3-247783, 
 log 17092 is 4-247783, 
 l..gl7()920 in 5-247783. 
 
 400. "We must now treat of the logarithms of numbers less 
 than unity. 
 
 Since 
 
 1 = 
 
 = lO*^, 
 
 
 •1 = 
 
 1 
 
 'io~ 
 
 lo-s 
 
 01 = 
 
 1 
 
 "loo" 
 
 = 10 '', 
 
 m 
 
 
 m 
 
 t 
 
 
 i 1 
 
 m 
 
 I II 
 
 l;!i 
 

 '1^ 
 
 t 
 
 I 
 
 IS 
 
 I*' 
 
 334 
 
 C?yV LOG/.RITIfMS. 
 
 the lor^nrithm of a numbor 
 
 Itctwecii 1 ami '1 lies between and -1 
 
 bet 
 
 wceii "i 1 
 
 uid "Ol - 1 and 
 
 between •<>! and -OOl -2 and -3, 
 
 and so on. 
 
 Hence tbe logarithms of all numbers less than unity are 
 
 neLfative. 
 
 We do not n-quire a separate taole for these logaritlims, for 
 ^leducc llu'ui from the logaritlnus of numl.»errj greater 
 
 we ('an ( 
 
 than unity by the following proce. 
 
 ]uir-6 =log 
 
 6 
 
 lo 
 
 
 
 =lo[i 6-lo<r 10 =W6-1. 
 
 log -06 = log J j 1^-^ = log 6 - log 1 00 = log 6-2, 
 
 log •006 = log j,||,„ = log 6 - log 1000 = log 6 - 3. 
 Now the l(»garithni of 6 is -7781513. 
 
 Hence 
 
 log •(! =^ - 1 + -77^1513, which is written I'7781513, 
 l,,n- -OH =:: _ 2 + •77S1.")13, which is written 2'778ir)13, 
 lo" -006= - 3 + •778ir)13, whicli is written 3-7781513, 
 
 the characteristics only being negative and the mantissa' 
 positive. 
 
 461. Tlius the same mantissa} serve for the logaritlun.. f 
 all numbers, vhcther greater or less than unity, which have tne 
 same signiticant digits, and ditfcr only in tlie position of the 
 place of units relatively to those digits. 
 
 It is best to regard the Table os a register of the h»garithms 
 of nund)ers which have one significant digit before tlie decimal 
 point, 
 
 I. 
 places 
 
 II. 
 
 phices t 
 
ox LOG ARl 1 If MS. 
 
 335 
 
 No. ! I,nr 
 
 tlims 
 hmal 
 
 For instance, wlien "we read in the tables 144 | 1583625, we 
 interpret the entry thus 
 
 Xky^ 1-44 is •l.-)83(;25. 
 
 We then obtain the following rules Ibr the characte^-istic to 
 be attached in each case. 
 
 I. If the decimal point bo shifted one, two, throe... h 
 places to the ri},dit, prolix as a characteristic 1, 2, 3 ... n. 
 
 II. If the (locinial point bo shifted one, two, three ... /i 
 places to the left, prefix as a characteristic 1, 2, 3 ... u. 
 
 Thus log 1-44 is -1583025, 
 
 .•. log 14-4 is 1 -1583025, 
 
 lo^ 144 is 2-1583625, 
 
 log 1440 is 3-1583625, 
 
 log •144i8 1-1583025; 
 
 log -0144 is 2-1583025. 
 
 log -00144 is 3-1583025. 
 
 462. In calculations with negative characteristics we follow 
 the rules of algebra. Thus, 
 
 (1) If we have to add the logarithms 3-04028 and 2-42367, 
 wt> tirst add the niantissie, and the rosult is 1-00995, and then 
 add the characteristics, and this result is 1. 
 
 The final rosult is T + 1-00995, thai is, -00995. 
 
 (2) To subtract 5-6249372 from 3-2450973, wo may arran<;e 
 the numbers thus, 
 
 -3 4- -2450973 
 -5 + -0249372 
 
 1 + -0207601 
 
 the 1 carried on from the last sul)traction in the decimal places 
 changing -5 into —4, and then -4 subtracted from -3 giving 
 1 as a result. 
 
 Hence the resulting logarithni is 1-02O7601, 
 
 ' 
 
 i '■•! i 
 
ill' 
 
 1 
 
 i t 
 
 i i t 
 
 ll 
 
 ii 
 
 $^ 
 
 ON LOGARTTIIMS. 
 
 (;j) To imilliply ;j-74825(]9 l.y :>. 
 
 3-7482r)(;<) 
 5 
 
 the 3 carrierl en from \\w liist innUiplicatinn of the flecimal 
 places Ix'in;::; added lo — 15, and thus .giving — 12 as a result. 
 
 (4) T.) divide l4-2Jr)(!7:ir; l.y 4. 
 
 Increase the iici^Mtive charactevistio so that it may be exactly 
 divisilile Ijy 4, iiiakin;^' a ])roper compensation, thus, 
 
 14-24r)()7;i() = T() I 2-2450730. 
 
 r4-24r)07:K; ■ld+2-24r)0730 - 
 Then - ^ = ^ =4 + -5014184; 
 
 and so the result is 4-5614184, 
 
 Examples.— clx. 
 
 1. Add ;M651553, 4*7505855, 6-0879746, 2-6150026. 
 
 2. Add 4-0843785, "5-005()657, 3-8905190, 3-4075281 
 
 3. Add 2-5324710, 30050057, 5-89051 90, -3150215. 
 
 4. From 2-483209 take 3-742891. 
 
 5. From 2-352078 take 5-428{; 19. 
 
 6. From 5-349102 take 3-624329. 
 
 7. :Multij»ly 2-4596721 hy 3. 
 
 8. Multiply '7-429683 hy 6. 
 
 9. Multiply 9-2843017 by 7. * 
 10. Divide 6-3725409 by 3. 
 
 ir. ])ivide 14-432902 by 6. 
 
 12. Divide 4-53027188 by 9. 
 
 463. We shall now explain how a system of logarithms 
 calculated to a base rt may be transformed into another system 
 of which tiie base is h, 
 
 of 
 
O.V lOCARI IIIMS. 
 
 337 
 
 Let m, be a iiuihIkt kA wliich tlic lugaiilluu in the fust 
 system is x and in tiie S'^contl \j. 
 
 an 
 
 Then 
 d 
 
 m — a' 
 
 m 
 
 ^^h' 
 
 Hence 
 
 6"- 
 
 a' 
 
 6-1 
 
 2/ 
 
 = lou^ 
 
 .->■»' » 
 
 ^^Pl 
 
 Vyi.h 
 
 V 
 
 louJy 
 
 Hence if we mnlti]»ly tlu' Uigiiritlini of any rnunber in the 
 
 system of whirh tlie bast; is a hv , - , , we sliull (obtain the 
 
 ^ " h.W>' 
 
 h^rrarithni of the same number in the system of whicli the base 
 is h. 
 
 This constant muhii)lier ,- , is tuUcd The ^Eodulus of tha 
 
 sijdeni of xvlwh the banc is J) with reference to the system of 
 which the base is a. 
 
 464. The common system of h);^aTrithms is used in all 
 numerical calcuhitions, but there is an(jtlier syst!;m, whicli we 
 must notice, em]iloye(l by the discoverer of logaritiims, Napier, 
 and hence called The Napierian Sys'ikm. 
 
 The base of this system, denoted by the symbol (.', is the 
 number which is the sum of the series 
 
 of which sum the first eight digits are 2*7 182818. 
 
 405. Our common logarithms iue formed fr(jm the Loga- 
 rithms of the Napierian System by multiplying each of the 
 [s.A.J • Y 
 
 ;»i. 
 
^^^ 
 
 ox LOGARirilMS. 
 
 I* 
 (1 
 
 liittfT liy a conmioii niulti])!!.'!* callcl Tlir Modulus of the 
 ( 'nil 111 11)11 System 
 
 Tliis niodulus is, in uccoiduncf; willi the conclusion of 
 1 
 
 All. 4(;;>> 
 
 ' lo- 10* 
 
 Tlint is, if 1 1111(1 N 1)(3 the logaritliiiiH of tlio same number in 
 tlio common and Xai^icrian systems respectively, 
 
 1 
 
 /- 
 
 locr, 1(> 
 
 .N, 
 
 Now log, 10 is 2-3()258r>().') ; 
 
 loLT^ 1(> 2 .3{)258i)()9 ' 
 
 and so the modulus of tlic coiunion system is •43429448. 
 
 466. To prove that loga/> x log,/t — I. 
 
 Let a3 = log,/>. 
 
 Then 6 = a*; 
 
 1 
 
 ,-. If — a : 
 
 1 
 
 = 10'V^ 
 
 f64' 
 
 Thus 
 
 log„/) X logi,a = w" X 
 
 l'" 
 
 467. The following are simple examples of the method of 
 applying the principles explained in this Chapter. 
 
 Ex. 1. Given log 2-=-3()l(»o()(), log 3 =--4771213 and 
 log 7 - •8450980, hnd log 42. 
 
 Since 
 
 42 = 2x3x7 
 log 42 = log 2 + log 3 + log 7 
 
 = -SOIOSOO + -4771213 + '8450980 
 =r. 1-6232493. 
 
 I 
 
ON LOaAR/T/IMS. 
 
 339 
 
 Ex. 2. (Jiveii lug2--301(»:i()() and lot,' ;j.^ -4771213, find 
 tlie logaritliiiis of (34, 8l ;vinl ix;, 
 
 log ()4 = log 2*'== Clog 2 
 
 « 
 
 ;. l()u(i4-^l-bOUi8UO 
 
 I' 
 
 I 
 
 log 81 = log 3«- 4 log 3 
 
 log 3= -4771213 
 
 -. log 81 = 19084852 
 
 ajiti 
 
 1 
 
 log OG -: log (32 X 3) = log 32 + log 3, 
 
 l.)g32-log2'' = r)l(.g2; 
 io''-96 = 5loi'2 ^](ii,'3-- ini'^iaOO 4- -4771213= 1-9822713. 
 
 Ex. 3. (iive-u 1 
 
 :/(G-25). 
 
 Oi-D — 
 
 "(K), liiul the loLiaiitiini of 
 
 I 1 
 
 1 , 625 1 
 
 log (6-25) ' = „ log 6-25 = „ log -~^^^ = _ (log 625 - log 100) 
 
 
 5 1 
 
 pi 
 
 - (log 5^ - 2) = ^- (4 log 5-^2) 
 
 - i (2-7958800 - 2) = '1 136657. 
 
 EXAMPLES.— Clxi. 
 
 1. Given log 2 = -3()l():3(Kt, liii.l log 128, Iolj 125 and 
 log 2500. 
 
 2. GivL'U log 2 -'3010300 and log 7 --8450980, tind the 
 louarithius of 50, 'OOo and 196. 
 
 -t 
 
 3. Given log 2 = •301(1300, and log3= -4771213, lind tlie \ i 
 logarithms oi 6, 27, 54 ami .■')7(!. ; 
 
 4. Given log 2 - •3()103(H>, log :i = -4771213, hjg 7 = \S450980, 
 lind log 60, log -03, log 1-05, and log •0000432. 
 
 
 ! 1 
 
 I 
 
 I 
 
 I 
 ] 
 
 Hi-' 
 
■i, 
 
 340 
 
 O^r LOGARITHMS. 
 
 5. Given lot,' 2 =-':i()l (>:}(•(», log 18= 1-2552725 and 
 
 log 2I = 1-3222]S):?, tin<l log-(KM>75 and log 31-5. 
 
 6. Given lou 5 = -6989700, find the logarithms of 2, -064, 
 
 and 
 
 1 
 
 /2'*"\ 
 
 vn-"/ 
 
 7. Given log2 = '3010300, find tlie logarithms of 5, -125, 
 
 and 
 
 r.00\i5 
 
 V 2 '"/ 
 
 8. What are the logarithms of -01, 1 and i{~>0 to the base 
 10 \ Whai, to the base -01 ? 
 
 9. What is the characteristic of log 1593, (1) to base 10, 
 (2) to base 12? 
 
 4'' 
 10. (Jiven . , =8, and .r — 3?/, find ,/• and ?/. 
 
 II. Given Id'-- 4 = -6020600, ](.g 1-04 = -0170333 
 
 >•"!-) 
 
 ((f) Find the logarithms of 2, 25, 83-2, (-625)'^ 
 
 (A) How many digits are there in the integral part ot 
 
 (I -04 )""««? 
 
 12. Given log 25 = 1-3979400, log 1-03 = -0128372 : 
 
 > *"& 
 
 («) Find tiu'. logarithms of 5, 4, 51-5, (-064)^ 
 
 (/>) llow many digits are there in the integral part of 
 (l-03y»«»? 
 
 13. Having given h)g 3- -4771 213, log 7 = -8450980, 
 
 log II .-1-0413927: 
 
 find the logarithms of 7623, and ^' . 
 
 tiUU o3«7 
 
 14, Sulve the e<jnations ; 
 
 8 
 
 ( I ) 4( >96^ 
 
 64^ 
 
 (2) i^-^^' 
 
 (3) a'.\f = m, 
 
 f4) (r"P = c. 
 
 (5) a^^jy^c-^-x^ 
 
 (6) a^V" =(;'-=^. 
 
ON LOGARITHMS. 
 
 34 1 
 
 468. We have cxplaiind in Arts. 451) — 4()1 the advantages 
 of the Common System ol" Logarithms, whidi may be t^tated in 
 
 ;omm 
 a more uoner 
 
 d t'orm tlin.'^ 
 
 ni 
 
 Let A he any sefj[nence ot" figures (siidi as 2';J5!)l()), havi 
 one digit in tiie integral ]iait. 
 
 Tlien any number K liaving the same se([uence of figures 
 (such as 2:35-918 or •()(>2:i5<)l(;i is of the form A x lU", wliere n 
 is an integer, [tositive or negative. 
 
 Therefore^ log,„.V = log,„(.l x l(>") = h)g,„. 1 \- n. 
 
 Now A lies between 1<>*^ and J(i', and thcivfore h>g yl lies 
 between and 1, and is tiierefore a proper fraction. 
 
 But logjo'^''^^ *^'^^^ ^",-in'^ differ only by tlie integer ii ; 
 .'. log,,,. I is the fractional part of log„jiV. 
 
 Hence the Iniidrifhitia of all tivmherii Jiaring THE same 
 SEQUENCE OF EKJUKKs liiirc the same wantii^>i<(. 
 
 Tiierefore one rcfiister serre^i for tJir )uanfis>ia of loiian'tJnns of all 
 such numbers. This renders the tables more coniprehensirc. 
 
 Af'ain, cousiderinL;; all numbers Avlnch have llio same 
 
 se([uen 
 
 ce o 
 
 f ti'^ires, the nundxr (•(•ntaining ("'n di-^its in the 
 
 integral part= 10.^1, and therefore the characteristic of its 
 logarithm is 1. 
 
 Similarly the number coidaiidng in digits in the iidegral 
 part=i()"'. A, and therefoie the characteristic of its logarithm 
 
 IS 7/1. 
 
 Also nund)ers Avhich have no digit in the integral part and 
 one cypher after tlu; decimal point are e([ual to A . 10'' and 
 A . 10"^ respectively, and therefore the characteristic? cd' their 
 logarithms are - 1 and -2 resjiectively. 
 
 Sinnlarly the nund)er having in iiji>hers following the decimal 
 
 .". the charaderislLC of its hvjarithin is ~{iii I- 1). 
 
 Hence v:c see that the rharncteristirs of the h^ijaritlnns of all 
 numbers ran he ileterniincil hij inspection anil thenfare need not be 
 rcjidercd. This renders the tables less Indkij. 
 
 .% U'i 
 
 a 
 

 I 
 
 
 51 , 
 
 
 342 
 
 ON LOGARITHMS. 
 
 4(59. The imtliod of using TuMes of Logaiitliiiis does 
 not lull witliiu the scope of this treatise, hut an account of 
 it niuy he found in the Author's work on Elementary 
 
 TllI(;()NO.METI?Y. 
 
 470, W'v proceed to give a sliort exphmation of the Avav 
 in \vhi<h Logaritiinis are applied to tlie solution of (juestiona 
 relating to Compound Interest. 
 
 471. Suppose ?• to represent the interest on £1 lor a year, 
 then tlie interest on P pouinls lof a year is represented hy 
 Pr., and the amount of P pounds lor a year is represented 
 byP + /V. 
 
 .% i 
 
 
 47'2. To find the amoitiH of a (jiven sum for any time at 
 compound interest. 
 
 Let P he the original principal, 
 
 r the interest on £1 for a vear, 
 7t the nuiiiiier of years. 
 
 Then if l\, P,,. 7'., ...P„ he the amounts at the end of 
 1, 2, '.i ... ti years, 
 
 P, = P +rr=P {\ + r}, 
 
 J\^=.I\ + P^r^I\{l + rj^P(l ir)- 
 
 /^, = / ', + /'.,/• = /'., ( 1 + ,-, = p ( 1 + r)-\ 
 
 P,, = P,i+r/: 
 
 473. Now suppose J\. V and r to he given : then hy the aid 
 of Logarithms we can lind », for 
 
 h.g /'.. ing;/>(i + ry; 
 
 ■--log /' I /* h>g I -\-r) ; 
 logl^lfr) 
 
 ■1 
 J- 
 
t : 
 
 OX lAXrAKII IhMS. 
 
 343 
 
 474. If the interest l)e ]tiiyal)](.i at intervals other tliaii a 
 year, the formula P„ = 7''(l 4- r)" is a[iplicable to the solution of 
 the question, it V)eiiig observed that r represents the interest 
 on £\ for the period on which the interest is calculated, half- 
 yearly, quarterly, or for any other period, and n represents tlie 
 number of such periods. 
 
 For example, to find the interest on P pounds for 4 }ears 
 at compound interest, reckoned quarterly, at 5 per cent, per 
 annum. 
 
 Here 
 
 1 
 
 5 1-25 
 
 "=4"^()U-100="^12^' 
 ?i = 4 X 4 = 16 ; 
 •. P„ = P(l + -0125)i«. 
 
 Examples.— clxii. 
 
 N.B. — The Logarithms required may be found from the 
 extracts from the Tables given in pages 329, 330. 
 
 1. In how many years -will a sum of mmiey double it.>elf 
 at 4 per cent, compound interest ? 
 
 2. In liow many years will a sum of money double itself 
 at 3 per cent, compound inteiest { 
 
 3. In how many years will a sum of money double itself 
 at 10 per cent, compound interest \ 
 
 4. In how many years Avill a sum of money treble itself 
 at 5 i)er cent, compound interest / 
 
 5. If £V at conq)ound intt-rest, rale /•, doiiltle itself in u 
 years, and at rate 2?* in ?)i years: show that m : n is greater 
 than 1 : 2. 
 
 6. In how many y»ars will X'lOOO amount to £1800 at 
 6 prr cent, compound intiavsl \ 
 
 7. In how many year^ will £V doiililc itxlf at (i per cent, 
 per ann. compound intereac payable half-yearly \ 
 
 n 
 
 
 III 
 
 ! I 
 
 \ 
 
 I ' ' fi 
 
 \\ 
 
if 
 
 m 
 
 t 
 
 ; * 
 
 ' 
 
 APPENDIX. 
 
 47o. The lollowiiig i.s iiiiotlKT incilnxl of pMving the prin- 
 cipiil theorem in Pcnmitations, to wliicli letorence i.s made in 
 the note on p;)i;e 2>^i). 
 
 2o prove that the vnnthci- of jH.naiiiations of n Uihuja tahen r at 
 a time is ii . (n - 1) (ii-r-f- 1). 
 
 Let there he n thiims a, h, c, d 
 
 It" n thinj^.s he tukeii 1 at a time, the numher of ])ornmtation- 
 
 is of 
 
 coiU'.se n. 
 
 No'.v take any 
 
 o 
 
 f them, as r/, then n-\ are left, and 
 
 any one of these may !)»• put after a 1o form a permutation, 
 2 at a lime, in wliich a stands tirst : and hence sin('e there are 
 things which may heiiin and each of these n mav liave n- 1 
 
 n 
 
 n 
 
 put after it, there are altogether n{n — \) permutations of 
 thin<fs taken 2 {jt a time. 
 
 Take any one of lliese, as ah, then there are 7?. -2 left, and 
 any one of these may he put after ah, to form a permutation, 
 3 at a time, in Avhieh ah stands first : and hence since there 
 are n{ii - 1) things Avhicli may hegin, and eacli of these n{ii- 1) 
 may have n - 2 put after it, tluire are altogether )i {n — 1) {ii - 2) 
 permutations oi n things taken 3 at a time. 
 
 If we take any one of these as ah(\ there are n-3 left, and 
 so the numher of permutations of n things taken 4 at a time is 
 ?t.(>i-l)(H-2)(H-:3). 
 
 So we see that to tind the numher of permutations, taken 
 r at a time, we mu^t nuiltiply the numher of permutations, 
 taken r—\ at a time, hy th<; numher formed hy suhtracting 
 r—\ from n, since th's Avill h(! the numher of endings any one 
 of these permutations may have. 
 
 Hence the numher oi* permutations of ;; things taken 5 at a 
 time is 
 
 n (ii - 1) ()i - 2) > - 3) X (». - 4), or n (n - 1) (a - 2) (//. - 3) {n - 4); 
 and since each time we multiply hy an additional factor the 
 nundjer of factors is eipial to the numl)er of things taken at a 
 tim«*, it follows that the numher of permutations of ti things 
 tukeii r at a time is the ]"roduct of the factors 
 
 n.{ii' ]){ii -- -J.) (,n-r-\l). 
 
 I. 
 
 5- 
 
 9- 
 
 13- 
 
ANSWERS. 
 
 1 . 5a + 7') + 1 2r. 
 4. H(i + 26 + 2c. 
 7. 126 + Ik. 
 
 i. (Page 10.) 
 2. « -f 36 f 2c. 
 
 2.'J - I a 
 
 li. (Pa-(i 10.) 
 
 I. 2<i. 
 
 «) 
 
 :<( 4 f)./-. 
 
 3. :i. 
 
 ^-3r;. 
 
 4a + 64- 2c. 
 10«- 76- 
 
 7. 4. 
 
 lii. (Page 10.) 
 
 ^. 2f^ 4- 26 4- 2c. 
 
 4-36-2. 6. 0. 
 
 4. 8./- f 5//. 
 8. 13r-//-6-i 
 
 I. 26. 
 5. 2?-. 
 
 I. 4((-h. 
 
 2. :c 4- 2?/. 
 6. 26 4- 2c, 
 
 a 4- 5c 4- f^. 
 
 (f-36-c. 
 
 iv. (Pago 11.) 
 
 46 
 
 (< 
 
 I 6 -4( 
 
 G 
 
 K 2'' f ('. 
 
 5. 14.': + li. 
 
 9, 2((-li. 10. I^Jff. 
 
 13. ut)a-276 4-()r. 
 
 7. G.';-(^. 
 
 II. c. 
 
 0-. 
 
 4. 2// 4- ii. 
 8. 3^4-; 
 
 4. 26, 
 
 12. u;4-3((. 
 
 'Hi 
 
 A<Uliti(»n. 
 I. 7«-26. 
 4. -()6 ^)c^-3(/. 
 7. 7«f46-4c. 
 
 [S.A.] 
 
 V. (Page 10.) 
 
 2. ~1()6 4-()C. 
 
 2(f. 
 
 8. 7u - A f 7t 
 
 3. -11./--8//-6-.'. 
 9. ~G//4 2.v. 
 
?! 
 
 t. 
 
 
 I 
 
 m 
 
 m 
 
 tifr 
 
 hi 
 I J. 
 
 346 
 
 Sulid-action, 
 
 A.VSIVA'A'S. 
 
 
 
 
 
 
 
 I. 
 
 2n + 2h. 
 
 2. a - c. 
 
 J- 
 
 2//-2/>4-2c. 
 
 4- 
 
 Sx- 17 II ^rb. 
 
 5. 7a-l(Sb + 20c. 
 
 6. 
 
 5(( - lib - 8x 
 
 7. 
 
 - ;ia 4- 3/* --4c'. 
 
 8. 2b + 2c -lb. 
 
 9- 
 
 Ux-ly + 4 
 
 10. 
 
 6(r - h -j- Oc. 
 
 II. 12jj - 9(/ + 2/-. 
 
 
 
 1. 
 
 ;3'v/. 
 
 5- 
 
 "". 
 
 9- 
 
 isOa-i/rV 
 
 13. 
 
 TG,'.-*//-*.^-'. 
 
 16. 
 
 \2(i-bc.nj. 
 
 19. 
 
 ab.c'jr.K 
 
 I. 
 
 4- 
 6. 
 
 <S. 
 
 TO. 
 12. 
 
 I. 
 4. 
 
 7 
 
 9 
 1 1 
 
 14, 
 
 16, 
 
 Vi. (Page 20.) 
 
 2. 12.'7/. " 3. 12,0-//-. 
 6. rt\ 7. 12«56'^ 
 
 10. 28rt7/>6"i'J. II. 3a^K 
 
 14. bldb^c-ijz. 
 
 17. 8ai^6V^. 
 
 20. 33«-"/ji'=„r-.f. 
 
 4. 3^(-7)tfi2. 
 
 8. 'Sba%cK 
 
 12. 2()(i-»/r'>,/'//. 
 
 15. 48./;«//i«V'. 
 
 18. 9mhiy. 
 
 Vii. (PaK<^ 22.) 
 
 a- + ab-ac. 2. 2;r' + ()«7' — 8(fr. 
 
 9rt''-15a*-ia'r'^ + 21(r. 5 
 
 3'/"7)-9a-*?>^ + 3rf-M. " 7 
 
 3. rt* + 3^3 + 4(t-. 
 
 a^6-2rt-7;-'+(fR 
 
 ^iiv'ii + 9iii-)i' + lOmn^. 
 
 1 S, ,,% + ^„->h- - 6aW' + Sa-'M. 9. x^i^ - jpif' + ;;■-//- - 7.''//. 
 
 iirhi - 3//t-u- + 3/» ///" — 7<,^. II. 
 
 104?j^^ - 13(k-'f/- + 4().r-)/ - Sx}/. 
 
 U4a''h^-72a^b^ + ma'V'. 
 
 ./•-^-12.'• + 2T. 
 .r-- 15.0 + oh'. 
 ic4 + a'--20. 
 ,.«-31.c'- H). 
 
 ;.'«-,'■- f 2,f - I. 
 
 a*' .<A 
 
 a'*-8l7/*. 
 
 viii. (Page 27.) 
 
 2. .'•- + 8,r 1 105, 3. X- - 2x - 1 20. 
 
 5. «■•- 8(^ + 15. 6. 7/- +7// -78. 
 
 
 x^-\2x'^ + b0x--Six-r4b. 
 lu. a" 3(r'--3(f' M3(r=-6((--Gt(+4. 
 
 ',•4, 
 
 !2. :*;* + .'-//■• f-//'. 13. 
 
 15. ./'- 5.r'fn,(;--l. 
 
 //•' 
 
 19. 
 20. 
 21. 
 23- 
 
 25- 
 
 26. 
 28. 
 29. 
 
 
 -^7- 
 
 5- 
 8. 
 
 10. 
 
 12. 
 
 13- 
 
 J 7. a'-l()/j'. 
 
 18. lUa^-6\ 
 
 I. 
 
 4- 
 7- 
 
 9- 
 
 10. 
 
 1 1. 
 12. 
 14. 
 
 15- 
 
A.VSIVI^RS. 
 
 347 
 
 -26 + 2c. 
 - 36 - 8x'. 
 c - 7j/ + 4,^. 
 
 3ory"/yf». 
 2()(?VyV//. 
 
 3^3 + 4a'^. 
 ■ - 7.CIJ. 
 
 + mui^b'K 
 
 2.'-- 
 
 - 120 
 
 ~ll - 
 
 -7«. 
 
 Ga+4. 
 
 //•■'. 
 
 
 -6\ 
 
 19- 
 
 ,r, _ 4^^4/, ^ 4,,ri/,2 + 4,,2/,3 .. i ;„/,» _ | o/. 
 
 20. rr' + r)a46 + r(^//--l()«-//'^.±12a6^ -!>//'. 
 
 21. a4 + 4aV+lfM 22. Slrt^ + na-V + rc". 
 23. :cH 4a-,/;-' +!()«*. 24. (/•' + /rM- f'' - 3(r6f. 
 
 25. ^^ + xhj - 9.';'y - 20.r2//3 + 2.o//^ f 15?/^ 
 
 26. a26'^ + c-'(?-'-((V-6y-'. 27. ri-«-«^ 
 
 28. a-^ - aj:- + /a';- - f.'j- - '^^.v; + rtcf - hex + ahc. 
 
 29. 1 ■''^. 
 
 jj- 
 
 .■)/ 
 
 7. 2. 
 
 I 
 
 4 
 7 
 
 9 
 
 10 
 
 1 1 
 12 
 '4 
 
 30. .';•' - //". 
 
 31. a^«-A-i" 
 
 34- -14. 
 
 35. ab + ac + bc. 
 
 38. m'^. 
 
 
 3^- 
 
 -47. 
 
 ix. (Page 28.) 
 
 I. -a%. 2. -((■'. 3. -a%\ 4. 12aW. 
 
 5. -30.rV. 6. -n^ + a-h-aJr. 7. - Gr/'' - S^/-*+ lOa^. 
 
 8. fl^ + 2(r'5 + 2'<-' + a. 9. - G.c^?/ + j^ir + ^'-'f ' 1 -?/^- 
 
 10. bn^^^ui-,i~l?.mn" + 1n\ 11. - 13/-3- 22/-^ + 9(>r+ 135. 
 
 _ 7 ,.4 + ,,.3. + 8x2;:;^ + 9.^2 + 9^3^ 
 
 12 
 
 13. of' + x-hf. 
 
 1 4. .0-* + ±c^ii -r 2;/y-7/- + 2,///'' + y\ 
 
 X. (Page 32.) 
 
 0" ■\-2(ix + f('^. 2. ;''2 — 2a.r + «2. 
 
 ,7.2 _ 6,,- + 0. 5. a;'' + 2./:V" + /• 
 
 3. ,r- + 4.>: -f 4. 
 6. rr«-2.':-//--H,A 
 
 r/-2 + tf + z^ - 2x11 + 2xz - 2yz. 
 m''^ 4- //" +]>'' + r- + 2iini - 2?/?jp 
 
 — 2;/M- - 27))) - 2nr+ 2i>i 
 
 {- 4x^ - 2:* 
 
 12./' + !). 
 
 I^v 
 
 .4 
 
 12- 
 
 }' 
 
 r)(u2-84.-c + 4..i 
 
 4,.» _ 28./:^ + 85./'- - 126a; + 81. 
 X ' -I- //■' + ;;'t + 2.(;2v2 _ 2xh- - 2v'^- 
 
 s si 
 
 £ 
 
34S 
 
 ansjv/':rs. 
 
 i6 
 
 17 
 18 
 
 19 
 
 20, 
 
 21 
 
 23 
 25 
 
 ■^-7 
 
 20 
 
 29 
 
 (t<i + //i -f r« 4- 2(rV>^ 4- 2«\-'* -I 2/*'^r'^ 
 
 .7;- + 4//- + 9.:;- + 4.''// - iSxz - 1 2//,^. 
 re-* 4- 4»/^ + 25;;'« - 4./;2i''< + KU-',:;-' - 20//' 
 
 2^2 
 
 a;3 + 3.';2 + 3a; + l. 
 a;3 + (lo2 + 12a; + 8. 
 
 22. .x^ - 3«.x2 + oo/x — a\ 
 24. x^- 3^024. 3.r - 1. 
 26. r6«-3tt4/,2 + 3^^254 _/yi 
 
 a3 + 3rt^6 4- 3a?;^ ^. ^j3 ^. ^3 + 3^2^ ^ Qabc + 3lrc 4- liac' + 'Shi-. 
 
 ^3 _ -sa^jj + 3rt52 _ &3 _ (.3 _ 3((,2j^. + (jftjc _ 3^-2(. + 3 
 
 ar 
 
 'Shc^ 
 
 1)U 
 
 ^-2m'n'^ + 7i\ 
 
 30. 771* 4- 2m^n - 2mn^ - 71^. 
 
 xi. (Page 34.) 
 I. r>;^ 2. fc^ 3. ."'j^//. 4. r/;'*?/;/'. 5. 6/>c. 6. 8^2. 
 7. UH<-?/'r«. 8. 121'm''7?y. 9. 12rt3.,:?/4. 10. Sa^/n"-^. 
 
 xii. (Page 35.) 
 
 I. .x2 + 2.r4.i. 2. if^-i/ + y-l. 3. a2 4- 2a6 4- 352. 
 
 4. x•*4-//^iu•-^-m2;)2. 5. 4(/7/ - 7./; 4- .r2. 6. 8a;V - 4.Ay2 - 2//. 
 
 7. 27?a"»/' - 1 87>i"'9t* 4- dvip. 8. 3.<j2//2 - 2x?/"' - 1/. 
 
 9. 13a26-9rt62 + 7/;. 10. ldPr^ + 12b'k^-V>c^. 
 
 I. 
 4. 
 7- 
 9- 
 
 xiii. (Page 36.) 
 
 -8. 2. 15rr\ 3. -21ry. 
 
 -67/?.'-«. 5. IVxr^. 6. «.2y2^.^^.^i^ 
 
 _ 2,(2 + 3r, _ a;2, 8. 2 4- 6(1^}) - Sa^¥'. 
 
 -\2x- + \).<{j-Sif, 10. -x^ + lr\t:h'^ + hy\ 
 
 xiv. (Page 38.) 
 
 I. CC4-5. 
 
 10. 
 
 + 4. 
 
 5. a;2 4-7.i'+12. 
 
 X- 
 
 4. r/^4-12. 
 7. x^ + x+l. 
 
ANSIVERS. 
 
 349 
 
 8. x^-Sx-^ + S-c+l. 
 
 II. x^-x+\. 
 
 16. ./;■--«.« + 5. 
 
 18. %ix^-'da\c + a}\ 
 
 12. 
 
 a; 
 
 
 10. 
 
 13- 
 
 2.« + l. 
 
 ' + 3t/- 
 
 15. a^-AaVy + ^a^r- ..h^ + hK 
 17. (t''-2(«-7>f a-f/r'-H^V. 
 
 19. 
 
 + 1. 
 
 20. 
 
 a- 
 
 21. a- + 2//. 
 
 ,.4 _ r,^ 
 
 %,'i- . 
 
 c"* - ary + a;-?/ - xy^ + ?/ 
 
 2J. 
 
 25. 
 27. 
 29. 
 
 31. 
 33- 
 35. 
 
 37. 
 
 39. 
 
 41. 
 
 44- 
 46. 
 48. 
 50. 
 
 53- 
 56. 
 
 59- 
 62. 
 
 -64-26--//'. 
 
 24. a 
 
 26. 
 
 ft 
 
 + 6 - c. 
 
 -h-\C-il 
 
 28. 
 
 a,"' - xij - x:. + If - ?/;; + ^2. 
 
 }} + 2q- r. 
 
 x^ + xhj + xhf 4- a'7/3 + if. 
 
 A'i _ ,,j)y-i ^ ^^.(iyi _ ,^:!y; ^. ^s 
 
 30 a'^-a%-\-a%-^-aW + h\ 
 32 2,t^ - 3jj'-^ + 2.C. 
 
 a:2 - 9a; - 10. 
 
 36. 24a;'- - 2ax - 3i 
 
 )((" 
 
 6«'^ -7.0 + 8. ^, J. 8a;3 + 1 2(1x2 _ 1 g^^-j^ _.. 2 7^^:$. 
 
 27.r'-;i(w.':2 4-48a''^.c-64«-l 40. 2a + 36. 
 
 x-Vid. 42. t<''^-462. 43. :/j2-3:c-y. 
 
 .7;2 - 3,t;^ - 2//-. 45. r" 4- 3.t;-y + 9.*;//-'+ 277/''. 
 
 rt^ + 2«26 + 4(t6- + 86^ 47. 27(f'i - 1 8a-6 + 1 2«6"-^ - 86-\ 
 
 ar* - 12:(;27/ + 183;^'-' - ^1f. 49. 3« + 26 + c. 
 
 rt- - 2aa; + 4.'j-. 51. x' + xy + y- ■ 52. lbx--4xy + y-. 
 
 54. rt:i;- + 4a-a; + 2«'\ 55. rt -x 
 
 57. 3./;- -a; + 2. 58. 4- (j,';+ Kt"- - lO.t;''. 
 
 60. ax + by-ab-xy. 61. bx + ay. 
 
 x^ + xy - y- 
 x~y- z. 
 x + y. 
 ./;- — ax + 6'-, 
 
 I. 
 
 4- 
 
 a;- + ax + 6. 
 a;- + ax — 6. 
 
 XV. (Page 40.) 
 
 2. 2/'" - (^ + "^'O y + ^^"" 
 
 5. a2-(6 + (0« + 6(Z. 
 XV i. (Page 42.) 
 
 3- ^^ + ex + (^. 
 
 
 u 
 
 ■Ml 
 
 Ira 
 
 4a» = 
 
 ' ! 1 j 
 
 h 
 
 H 
 
 I 
 
350 
 
 AJVSIV£RS. 
 
 2. m V n, in'^ + inn + n^, m^ + mhi + &n\, m^ + mhi + &c., 
 
 m^ 4 m^n + &c. 
 
 3. a-\, <i^ - « + 1, (<■* - n^ + &c., rt" - <t^ + &c., (i^ - a^ + &c. 
 
 4. ]/ + 1, 1/ + ?/ + 1, y* + r + ''^<^', 1/" + /' + '^^•j / + y'-^ ^^• 
 
 XV ii. (Page 43.) 
 
 1 . 5 -; (jt- - 3). 2. Sx (x^ fHx- 2). 3. 7 (T?/'-^ - 2// + 1 ). 
 
 4. 4a'// (.(;'^ - 3.''// + '2 if). 5 . X (./'.^ - ax'^ + bx + c) . 
 
 6. a^y (.(.i^?/ - m; + 9//0. 7. 27a-'6«(2 + 4a36''^-9rtV>3). 
 
 6. -iox^if {x-hf - 2x - 3ij). 
 
 xviii. (Page 44.) 
 I. {x-a){x-b). 2. {a-a-){b + x). 3. {b~y){c + y). 
 4. (a + 7)1) (^ + '0- 5' (f<^ + ^) (^-^ - !/)• 6. (a6 + cfi) (ic - ^). 
 
 7. {cx + my){dx-ny). 8. {ac - bd) {bx - dy). 
 
 xix. (Page 45.) 
 
 I. (a3 + 5)(;c + 6). 2. (;c + 5)(aj + 12). 3. (?/+ 12) (j/+ 1). 
 
 4. (y+1 !)(?/+ 10). 5. (m + 2())(7« + 15). 6. (m + 6)(w+17). 
 
 7. (tt + 8b) {a + h). 8. {x + 4)ii){x + 9m). 9. (y + 3n) {y + I6n). 
 
 10. {z + 4p){z + 2^)p). II. (.<;2 + 2)(,r'^ + 3). 
 
 1 2. (x"^ + 1) {x^ + 3). 1 3. (ay + 2) (.T?/ + 16). 
 
 14. (.'^Y + 3)(./;Y + 4). 15. (m- + 8) (w5 + 2). 
 
 16. (71 + 20(/) (/i + 7(/). 
 
 
 I. 
 9- 
 
 I. 
 
 4* 
 7. 
 9- 
 
 I. 
 
 3- 
 
 5. 
 
 7- 
 
 9- 
 II. 
 
 13- 
 
 
 xx« 
 
 (Plige 45.) 
 
 ^ (^-5)(,.-2). 
 
 
 2. (.t- - 19) {X - 10). 
 
 3. (7/-ll)(//-12). 
 
 
 ■ 4- iu-miy-io). 
 
 5. (71- 23) (7^-20). 
 
 
 6. (»-5G)(w-l). 
 
 7- 0.3-4)(.(;^-3). 
 
 
 8. (rt6-26)(t<6-l). 
 
 9. (//V'-5)(6V-6). 
 
 
 10. C^i/v lJ)(.';//^-2). 
 
AATSlVJiA'S. 
 
 nhi + &c. 
 
 h&c. 
 •&c. 
 
 2//+ I). 
 
 C). 
 
 9a-V>3). 
 
 (c + y). 
 
 >)• 
 
 ). 
 
 i)- 
 -2). 
 
 !. (x-^l2)(x-5). 
 
 4. (a + 2())((t-7). 
 
 7. (,;4 + 4)(,.4_l). 
 
 9. (m^' + 20) (7/1^-5). 
 
 xxi. (Pnge4G.) 
 
 2. (.0 + 15) (re -3). 3. ((f t 12)((t-l). 
 5. {h [• 2b) {h - U). 6. (/) + 30)(/>-5>. 
 8. {xy+U)(xy-n). 
 10. (?<+30)(7i-13). 
 
 xxii. (Pago 46.) 
 
 I . (x - 1 1 ) (.>: -f 6). 2. (.^ - 9) (./; + 2). 3. {ni - 1 2) ()7i + 3). 
 
 4. ()i-15)(n + 4). 5. (//-14)(y+l). 6. (2 - 20) (2! + 5). 
 
 7. (;/;5-10)(af +1). 8. {rd-30)(cd + 6). 
 
 9. (m% - 2) (iri-^n + I). to. (j^Y - 12) (jj^q^ + 7). 
 
 ^) (^ - 2/)- 
 
 • 
 
 
 xxiii. 
 
 (Page 47.) 
 
 rf?/). 
 
 I. 
 
 (./; - 3) {x - 12). 
 
 
 2. (x + 9){x-b). 
 
 
 3- 
 
 {ab-m)(ab + 2). 
 
 
 4. (,(;* - 5m) (a;'* + 2m) 
 
 
 
 {if+lO){y^-d). 
 
 
 6. Cc2+10)(:^-^-ll). 
 
 (?/+l). ' 
 
 7. 
 
 x(yy^ + 3ax + 4a'^). 
 
 
 8. (x + m) {x + n). 
 
 (m+17). 
 
 9- 
 
 (7/3-3)(l/-l). 
 
 
 10. (xij — ab) (x-c). 
 
 [u + 1««). 
 
 Il- 
 
 {x + a) (x - h). 
 
 
 1 2. (x - c) {x + d). 
 
 i). 
 
 ia- 
 
 {ab-d){h-c). 
 
 
 14. 4.{x-4>j)(x-:]!i). 
 
 16). 
 
 
 
 
 
 -2). 
 
 
 
 xxiv. 
 
 (Page 48.) 
 
 I. (a; + 9)2. 
 5. (a+lOO)"-^. 
 
 9. {x^+uy^. 
 
 2. (a; + 13)-. 3. (a; + 17)2. 
 
 6. (^- + 7)-. 7. (.'; + 5(/)2. 
 10. (a'7/ + 81)2. 
 
 4. (^ + 1)'^. 
 8. {m'^ + Sn% 
 
 XXV. (Page 48.) 
 
 I. Or-4)2. 2. (.r-14)^ 3. (./;-18)2 4. (^-20)2. 
 
 5. (;i'-50)2. 6. (,';'^-li)2. 7. (^.c-lo^)'^. 8. (m'-- 16/12)"^. 
 9. (^3-19)^. 
 
 . I 
 
 
 II 
 
r' 
 
 352 
 
 I I 
 
 16 
 
 17 
 
 19 
 21 
 
 23 
 25 
 27 
 29 
 
 31 
 
 33 
 35 
 
 ANSWERS. 
 
 XXvi. (Piige •'>0.) 
 
 I. (.r4-?/)(.r-?/). 2. (..; + c5)(.';-:3). 3. (2:/; -I- 5) (2.r - 5). 
 4. (fr-f-,f-)(rr— -x-). 5- (.'>M )(.';- 1). 6. (.c' -I- 1) (x-^ - 1). 
 
 (.v;Hl)(.f^- 1). 8- (7/r + 4)(m'-^-4). 
 
 (Cy + 7;;) (6,y - Ta). I o. {S^xAJ + 1 h//*} (O.n/ - 1 1 '//>). 
 
 {^^ -i, + c) (a - h - (•). 1 2. (.<; + m - ?i) (^ - vi -t- //). 
 
 (^f + /y 4- c + f') (rt + h~c- d). T 4. 2.r x 27/. 
 
 (,,;-i/ + ,'.')(* -?/-;;). 
 
 (rt - 6 + i/t + ?i) (rt -- ?' - m - ?i). 
 
 (,, _c + 6 + H) ((t-c -/>-<.?). 18. {(t-\-h--c) {a-h-vc). 
 
 (■,,; 4. ly + ^') (,,.; -f 7/ - rj). 20. ((fc - h + m - 11) (a -b-m + n) 
 
 {(ix-\-hD+l) {ax -\ hj-l). 22. 2ax x ^y. 
 
 {V+a-h){\-a^h). 24. (1 +rc-7/) (I -r'j + //). 
 
 26 
 
 28 
 
 ((t - & 4- c + ff.) {a-h-c- d). 30, 
 
 3rtx((^*; + 3) (ax-3). 32, 
 
 12(,.:-l)(2xfl). 34 
 10(K) X 506. 
 
 (rt + 2/)-:V) (rt-2/> + 3c). 
 
 (i + 7c)(l-7c). 
 
 ((t + 6 - c; - fO {<i'-h-c + d). 
 
 (9,f + 7//)(5x + i/). 
 
 xxvii. (Pag6 '^^O 
 
 2. (a-?0 (a- + ((&-+- 6^). 
 
 I. {a + h)(d'-ah + h'^). 
 
 3. (rt-2) (rt- + 2((-l-4). 
 
 4. (.t; + 7)(a;2-7^ + 49). 
 
 5. (/)-5)(?r + 56-r25). 
 7. ((6-6)(rt- + 0a-t-3(;). 
 9. (4(1 - lO/') (i6(t- + 40uh + b)Ob-). . 
 
 10. (9x + 87/) (81.X-- - 7'2xy -l- 04//-'). 
 
 11. {x + y) (a;- - ;-*j j/ + 2/''') (.^^ - V) {x^ + a: »/ + if) 
 
 6. (..• + 47/)(..^-4.'//-l-16;/^). 
 8. (2^ + 3ij) (■ix'^ - (ixy + 9y-). 
 
 
(2./- -5). 
 
 - 1> + c). 
 - 7/1 -h n). 
 
 6 + 3c). 
 
 1(\'/-). 
 / + 9r). 
 
 AA'SIVERS. 
 
 353 
 
 12. (,r-f l)(,;'-2_:,M.l)(,;-l)(.,/-5-f., + n. 
 
 1 3. (r< + ii) (f/*-' - 2rf + 4) (« ~ 2) (a^ + 2a + 4). 
 
 1 4. (3 + y) (!) - 3// + 7/-) 1 3 - 1/) (9 + 3 »/ + f). 
 
 I 
 
 3 
 7 
 
 13 
 
 16 
 
 17 
 
 27 
 31 
 
 xxviii. (Pago .11.) 
 
 u-\-h, 2. Take /i I'loin a and add .• to the result. 
 
 2.':. 4. rt — 5. 5. ./; -i- 1. 6. a-— 2, a; - 1, .v, .v + 1 , r 4- 2. 
 0. 8. 0. 9. (?rt. 10. f. II. x~)j. 12. .'•-//. 
 • 305 - (i/;. 14. .^• - 1 0, 15. a; + Tx;. 
 
 A lias a; + 5 shillings, B has 7/ -5 shillings. 
 cc-B. 18. i<7/. 19. 12-x-//. 20. nq. 21. 2.') -a". 
 ?/ — 25. 23. 25G//t^ 24. 4/^ 25. ./;-5. 26. //-f-T. 
 
 a- - 7/- 
 28. 
 
 32- 
 
 28. (^ -I- //)(-'•-//). 
 7. 33. 23. 
 
 29. 
 
 30- 
 
 34- 5. 35. U). 
 
 XXix. (Page 53.]^ 
 
 1. To a add &. 
 
 2. From the square of" a take the square of h. 
 
 To four times tlie square of a add the cuhe of h. 
 
 Take four times the sum of the squares of a and h. 
 
 Frum the s(|uare of a take twice h^ and add to the result 
 three times c. 
 
 To a add the product of m and i, and take c from the 
 result. 
 
 To a add 7H. From h take c. Muliijdy the results 
 together. 
 
 Take the square root of the cuhe of x. 
 
 Take the scjuarc root of the sum of the sqv • ;s of x and y. 
 
 Add to a twice the excess of ?> ahove c. 
 
 Multiply the sum of a and 2 by the excess of 3 above c. 
 [s.A.] 'A 
 
 J)- 
 4- 
 5- 
 
 6. 
 
 7. 
 
 Q 
 
 9- 
 10. 
 
 II. 
 
 t: 
 
 i 
 
 ^1^ 
 1 
 
/ 
 
 / 
 
 354 
 
 ANSWERS. 
 
 12. Divide the sum of the squares of a and h by four times 
 
 the product of a and h. 
 
 13. From the square 01 a.- subtract the square of y, and take 
 
 the square root of the result. Then divide this result 
 by the excess of x above y. 
 
 14. To the square of x add the square of y^ and take the 
 
 square root of the result. Then divide this result by 
 the square root of the sum of x and y. 
 
 XXX. (Page 53.) 
 
 I. 
 
 2. 
 
 2. 
 
 0. 
 
 3- 
 
 ^7. 
 
 4. 
 
 31. 
 
 5- 
 
 20. 
 
 6. 
 
 33. 
 
 7- 
 
 .05. 
 
 8. 
 
 27. 
 
 9- 
 
 14. 
 
 10. 
 
 120. 
 
 II. 
 
 210. 
 
 12. 
 
 1458 
 
 13- 
 
 30. 
 
 14. 
 
 5. 
 
 15- 
 
 ?>. 
 
 16. 
 
 4. 
 
 17- 
 
 49. 
 
 18. 
 
 10. 
 
 19. 12. 20. 4. 21. 43. 22. 20. 23. 29. 24. 41536. 25. 52. 
 
 
 xxxi. (Page 54.) 
 
 I. 0. 2. 0. 3. 2ac. 4. 2.177, 5- o? + }f; 
 
 6. 4x* + (6m - 6?i) .^'^ -(4m--t-9r>??i + 4?r).r2 
 
 + {^m-n - Q>m n-) x + 4m^ni 
 
 7. cr'^ + dr + e. ' 8. - a^ - ¥ ~ c^ + '2a^b'^ + 2a^e^ + 21"-^ 
 
 When c = 0, this becomes - a^ - b* + 2a-b-. When 
 h + c = a, the product becomes 0. Wlii;n a = J) = c, it 
 becomes 3rt*. 9. 0. 10. 34. 
 
 12. (a) {a + b)x''- + {c + d)x. (/3) ((/ - /^ •'■" - ^" + r/ - 2) :c-'. 
 (7) (4 - (/) x^ - (3 + h) .r2 _ (5 + c) a-. (5^ n- - //- + f2(^ + 2b) x. 
 (e) (^jj/-^ - }?-) .1'' + (2?/)f2 - 2^7) X'' + (2)11 - 2ii) x-, 
 
 1 3. x'^ - (a + /» + (•) a,'- + {ab + ac -f be) x - tibc. 
 
 1 4. .'-'^ + (ft + b^-c) x'^ + {ab +ac-r he) x + abc. 
 
 J 5. (,, + ft + cf = a^* -H 3a-/i + 3^f //-' + 6-' 4- v^ + 3f r-V 
 
 + 6/^/;r + 3//"V-l 3rtr-' + 3/jr-'. 
 
 r« ^b- cf = «/^ ^- '^(i~b + 3«6-' -1- ?i^ - .■■' - :W^(' 
 
 -Qabc-Zb'-c + :iu,-+:)b/\ 
 
 I. 
 
 8. 
 II. 
 
 17- 
 
 23- 
 29. 
 
 I. 
 
 6. 
 10. 
 12. 
 
 '5- 
 19. 
 
 20. 
 
lour times 
 
 /, and take 
 this result 
 
 (1 take the 
 i result by 
 
 6. 33. 
 
 12. 1458. 
 iS. 10. 
 
 36. 25. 52, 
 
 5. a^' + hl 
 
 b-. When 
 
 (( = ], = c, 1 1 
 
 b. 34. 
 
 (■ + (I - 2) :c-. 
 \- (-la + 2h) X. 
 
 3«r2 + 3k2. 
 
 f 3(h-4-3/-,'', 
 
 ANSIVERS. 
 
 355 
 
 (^ + ,. __ „):! =. _ fe:j 4. 3f,;-'ft _ 3^?,-i + fts + gS + 3rt'^(; 
 
 (c + r? - hf = «•■< - 3rt*'^6 + 3r<6- - ft"^ + c^ + 3«-c 
 
 - 6rt/jtf + 36-f + 3ac- - 3/)r-. 
 The sum of the last three subtracted from the tirat gives 
 24a6c. 
 
 t6. 9fi2 + 6ac - 3a6 + 4&C - 6^2. 17. r/T'-,-'i'». 
 
 1 8. 2«c - 26" - 2rt(i + 2/>(i The value of the result is - 26c". 
 
 1 9. 06 + ,'•?/ + (6 + 1 + 2a) X + (2rt - 6 - 1) ?/. 
 
 20. 9. 21. f<6 + .(:- + (a-6+ l).'c-(a + 6+ 1) //. 
 
 22. 2. 23. (7m + 4?i + l)a;+(l -6h-4/?0?/' 
 
 2 5 . -W + 6«f + 2«?> + 96c - 66'-^. 26. 3 ; 1 28 ; 3 ; 1 1 «. 
 
 27. 9. 28. 44. 29. 20. 30. 35. 31. 18. 
 
 xxxii. (Page 60.) 
 
 I. 
 
 3. 
 
 2. 2. 
 
 3- 1. 
 
 4- 
 
 7. 5. 2 
 
 6. 2. 
 
 7- 
 
 3. 
 
 8. 
 
 4. 
 
 9. 9. 
 
 10. 
 
 An 
 
 S-. 54. 
 
 
 
 
 1 1. 
 
 2. 
 
 12. 9. 
 
 13. ^>- 
 
 
 14. -7. 
 
 15. 3. 
 
 16. 
 
 
 17- 
 
 2. 
 
 18. 8. 
 
 19. 10. 
 
 
 20. 6. 
 
 21. 4. 
 
 -> -» 
 
 10 
 
 
 3. 
 
 24. 15. 
 
 25. 1. 
 
 
 26. 2. 
 
 27. 3. 
 
 28. 
 
 4. 
 
 29. 
 
 6. 
 
 30. -1- 
 
 
 
 
 
 
 « 
 
 XXXiii. (Page 62.) 
 
 I. 70. 2. 43. 3. 23. 4. 7,21. 5. 36,26, |s, 12. 
 
 6. 12,8. 7. 50,30. 8. lu, 14, is, 22, 2(5, 30. 9. i'6s. 
 10. 12 ,><hilliugs, 24 shilliiigH. 11. 52. 
 
 12. .4 has .£130, B £m), C i'13o, J) i'90. 
 
 13, 152 men, 76 women, 38 children. 14. i;35(>, i;i5(», i720. 
 15.21,13. 16. iU 15«. 17.84,26. 18.62,28. 
 
 19. The wife i'400O, each Hon, i'inuu, each dauj^hter i;50(». 
 
 20. 49 gallons. 21. i;i4. 124, i'38. 22. 31,17 
 
 w if 
 
 ii 
 
 i, V 
 
 A' ■■* 
 
 ^11 
 
 m 
 
 I 
 
 r 
 
If 
 
 23. .£21. 
 
 27. fiO, 24. 
 
 32. 57, 10. 
 
 36. 2(X), 100, 
 
 24. 48, 36. 
 28. S, 12. 
 33. 4. 
 37' '^'^ 20. 
 
 xxxiv. 
 
 25. 50, 4(K 
 
 29. S8. 30. IS. 
 
 34. 80, 128. 
 38. 53, 318. 
 
 ;i. 40. 
 
 39. 5, 10, J 5. 
 
 I. a'^h. 
 6. a-6^. 
 
 (rage G^.) 
 
 2. rZ-j/'-^g. 3. 2x-i/. 4. I'yin-i^j 
 7. 2. 8. 17j)(/. 9. 4/-J/ 
 
 2-2 
 
 5. \8ahcd. 
 
 10. Sdc'-^i/^. 
 
 XXXV. (Pago CO.) 
 
 1 . a - h. 
 6. 1 - 5(fc. 
 
 2. «--/r. 
 7. X + !l. 
 
 a — .'.'. 
 
 8. :r.-ij. 
 
 4. '<+.'•, 
 
 9. .'; - 1 . 
 
 5. 3.':+l. 
 
 10. l+((. 
 
 XXXV i. (Pago 70.) 
 I. 3453. 2. 30. 3- •)-«• 4. 355. 5. 23. 6. 2345. 
 
 (Page 74.) 
 
 I. cr.-\-4. 
 
 - ',' — '\ 
 
 g. x-y. 
 
 13. .7H-7/. 
 
 17. bx-ij. 
 
 20. .' ■-' + .'•// + ;/■• 
 
 23. lU-if. 
 
 26. 3 ((t -.'■). 
 
 2Q. ,i;2+)/-. 
 
 3. .v-/. 
 
 7. ft;-4//. 
 
 II. .'•-//. 
 
 15. 4'' F//. 
 
 XXXVll. 
 
 2. r^ + 10. 
 6. .'0 + 3//. 
 10, .''j + 7/. 
 14. (I + h-- c. 
 J 8. x^ + ,'.•■"• - 4,/'- + .'• -f 1 . 
 21. x" A'X-'-j:- 1. 
 24. 3.'- 11?/. 
 27. 3.'' -2. 
 30. ;'-f-3. 
 
 4. ;'■+ 12. 
 8. x-lby. 
 
 1 2. rr -f ?/. 
 
 16. 3.''-7/, 
 
 19. .(;'-' -2./' 4 4. 
 22. 3f(-4- 2((/; - /''. 
 25. 'Sa-h. 
 28. 3,r + (r. 
 31. (3rt + 2.')''- 
 
 xxxviiL (Page 7().) 
 
 I. ,7:4-2. 2. .r-1. •' 3- •'^ + 1- 4- .'/-I. 
 
 5. x^-2x+'). 6. .'■ 2. ■ 7. //--2// + 5. 
 
ANSWERS. 
 
 357 
 
 I i 
 
 ). 42, IS. 
 31. 41), 
 19, 22. 
 5, 10, 15. 
 
 . Ibahcd. 
 
 5. :u + l. 
 
 6. 2345. 
 
 + 12. 
 
 + ?/• 
 
 2.t- + 4. 
 
 I- 2(f6 - /' '. 
 6. 
 
 2.')''. 
 
 4. //-I. 
 
 I. 
 
 3rt* 
 
 5. ^-. 
 
 9* \lA,f 
 
 13. 
 
 17- 
 
 20. 
 
 23- 
 
 27. 
 
 3?/ - 5.' ■; 
 
 3a^>_ 
 26c + G 
 
 5 
 
 2x - 2// 
 
 xy 
 
 12* 
 
 xxxix. (Page 81.) 
 
 10. 
 
 2.>: 
 • 9' 
 
 5/7? 
 
 5h 
 Ua 
 
 2az' 
 
 P 
 
 a 
 
 a + h 
 
 12. = 
 
 4- 
 
 5:^ 
 
 8. 
 
 bJPc 
 4't--^ 
 
 2 
 
 ma; 
 
 3w'''p - x' 
 
 14. 
 
 2rt + X 
 
 4ax' - x 
 
 
 16. 
 
 18. 
 
 c - 2a 
 c + 2rt' 
 
 21. 
 
 b-' 
 
 7ax - 7bij 
 
 02 — - 
 
 a' 
 
 2.^ - \iy' 
 
 3 
 19. 5. 
 
 2 
 
 "■^' 2(«^c* 
 
 28. 
 
 "5- 2c' 
 
 9ahx-12cdx 
 2a + 2/; 
 
 26. 
 
 a' 
 
 x 
 
 Xl. (Page 82.) 
 
 
 a + 5 
 
 
 x-5 
 
 
 .x- + 1 
 
 I. 
 
 a + 3' 
 
 2. 
 
 u;- 3* 
 
 3- 
 
 x-r 
 
 4- 
 
 .X - 3;/ 
 
 x + 7)i' 
 
 5. 
 
 r-'2-x- + l 
 
 6. 
 
 x^ + ?/•* 
 x^ - if 
 
 7. 
 
 x-2 
 
 x+i' 
 
 8. 
 
 a;-3 
 «+l* 
 
 9. 
 
 x^ - bx + 6 
 3.c2 - 7.>- 
 
 10. 
 
 .x2-5x + G 
 "3rc''* - 8.r "• 
 
 II. 
 
 X^ + 03^ - ?/2 
 
 \ 
 
 
 12. 
 
 a^ + ba + 6 
 
 13- 
 
 7>2 + 56 
 
 14. 
 
 r,i^ + 4771 
 w'-^ + 7// - (* 
 
 15- 
 
 a'-a+l 
 
 a^ + (i + l' 
 
 16. 
 
 3rt,r - 7a 
 Ix- - 3,i; 
 
 17- 
 
 I4.i;-G 
 9ax-2\a 
 
 18 
 
 l()(f-14a2 
 
 '9- 
 
 
 
 
 iH 
 
 *ii 
 
 
 1 
 

 
 358 
 
 ANSWERS. 
 
 
 rt2-a+i 
 
 20. 
 
 o?-2a + 2" 
 
 23- 
 
 ro^ - 2x + 2 
 
 26. 
 
 4a;2 + 9x+l 
 
 2x''^~3a;-2' 
 
 29. 
 
 fl:-3 
 
 32. 
 
 a — h-c 
 
 a + b - (■' 
 
 35. 
 
 X^ + 4: 
 
 x'^ + x+V 
 
 38. 
 
 a;- -2.x + 3 
 
 2x2 + 5x-3' 
 
 I. 
 
 7x2 
 12J/2- 
 
 5s 
 
 aaj. 
 
 Zmnxy 
 
 I. 
 
 a-b 
 
 a' 
 
 . Sx - 1 
 21. ., . 
 
 22. 
 
 a — 5 
 a~-"3' 
 
 . 27. 
 30. 
 
 33. 
 36. 
 
 39- 
 
 3. 
 
 25. 
 
 2./;2 + 3x-5 
 T.t; - 5 
 
 2x - 3a 
 
 4«2 + 6mK + 9rt^- 
 
 
 ^^- x-2' 
 
 7n- 1 
 
 
 x^ + 6x 
 
 lll + l' 
 
 5a + 2b 
 3aT2/»' 
 
 x^ 4 a:2 - 2 
 
 2:/:2 + 2x+l" 
 
 x^- 2x2 -2x4-1 
 4x2 - 7x - 1 
 
 J'- 
 
 x + 3 • 
 
 34- 
 
 x-5 
 2x + 3* 
 
 37- 
 
 x2 + x- 12 
 3x 4 5 
 
 An 
 
 a2 — oa -1- (i 
 
 3«2-8a 
 
 Xli. (Page 86.) 
 
 1 
 
 ^' 2- 
 
 j< 
 
 6. 
 
 10. 
 
 9* 
 
 5km^ 
 
 2x3 
 3w3' 
 
 3 
 
 8* 
 
 4. 
 
 8. 
 
 61/ 
 9rtx* 
 
 8fl2r2 
 9rf2 • 
 
 xlii. (Page 8G.) 
 
 (x-l)(x-6) 
 ^ ~ x« "'• 
 
 7. 1. 8. k 
 
 'I- - w + ?i 
 
 II. 
 
 4 
 
 -> 
 
 - 3* 
 
 
 •-> 
 
 (X + 2) (X - 4) 
 .X (x - 2 1 
 
 'X-H 
 
 5- x-3- 
 
 
 6. 
 
 (x-2)(x-5) 
 
 X2 
 
 9. 
 
 x-</' 
 
 
 JO. J. 
 
 c-a~b 
 
 X - // - z 
 
 x-^-m-n 
 
 i^.. 
 
 X + .7 + : 
 
 
ANSWER,^. 
 
 359 
 
 ) 
 
 f :3.x - 
 
 5 
 
 ./J -5 
 
 
 x-^ 
 
 
 x-2' 
 
 
 '^ + 6x 
 
 
 K + ?j' 
 
 
 c-5 
 
 x + :y 
 
 
 [+_x - 
 3:6-4 5 
 
 12 
 
 1 r 
 
 ' — 5a -1 
 
 (i 
 
 ia- ~ 8a 
 
 
 (■^-4) 
 0<;-5) 
 
 - a - 6' 
 
 xliii. 
 
 (p 
 
 acre 
 
 ^"■; 
 
 I. -7 
 
 lOac 
 
 8.):?/ 
 
 36x 
 
 o 
 
 ^// 
 
 3- -I 
 
 4- o/,",,-.. 
 
 36 
 
 H.v; 
 
 14' 
 
 x-^2 
 
 xliv. (Page 80.) 
 
 I. 12<rV. 
 
 2. 12.X2//2. 
 
 5. 4(^/;^ 6. «2//2^., 
 
 9. 20j;2^2^. 10. 72axh/. 
 
 3. 8a ^/r. 
 
 7. rt''.l--?/2. 
 
 4. 
 
 r^-,'' 
 
 8. 1()2«-,.'. 
 
 I. x^{ii-^x). 
 
 4. 4.T2-1. 
 
 7. (.r'-l)(x + l). 
 
 9. (x + l){x^-l). 
 
 13. (2rt.-l)(8a3 + l). ■ 
 
 15. (a. + &)2 («-?>). 
 
 [7, 4(1-0-2). 
 
 19. {a- h) {(I - c) {h - r). 
 
 -I. (./• + ?/)2(a:-i/)2. 
 
 23. a:2(x2-,y^). 
 
 25. U{x-yY(x^+if). 
 
 xlv. (Pago 91.) 
 
 2. x^-x. 3. a (((--¥). 
 
 5. a^ + 6^. 6. a-'---l. 
 
 8. (r^+l) (.'••'+ 1). 
 
 10. x^-l. 
 
 12. .'r(/+l)<'r"- 1). 
 
 14. 2..J2 + 2^7/. 
 
 16. ^2 -/>-'. 
 
 j8. ,r^-l. 
 
 20. (x+l')(x + 2 {.. +^). 
 
 2 3. (rt + 3)(a2_l). 
 
 24. (;t'+l)(7: + 2)(;/- + 3)(./; + 4). 
 
 26. 120x'?/(x2~?/^). 
 
 xlvi. 
 
 I. (,'; + 2)(.f + 3)(.<;-l 4'). 
 3. (.c+l)r''l-2)(.'- + 3). 
 5. (./;-ll)(x' + 2)':r - 2\ 
 
 (Page 93.) 
 
 2. (a - 5) (/< + '4) (a - 3). 
 4. (x + 5) (x + G,) (.< + 7). 
 6. (2x-\-l)(x^l){c--2). 
 
 ' ^ik 
 
m 
 
 
 m 
 
 'I, 
 1 
 
 PS 
 
 3fc 
 
 ANSWERS. 
 
 7. (^^'-^ + y) (.''^ + !/)(.'>' + 2/2) (X- 7/). 
 
 9. (7a;-4)(3i-j-2)(a;2-3). 
 II. (a'-^- 6^) (a + 26) (a- 2?>). 
 
 8. G/;~5)(.(;-3)(;C + 5). 
 
 Xlvii. (Page 94.) 
 
 I. (.>J-2)(u;-l)(rc-3)(a;-4). 2. (.0 + 4) (a: + 1) (a: + 3). 
 
 3. (^-4)(x--5)(a;-7). 4. (3:^-2) (2u; + l)(7.c-l). 
 
 5. (.':+l)(rf-l)(aJH-3)(3:/j-2)(2a;+l). 
 
 6. (^:-3)(.c'^ + 3a; + 9)(a-12)(:c2-2). 
 
 I. 
 
 \bx 16,« 
 20"' '20' 
 
 xlviii. (Page 95.). 
 
 9.0 - 21 4a; - 9 
 
 Ax. - 8// 3.«2 _ 8^;.y 
 ^ lOx-" ' "l(U2"' 
 
 48ft;' -60r- 15a -10c 
 
 eoa^o • 
 
 4- 
 6. 
 
 18 ' ~ 18 • 
 2()(( + 25/j 6rt2_M 
 
 :</,-' ' 
 
 ft-^6' 
 
 aW ' 
 
 7- 
 9- 
 
 3 - 3.r 3 + 3« 
 
 !-.(;-' l-x^- 
 
 5 + 5.C 6 
 
 1-^2' i_^2- 
 
 g 2 + 2/r 2-27/2 
 ah + a.r h 
 
 IT. , 
 
 a — c 
 
 n. 
 
 (« - /)) (ft - c.) {a - c)' (rt, - 6j (/) - c) {a - c)' 
 
 c{ h-c) b(a-b)_ 
 
 (the (a - 1) (a - c) (h-cy abc (a -h) (« - cYilT^c)' 
 
 i , 
 
 15./--f 17 
 15 
 
 xlix. (Page 98.) 
 
 7lrt-2o6-56c 
 84 • 
 
 22x + 9y 
 
 42 
 
 4- 
 
 
 5. 
 
 27,f2-2./;''i/-i6x7/-i%2 
 ^ 12:i;2 
 
 II 
 
AATSIVE/?^. 
 
 3^» 
 
 6. 
 
 
 35, t-^-h23^^ /) +J 1 he - 42c^ 
 
 8(K>;3 -t 64r- + 84,'' -I- 45 
 4a-c - 3ac- - 3rtc + 7c^ 
 
 1 1 i/- - 8.«-?/2 - 4^;?/ - 7x^ 
 
 T O -!- - - 
 
 '3a^ - la% + 4a%c - tmh'^c + ahc'^ - h-c^ 
 
 II. 
 
 aWc^ 
 
 ((•-'(-' 
 
 i 
 
 i 
 
 I 
 
 . i :M 
 
 
 4. 
 7. 
 
 10. 
 
 4a;2/ 
 
 loi + y){o:-y)' 
 
 1 
 
 (a + x)la- x)' 
 
 1. (Page 99.) 
 4 
 
 - (,_7)(;,_3y 
 -1 
 
 8. 
 
 (x-^/)'-^- 
 
 li. (Page 100.) 
 
 I. 
 
 5- 
 
 2 
 
 l-a 
 
 x + y 
 
 2. 
 
 4a; 
 
 - rt + b.r, 
 0. - -, . 
 
 2a; 
 
 8i 
 4- -«— 
 
 a8->* 
 
 3a;^-24.'"- + f)0y-4f) 
 (;r^2)(./;-3)fi;-4)* 
 
 e 
 
 6. 
 
 3:c' + 20a;--32.:(:-235 
 (x + 4)(.(;-3)(.(; + 7) ' 
 
 3,/r - 2rrv' - fi^-i 
 
 10. 
 
 {.r+l){x + 2)^K + 3y 
 
 II. 
 
 3.»;-' 
 
 14. 2. 
 
 12. 
 
 
 r-d 
 
 (a -f c) ((t + (^) (rt + e)' 
 
 16. U. 
 
 J7. 
 
 13. <>■ 
 
 
 II 
 
 IM 
 
i' 
 
 \62 
 
 1 8. 0. 
 
 ANSIVERS. 
 
 19. 
 
 a -f // 
 
 20. 0. 
 
 21. 0. 
 
 I 
 
 I ^- 
 
 x-y 
 5. 0. 
 
 8. 
 
 1 
 
 1-X4- 
 
 lii. (Page 103.) 
 
 6. 
 
 " 2+cc' 
 • 1 
 
 3- 
 
 3;r; 
 
 ,v.a 
 
 4. ., 
 
 !/ + 6 
 
 (a; + a.)(.i; + />)' 
 
 2 
 
 a^ - 2a¥ + 2a^h + If' 
 1 
 
 7. 
 
 (x-r:)(?/-^5)* 
 
 10. 
 
 ahc 
 
 liii. (Page 110.) 
 
 I. 
 
 2.'(;+ll 
 
 (« + 4) (.T + 5) (;•/; + 7)' 
 2;<;-17 
 
 2. 
 
 6. 0. 
 
 (x-4)(;^+ri)(a;-13)' 
 
 1 Ix-^ - ,/;- + 2bx - 1 
 3(l-x'») 
 
 4- ,^3' 5- 
 
 2 (.. - H) 
 {:o-6)(.<--7)(.--9/ 
 
 11 {m + /i)^ 
 
 7. 
 
 8. 0. 
 
 1 
 
 ^' l~+x 
 
 liv. (Page 107.) 
 
 I. 
 
 16. 
 
 2. 
 
 12. 
 
 3- 
 
 15. 
 
 4. 
 
 2.S. 
 
 5- 
 
 63. 
 
 6. 
 
 24. 
 
 7- 
 
 60. 
 
 8. 
 
 45. 
 
 9- 
 
 :',(!. 
 
 10. 
 
 120. 
 
 II. 
 
 72. 
 
 12. 
 
 OH. 
 
 13. 
 
 04. 
 
 14. 
 
 12. 
 
 «5- 
 
 28. 
 
 16. 
 
 1. 
 
 '7- 
 
 s. 
 
 18. 
 
 0. 
 
 19, 
 
 
 20. 
 
 4. 
 
 21. 
 
 5. 
 
 22. 
 
 i. 
 
 23- 
 
 J. 
 
 24. 
 
 3 
 2" 
 
 25. 
 
 loo, 
 
 26. 
 
 24. 
 
 27. 
 
 2 
 
 28. 
 
 6. 
 
 29. 
 
 24. 
 
 30- 
 
 4. 
 
 Iv. (Page 108.) 
 
 I. 10. 
 
 
 4. 1. 
 
 S. s. 
 
- y 
 
 II. 0. 
 
 1 
 
 
 
 
 .^iVj/ri^iV^-. 
 
 
 .b-b 
 
 6 
 
 1 
 
 9' 
 
 7- 
 
 9. 
 
 8. 2. 
 
 9. 11. 
 
 10. (•). 
 
 II. 2 
 
 1 '^ 
 
 12. 
 
 13- 
 
 H. 
 
 14. 7. 
 
 15. 9. 
 
 16. 7. 
 
 17. 7 
 
 1 1 8. 
 
 9. 
 
 19. 
 
 9. 
 
 20. 9. 
 
 21. 10. 
 
 
 
 ll: 
 
 _1_ 
 
 abc 
 
 I 
 
 1+^' 
 
 5- 
 
 63. 
 
 0. 
 
 120. 
 
 5- 
 
 28. 
 
 0. 
 
 4. 
 
 5- 
 
 loo, 
 
 0. 
 
 4. 
 
 . H. 
 
 Ivi. (Pago 109.) 
 
 a 
 
 a 
 
 + h' 
 
 he, — dm 
 
 3(' - 2 
 bl) 
 
 h {a + c) 
 
 .i^b~bc + <l 
 
 (I 
 
 +/ 
 
 (ibd + db 
 
 4- 
 
 a-.'^ • 
 
 
 
 5' \+a ' 
 
 * 3rt-12^/" 
 
 7' 
 
 3rt/) — 2/(; - 
 4m;- T 
 
 -3 
 
 • 
 
 
 8. 1. 
 
 
 10. 
 
 a 
 "2' 
 
 
 1 1. 
 
 2. 
 
 b 
 12. 0. 13. - , 
 
 •^ (I - 1 
 
 14. 
 
 3(t + 1 
 2a + h' 
 
 
 15- 
 
 18rt-l 2/) 
 4(1 + 3 • 
 
 16. 17. ^ 
 
 18. 
 
 ahd + ac 
 ad + d ' 
 
 
 19. 
 
 b-\. 
 
 b 2d-' 
 
 20. -. 21. , ■ 
 
 
 1. 
 
 -3- 
 
 bm 
 
 24. 
 
 3rt"V)c + 2r( ••*/>+ + iih^ 
 
 25. 
 
 he 
 c'-b' 
 
 
 
 26. -. 
 
 c 
 
 ab-\ 
 
 "^- bc + d' 
 
 28. 
 
 a (vi - 3c' + 3(( 
 
 ) 
 
 • 
 
 29. -^-. 
 
 a-e (c - d) 
 
 Ivii. (Vii-ci 111.) 
 
 I. 2. 
 
 6. !. 
 
 11. 9. 
 16. 12. 
 
 2. 15. 
 
 ^- 2- 
 12. 19. 
 
 . 1. 
 
 4. 
 
 13* 
 
 '• 10- 
 
 1^ 
 
 If 
 
 '1 i Mi 
 
 ir? 
 
 / 
 
 
 
 
 
 
 
 \i 
 
 8. 
 
 G. 
 
 9- 
 
 - 7. 
 
 10. 
 
 35 
 
 
 13. 
 
 1. 
 
 14. 
 
 4. 
 
 15- 
 
 !■! 
 
 
 1 
 
 
 1 
 
 
 
 [:|| 
 
 lo. 
 
 2- 
 
 19. 
 
 b' 
 
 20. 
 
 ■.i. 
 
 1 
 
■i .MW P .i 1 !■. n ' ■ i Hin wirpiUjjpi 
 
 ' m 
 
 3^4 
 
 ANSWERS. 
 
 Iviii. (Page 113.) 
 
 459 
 
 .1. 
 
 20. 
 
 2. 
 
 3. 
 
 3- 40- 
 
 4. 
 
 46^* 
 
 5- 
 
 60. 
 
 6. 
 
 10. 
 
 7. 
 
 5. 
 
 8. 20. 
 
 9- 
 
 3. 
 
 lO. 
 
 I 
 
 II. 
 
 8. 
 
 12. 
 
 lUO. 
 
 13. ^• 
 
 14. 
 
 •1. 
 
 15. 
 
 5. 
 
 1 6. 
 
 5 
 6' 
 
 17- 
 
 5. 
 
 
 
 
 
 
 I. 100. 
 
 6. 2-, 4/^. 
 
 lo. 960. 
 
 14. 540, 36. 
 
 lix. (Page 114.) 
 2. 240. 3. 80. 4. 700. 
 
 7. 24,76. 8. 120. 
 
 II. 36. 
 
 12. 12, 4. 
 
 5. 28, 32. 
 
 9. 60. 
 ;. ^1897. 
 
 15. 3456,2304. 16. 50. 17.35,15. 
 
 18. 29340,1867. 19. 21,6. 
 
 21. A lias ^1400, B lias £400. 
 
 20. 105^, 13l| 
 
 22. 28, 18. 
 
 23 
 
 in {nh - a) n {mb - a) 
 
 24. 
 
 a +h a— h 
 
 55. 18. 
 
 n - m ' m -n '^' 2 ' 2 
 
 26. .£135, i'297, i*432. 27. i7200. 28. 47, 23. 
 
 29. 7, 32. 30. 112, 96. 31. 78. 
 
 34. 20. 35. 42 years. 
 
 32. 75 gallons. 
 
 33. 40,10. 
 
 2,7. 20 (lays. 
 
 41. 4, , (lavs. 
 
 44. 2 lioniN. 
 
 36. 1^- days. 
 38. 10 days. 39. 6 houirf. 40. 1^ days. 
 
 42. 1^ hours. 
 
 ahc 
 
 4:;, , , lllllllUUS. 
 
 ab + ac + he 
 
 43. 48'. 
 46. 48^. 
 
 47. 51. J, 61.., 47. gallons. 
 
 O 'J o 
 
 1 
 
 48, 9^ miles from Ely. 
 
 55- 
 56. 
 
 57. 
 
 58. 
 62. 
 65. 
 68. 
 
 71. 
 
 I. 
 
5. fio. 
 
 o. 
 
 9" 
 
 5. 5. 
 
 . 28,32. 
 9. 60. 
 ^1897. 
 
 J. 35, 15. 
 
 2 
 3* 
 
 131^. 
 
 ANS]l^EKS. 
 
 365 
 
 49. 14 uuIl'S. 
 
 52. 42 hour.s. 
 
 50. 
 
 <(€ hi 
 b' a' 
 
 51. 11 
 
 13 
 21" 
 
 53. 31).^^ miles. 
 
 54. 50 liour.s. 
 
 55. (1) 38jypiistl. (2) 54^^^ past 4. (3) loj^^^'pa.st 8. 
 
 3' 
 
 56. (1) 27y^ imst 2. (2) 5^^ aii.l also 38^^^ past 4. 
 (:)) 21 j ^ pn^^t 7, and also 54 ^^ past 7. 
 
 57. (1) 10^^ past 3. (2) 32^ past 6. ' (.3) 49-^j past 9. 
 
 58. (JO. 59. i'3. 
 
 62. X600. 
 
 65. 90', 72', 60'. 
 
 68. 2, 4, 94. 
 
 71. 30000. 
 
 6°- i, 
 
 61. 18i days. 
 63. ;£275. 64. 60. 
 
 66. 1 26, 63, 56 days. 67. 24. 
 
 69. 200. 
 72. i:200000000. 
 
 K' 
 
 70. 2^ 5 
 
 ih n '-■ 
 
 II 
 
 73- '"iO. 
 
 "1 ^ 
 11 
 
 il 
 
 f 4'M 
 
 25. 18. 
 
 47, 23. 
 5 gallons. 
 
 1- days. 
 
 ^29 ^^^''• 
 48'. 
 
 6. 48'^. 
 
 nil Ely. 
 
 I. 
 
 a;^ + ax + 3a 
 
 X 
 
 .^2 ^ ^2 
 x(:x-y) 
 
 I. 
 
 8 - 13x 
 70 • 
 
 Ix. 
 
 (Page 127.) 
 
 a;3 + 5.i2+l 
 5- 2x=^-cr'<+r 
 
 2,/^4-6rf2/, + 6a6- + 263 
 
 ((< - />) («.- + />•-') 
 
 Ixi. (Page 128.) 
 
 2. 
 
 a^ + ?/ 
 xy ' 
 
 6. 
 
 3. a;(l-a;). 
 
 u'^-x + l 
 
 7. 
 
 r^ + ?/ 
 a- - y. 
 
 a^ + a+l 
 a 
 
 ^ 
 
 IVM 
 
 I Mill 
 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
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 (716) S7a-4S03 
 
 
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?66 
 
 /J XS J VERS. 
 
 8. 'X. 
 
 13. 
 
 X 
 
 
 10 u;. 
 
 1 1. 
 
 
 14. 7/1-1. 
 
 15. T/ 
 
 12. rc- 
 1 
 
 c {a -b- (.)' 
 
 II. 
 
 Ixii. (Page 129.) 
 
 1 3^ 1 5 
 
 ^' 2'^ 2a'^ cl''^ 2<^ 
 
 X 3 3 w 
 
 3. - .,- + ---V 
 ^ )/ y X x'- • 
 
 (5w 4r/ 12r 24s 
 c _--', ^- -^ _ 1 . 
 
 ■'' qr^ prs ^J(/s 2W' 
 
 (I h c d 
 
 2. + -\ 1- . 
 
 *" (/ (• (I a 
 
 a^ a'^ a 1 
 
 ^ x^ _ x^ 3x 1 
 ' l(To 40"''40~8" 
 
 I. 
 
 Ixiii. (Page 131.) 
 
 1. 2-2rt + 2rt2-2(/^ + 2r.'> 
 
 , 2 4 8 i<; 
 
 2. 1 + -0-— 3 + - , 
 
 Ill m- ?/r in' 
 
 _ 2b 2}r _ 2&3 2b^ 
 3- ^ a ^Z^ a3*^ "a^ 
 
 ", 2.^2 2x> 2x« 2.t'< 
 
 4. 1 + --, +- . +-7; +--^ 
 
 ^ a** «' a" rt'" 
 
 .7;''^ iC^ .T* xj' 
 ■' a a- (r (V 
 
 , h hx bx' ft.r'' hx^ 
 a (r rt' 0^ ((■' 
 
 7. l-2.f + 6j;2-lC,(;^ + 44a;^ 
 
 8. \+2x + x'^-y^-2x* 
 
 9. l+3?)4-66"'^4-12&H-'lM 
 
 10. x^- 6.1 + 0^ + , 
 
 XX- 
 
 I. 
 
 3- 
 5- 
 
ANSWERS. 
 
 367 
 
 a^ a?h a-lr (C-l? a%* 
 a; x^ x'^ x^ X-' 
 
 , 2x dx^ 4j;3 5.V* 
 
 12. 1 + ., - .. +- , 
 
 a ((- tv^ «' 
 
 13. x^ - '^ax- + 2a-x + \n\ 
 
 14. //(»- l(>m--41w-95. 
 
 36' 
 
 1 
 
 8* 
 
 IxiV. (Pag«! 132.) 
 
 , x^ r-i 23,>' 1 
 I. •+■ 4- + 
 9 4^120 20 
 
 v4_ 
 
 1 
 
 ^r 4i)«(- 7>i ] 
 2. — 4- 
 
 ■20 tJ(Mi GO 1.") 
 
 3- ^ ' ^.4- 
 
 12 11 
 
 .1 J — I 
 
 a'* ac 0- c^ 
 
 4. ''^' + 1+^4- 
 
 5. 
 
 1 _ 1 
 
 5^ 7 107 5 7 
 
 o - 
 
 /,.6 /v».i ^...» 
 
 10. ,,- 4- -.-4. 
 
 I. x-~. 
 
 X 
 
 Ixv. (Page 134.) 
 1 
 
 2. a + 
 
 b' 
 
 n m I 
 
 3. m2 - - + -T 
 
 ^ c^ c"^ c I 
 
 4- '''^;??+r/^^^^^r/- 
 
 a:- 
 
 r' 
 
 5. %?'. 
 ?/ ^' 
 
 3 
 
 I 
 
 ^'. .,+ ,+,.> 7. ^..-2 4-".> 8. '{r^~5x'^+\.r + U. 
 
 If ((I) b- y- .<- 1 4 
 
 9- r.-l+ ... 
 
 ^°- a' ah'uclr bc'^c-' 
 
 Ixvi. 
 
 I. '05.^2 -'HSx- '021. 
 3. •l2/" + '13x>/-'14j/-'. 
 5. 0.. 
 
 (P:.-e 135.) 
 
 2. •0].r2+l'25u;-:il. 
 
 4. •)72.<;-- •05,;7/--31Jy-. 
 
 6. -^00703, 
 
 i| 
 
368 
 
 ANSWERS. 
 
 \ 
 
 Ixvii, (Page 135.) 
 
 ^ «! (t, (ty / 
 
 C-M). 
 
 2. u;//^ 
 
 
 4. ('t + 6) I (a + fiy^-c(a4-6)-rf + ^^-^J. 
 
 I. 46. 
 
 Ixix. (Page 138.) 
 
 2. — j= r- and . 
 
 2a^3 
 
 4- 
 
 6. 
 
 37a;2^7r-l});.2 
 24 • 
 
 60.^'' + 42(t.(:" - ] 07 a V + lOa^.r -f 14(i^ 
 12 • 
 
 11 
 5- -9. 
 
 a;3 ^. a>2^^ + 2?/^ re -8 a;* 
 
 O, >—;,-' r.T — • TO. — Q. II. 4 
 
 x{x^-y^) x + S 1 - .r 
 
 12. 
 
 r-6 
 
 13. z^-;.. 
 
 14. 
 
 ff& + <<c + he + 2a + 26 + 2r + 3 
 abc + ab + ac + hc + a + h + c+ I' 
 
 1 h__hl ^^ 
 
 ^' a ax a^x ax^' 
 
 20. 
 
 23. '•. 24. 0. 
 
 rt^f^n 1 
 27- ^2 - • 
 
 
 18. 
 
 21. 
 
 8rt-6-' ^a^ + ft^) 
 
 1 f/ 4- /' - ^' 
 
 2(.o+l)-" ""• a-lHv' 
 
 30. 1. 
 
 31. ;^- 
 
 -2 + 5aH-17x2-nr^-21,r'« 
 ^^' (3-2x-7x-V« 
 
 
 33. •> . ..o. 
 
 34. ^- 
 
 35. 
 
 
 4'- 
 
 /■ 
 
 46. 
 
 r 
 
 V 
 
 49- 
 
 2 
 
AX.'^IVERS. 
 
 3<59 
 
 35- 
 
 2a 
 
 a 
 
 + h 
 
 'I. x- + 3,»; + 3- + 
 
 \6. 0. 
 
 ^ 1 
 
 39- 
 
 X %" 
 
 46. 
 
 V - i 
 
 49. 2(6- - aa; - ay. 
 
 
 47- 
 
 1 
 
 (,r;2-|-l)(,7~^+lV 
 
 50. 
 
 u 
 
 a 
 
 + h + < 
 
 40. 
 
 (I 
 
 + b-< 
 
 44. 1. 
 
 48. 1. 
 
 51. (a'-¥y 
 
 \xx. (Page 14.5.) 
 
 2ap 
 
 .0 = 10 
 
 //^3. 
 
 // = «. 
 
 .';=10 
 )/ == 35. 
 
 2. it' 
 2/ 
 
 = 9 
 
 = 7. 
 
 5. ;t=19 
 
 y- 
 
 o 
 
 8. .<; = 2 
 
 -8 
 
 — r. 
 
 // = b 
 
 il ^ ''^' 
 
 y = '3. 
 
 a 
 
 
 
 
 Ixxi. 
 
 (P. 
 
 ige 145.) 
 
 
 
 I. 
 
 .r-12 
 
 2. .0-9 
 
 
 3. re = 49 
 
 4. .,;=13 
 
 
 
 // 1. 
 
 7/ = 2. 
 
 
 7/ = 47. 
 
 ?/ = 3. 
 
 
 5- 
 
 .,, = 40 
 
 6. .T = 7 
 
 
 7. X^i> 
 
 8. .'3 = 6 
 
 
 
 7/ = 3. 
 
 y = ± 
 
 
 7/=l. 
 
 2/ = 4, 
 
 
 9- 
 
 x = 7 
 
 
 
 
 
 ' 
 
 
 v/=17. 
 
 
 
 
 t 
 
 
 
 
 Ixxii. 
 
 (I 
 
 'ago 1 It;.) 
 
 ' 
 
 
 I. 
 
 .': = 23 
 
 2. .»•::--. 8 
 
 
 3. --JJ 
 
 4. .»; = .5 
 
 
 
 //=10. 
 
 i/-4. 
 
 
 .'/ -= 2. 
 
 // = 9. * 
 
 
 5- 
 
 x = 2 
 
 6. a- = 7 
 
 
 7. .1-12 
 
 8. .'; = 2 
 
 
 
 // = 2. 
 
 7/ = 9. 
 
 * 
 
 
 j/ = 9. 
 
 7/ = 3. 
 
 
 9- 
 
 /=3 
 
 7/ = 20. 
 
 ■ [S.A.J 
 
 
 
 ,- J "\ 
 
 2 a < 1 
 
370 
 
 AiXSJVERS. 
 
 I. X- 
 
 // 
 
 5. X 
 
 y 
 
 9. X ■■ 
 V- 
 
 I. .' 
 
 // 
 
 5. X 
 
 II 
 9. .-• 
 
 V 
 
 13. .-: 
 
 D 
 
 — I 
 = -2. 
 
 -- - 5 
 -14. 
 
 -2 
 
 1. 
 
 = H 
 = 12. 
 
 = 9 
 = 140. 
 
 = 12 
 = 6. 
 
 = « 
 = 5. 
 
 I. X- 
 
 11 = 
 
 4. x = — 
 
 Ixxiii. (Pago 147.) 
 
 .2. ..=0 3. x-=12 
 
 .'/ =^ - 3. y= - 3. 
 
 6. .'•= -3 7. ;/: = ? 
 
 //=-2. 2/=-5. 
 
 Ixxiv. (Page 148.) 
 
 .2. .'• = 20 
 //-30. 
 
 6. ..J --=4 
 // = 9. 
 
 10. .'-=19 
 ij = 3. 
 
 14. .':=19. 
 2 
 
 */=-17 
 
 3. ;c = 42 
 ?/ = 35. 
 
 7. ,'; = 5 
 y = 2. 
 
 II. .'• = 6 
 i/ = 12. 
 
 1 
 1 
 
 2/ = 
 
 5" 
 
 IXXV. (Page 140.) 
 
 4. 
 
 ./; = 
 
 : -2 
 
 
 .'/ = 
 
 :19. 
 1 
 
 8. 
 
 «.', ■""" 
 
 2 
 
 J 
 
 
 .'/ = 
 
 3 
 
 4. re =10 
 ?/ = 5. 
 
 8. x = -iO. 
 
 y 
 
 12. .t: 
 
 60. 
 
 3^01 
 
 '708' 
 278 
 59 • 
 
 67 ->j/ 
 ?»7 - ?t^J 
 
 2. 
 
 ce + hf 
 "^ hd f a<j 
 
 em + hn 
 
 -^ ae + be 
 
 '"/- ep 
 
 mq - lip' 
 
 
 cd - af 
 ' /;(/ + ae 
 
 an — cm 
 ac + be 
 
 dc 
 c->^-d 
 
 5- 
 
 n'r + nr' 
 
 x= , -, 
 mn +mn 
 
 . a f h 
 6. a= ^-- 
 
 //- 
 
 
 ;/ = 
 
 7. .1 = 
 
 !/ = 
 
 ^c(J-hc) 
 f-hd 
 
 _ c {ac - d) 
 tf-hd ■ 
 
 8. 
 
 mr' — mr 
 mn' + m'n 
 
 1 ■ 
 
 a 
 
 ~h 
 
 !/ = 
 
 9. a; = 
 
 a 
 
 ca 
 
 y=. 
 
 2h' - 6it-N-J 
 3(t 
 
 10 
 
 36 
 
ANSWERS. 
 
 371 
 
 a 
 
 10. x= ,- 
 
 be 
 
 y- 
 
 a + -2b 
 
 II. x^ 
 
 a' 
 
 V = 
 
 b + c 
 }j2 - r^ 
 
 (( 
 
 12 X- 
 
 i/ = 
 
 hni 
 
 b— m 
 
 bm 
 
 b + 111' 
 
 "lOH 
 278 
 59 • 
 
 I. x = 
 
 2 
 1 
 
 2fr 
 
 y= 
 
 X — 
 
 m + a 
 
 26 
 m-n 
 
 1 
 
 (I 
 
 1 
 
 Ixxvi. (Pago 151.) 
 1 
 
 b-2a 
 
 •'~3a-// 
 
 61 
 5- ^'=92 
 
 V- 
 
 103" 
 
 8. x=~ 
 n 
 
 2/ = 
 
 11) .1 
 _n- - u- 
 
 bd-ac 
 _b'i-a^ 
 
 6. x = 
 
 V 
 
 I 
 
 m 
 
 I. x = l 
 
 y = 2 
 
 5. .r==l 
 ?/--2 
 1^ = 3. 
 
 9. :(;-=2 
 «=10. 
 
 I. 16, 12. 
 4. 31, 23. 
 
 Ixxvii. 
 
 2. X = 2 
 ?/ = 2 
 
 c = 2. 
 
 6. a: = l 
 
 ;^ = 6. 
 10. x=2() 
 
 z = b. 
 
 (Page 153.) 
 
 3. a; = 4 
 i/ = 5 
 ;;; = 8. 
 
 « = 
 
 2 
 3 
 
 !/=-« 
 
 36 
 
 3' 
 
 Ixxviii. (Page 155.) 
 
 2. 133, 1J3. 
 5. 35, 14. 
 
 % 
 
 4. a; = 5 
 7/ = 6 
 
 ^' = 8. 
 
 8. a; = 5 
 y^6 
 
 « = 7. 
 
 3. T'li."), 6-25, 
 6. 30. 40, 50. 
 
 1 
 
372 
 
 AA\SIVERS. 
 
 7- 
 
 £ 
 
 lO. 
 
 4 
 
 14. 
 
 G( 
 
 18. 
 
 1( 
 
 21. 
 
 1. 
 
 24- 
 
 2( 
 
 29. 
 
 8 
 
 34. 
 
 5! 
 
 39- 
 
 2 
 3' 
 
 43- 
 
 i; 
 
 46. 
 
 .3 
 
 49. 
 
 2 
 
 52. 
 
 ►- 
 rf 
 
 55. 
 
 1 
 
 59- 
 
 4 
 
 61. 
 
 1 
 
 l>, i*14(M''2<H). 8. 22.s\, 2(;..\ 9. £200, .y::'.(K), i'2(:(). 
 1,7. II. i7, 11. 12.35,11,98. 13. i!)0, >.r.o. 
 
 >, 'MS. 
 ) barrel" 
 
 15. (J, 4. 16. 10, lo. 
 
 19. 
 
 ').s\ 10^/., 12.s\ (v/. 
 J. 2;. 28. 
 
 30. /i 
 
 3.S., I.S-. ^(1. 
 
 2 2. 4.S-. (v/., 3.<J. 
 
 17. 5o:i, 1(172 
 20. i'20, i'lo. 
 
 26. 45. 
 31. 36. 
 
 27. 24. 
 32. I J. 
 
 35. 759. 
 
 36. 
 
 6* 
 
 37. 
 
 4 
 15' 
 
 23. 35, (55. 
 28. 45, 
 33' 33: 
 
 iS. -. 
 
 40. 
 
 19' 
 
 41. 
 
 35 
 41' 
 
 42. 
 
 19 
 
 40' 
 
 (UK). 
 
 44. i;5000, fi per cent. 45. £4000, 5 jteicHiit. 
 
 4' 
 
 18 
 
 47. 20, 10. 
 
 48. 3 miles an hour, 
 
 miles, 8 miles an hour. 50, 700. 51.450,000. 
 
 2, 60. 
 
 53- 1^ 
 
 ;).s. 
 
 54. 750, 158, 148. 
 
 5 and 2 miles. 56. The second, 320 strokes. 58. 50,30. 
 
 5 
 yd. and 5 yd. 60. )., 6, 4 miles an hour respectively. 
 
 8. 2r 
 
 II. .'• 
 
 22. 4.. 
 25. 
 
 10. 
 
 142851 
 
 I. '2xy. 
 
 5. 267ft-6,( 
 
 .5r('-7r' 
 9' lla;^</^- 
 
 Ixxix. (Page 1G4.) 
 2. 9a^6*. 3. lb;i"/iV. 4. Srt'-^ftV. 
 
 I.- 2rt+3/). 
 5. 3a6c-17. 
 
 6. V^a%^c^\ 
 16.c« 
 
 7- 
 
 3a 
 
 4/r 
 
 8. 
 
 1 
 
 2(a-- 
 
 10. 
 
 17y^' 
 
 II. 
 
 25rfc 
 1 8/r 
 
 Ixxx. (PagelG7.) 
 2. 4//'-3/"'. 3. a/*-fSl. 4. 7/' -19. 
 
 6. x-'-^-Sx' + S, 7. 3x- + 2.tM-l. 
 
 t. 
 
 5- 
 
.-INSH'EKS. 
 
 373 
 
 8. 2/-- - 3r f 1 . 9. -2,1- + u - 2. 10. I - 3.': + 2,/-'. 
 
 II. .'-'^ - 2x"'* + 3./;. 12. 2//'- - 3>/;i + 4.-;-. 13. rn-2/; + 3f. 
 
 14. rt^ + rt-6 + (t//- f R 
 
 16. 2.>':^-\-2ax + W. 
 
 1 8. 4a- - 5a6 + 8/u-. 
 
 20. 2/yv; - 3?//:- -I- 2a;"*. 
 
 22. 4/- - 3x-?/ + 2//". 
 
 25. o.c - 2)/ + 3;V. 
 
 15. ;c''-2x--2,<;-l. 
 
 17. 3-4a; + 7a;2-l(X«A . 
 
 19. 3(1- - 4((j)^ - 5i'. 
 
 2 1 . 5x'-(/ - 3^!/- + 2// '. 
 23. 3r(-2/)H-4c. 24. x^-'6j:-\b. 
 
 26. 2>t;'- - // + ?/-. 
 
 I. 2(i-^ 
 
 a 
 
 6- 
 
 4- 
 
 a b 
 , +-. 
 6 a 
 
 7- 
 
 2rt 36+". 
 
 4 
 
 10. 
 
 1 2 3 
 
 + -. 
 
 X y z 
 
 12. 
 
 ah~:icd + ~-' 
 
 14. 
 
 2vi 'Sn 
 
 ^ . 
 
 n m 
 
 16. 7j;--2./j- 
 
 3 
 
 2* 
 
 X 
 
 1 8. 3.(;'- — - - 3 
 
 Ixxxi. (Page 168.) 
 
 3 a 
 
 ■"> _ _. _- 
 
 « 3 
 
 3. ft^- 
 
 a- 
 
 S. X ""XT". 
 
 •^ 2 
 
 8. a;'^ + 4 + -.;. 
 
 6. a;- + .« - ,. 
 
 9. .ja'v; + 2a-- ,. 
 
 o 4 
 
 4 V 
 
 II. b»i- -+i. 
 
 ?t o 
 
 2x 3?/ s; 
 1 3. '' + . 
 
 a h c d 
 ^^' 3~4'^5~2' 
 
 17. 3x^ 
 
 ax 
 
 + bx. 
 
 I 
 
 111 
 
 } 
 
 
 Ixxxii. (Page 170.) 
 
 I. 2a. 2. 3.'"//-. 3. -bmn. 
 
 5. 76-V\ 6. -Wab'c\ 7. -12w')t^ 
 
 4. -6rt'6. 
 8. lla^6«. 
 
^14 
 
 A/VSIVERS. 
 
 Ixxxiii. (Page 172.) 
 
 I. a-k 2. 2(1 + 1. 3. ft + 86. 4. a-\-b\-c. 
 
 5 . X - // + ;j. 6. ;i.'J-' - 2x + 1 . 7. l-H + a'i 
 
 8. X- y + 2::. 9. a'-* - 4(( -I- 2. 10. 2)///^ - 3//i + 1. 
 
 II. .'J -t 2// - ;.. 
 
 I. 2r/-.3.,\ 
 
 4. '( -- 1). 
 
 I. ±8. 
 
 9. ±52. 
 
 12. 2in~'dn~r. 13. »i.+ l 
 
 Ixxxiv. (Piige 17;i.) 
 2. 1 - 2u. 
 5. .r+l. 
 
 IXXXV. (Page ITT).) 
 
 2. ±((6. 3. ±101). 
 
 6. ±8a-r. 7. ±0. 
 
 10. ±4. II 
 
 1 
 
 IIL 
 
 3. 5 + 4.r, 
 6. Ill - 2. 
 
 4. ±7. 
 8. ±129. 
 
 • *V('i-> 
 
 13- ± V6. 
 
 14. ±2v'2. 
 
 Ixxxvl. (Page 179.) 
 I. (>, -12. 2. 4, - IH. 3. ], -JT). 
 
 5. 3, - 131. 6. 5, - 13. 7. 9, - 27. 
 
 I. 7, -1. 
 5. 8,4. 
 9. 12, 10. 
 
 Ixxxvii. (Page 180.) 
 :., -1. 3. 21, -1. 
 
 6. 9, 5. 
 10. 14, 2, 
 
 7. 118, 110. 
 
 4. '2, -48. 
 8. 14, -30. 
 
 4. '.>, - 7. • 
 8. 10±2\'34. 
 
 Ixxxviii. (Page 181.) 
 
 I. 3, 
 1 
 
 10. 
 
 5- 
 
 4' 
 
 8. 8, 
 
 
 25 
 2* 
 
 4. 20, -■; 
 
 6. 9, -8. 
 
 
 7. 45, -82. 
 
 9 4. 15. 
 
 
 10. 290, 1. 
 
AXSIVERS. 
 
 375 
 
 b l-c. 
 
 *• 3' 3* 
 
 , 3 
 
 4- ^1 \{- 
 
 2 
 
 7. 8, 
 
 Ixxxix. (I\i;4<' l^L'.) 
 
 ] 3 
 
 - 5' 5" 
 
 3 5 
 
 5- ./ f 
 
 8. 7 
 
 46 
 
 7* 
 
 3- :^ 
 
 S> 
 
 6. !, -r. 
 
 
 xc. 
 
 (Pago 182 
 
 ) 
 
 • 
 
 1. 3,-3. 
 
 2. 
 
 10, . . 
 
 
 3. «,-^ 
 
 4. s,-^^. 
 
 5- 
 
 5 -^^ 
 
 
 6. 4, 1 
 
 7. B, -^. 
 
 8. 
 
 7 3 
 2' 14" 
 
 
 
 XCi. (Page 184.) 
 
 d h 
 
 m i III 
 3- 2' ~ 2* 
 
 u- + ah (i^-'iih^ 
 
 ''• a-h' 
 
 8. 
 
 c' Cb I 
 
 62 62 
 
 10. — , — • 
 
 ac ac 
 
 5. 1, -a. 6. /'. -" 
 
 9- 2 (a + 6) "' ^2 (a + 6; ' 
 
 + 
 be 
 
 2a '- b 3a + 26 
 II. - . , - - 
 
 ac 
 
 ac'^ + hd^ _ «t-j-6t/- 
 ^^' "2^'3(iV<5' 2a-3rfVc* 
 
 xcii. (Page 185.) 
 I. 8, -1. 2. fi, -1. ,3- 12, - 1. 
 
 5, 2, --D. 6. C, '^. 7- 5, 1. 8. 4, - 1. 
 
 4. 14,-1. 
 9. 8, - 2. 
 
 ■ m 
 
 ? 
 
37^ 
 
 
 ^fA:9/ri^/v\9. 
 
 
 1 
 
 10. 
 
 •i ^ 
 
 •^'"3- 
 
 _ 1 
 ... 7,3. 
 
 12. 12, -1. 
 
 13. 14,-1. 
 
 14- 
 
 3 f) 
 
 2' «• 
 
 ..13,-- 
 
 16. 5,4. 
 
 17. 36, 12. 
 
 1 8. 
 
 «,2. ^ 
 
 .# 25 5 
 '9- 18- -:i- 
 
 20. ', - 7- 
 
 21. 7,- ^. 
 
 22. 
 
 7, - 5. 
 
 =3. 3, -^. 
 
 I 2 
 24. 2» 3- 
 
 2 1 
 25. 3, (.. 
 
 -•6 
 
 in, - 14. 
 
 -7. -, 3. 
 
 28. 3, - ^ . 
 
 29. 2, ;. 
 
 ■ 
 
 2 -2'^ 
 ■'' 15- 
 
 3T. 3,-3-. 
 
 32.4,-3. 
 
 3 J- ••? 21* 
 
 58 
 34. 14,-10. 35. 2, '^3. 36. 5,2. 37. -a, -6. 38. -a,h, 
 
 39. a-{-b,a-b. 40. a-, -a^ 41. 1,- ,. 42. , , . 
 
 1. .'>; = 3(> or 10 
 
 4. .c = 22 or - 3. 
 i/ = 3 or -22. 
 
 I. r(; = 6 or 
 7/ = 2 or 
 
 4. x = 4 
 2/ = 4. 
 
 -2 
 -6. 
 
 I. .T = 4 or 3 
 //=^3 or 4. 
 
 4, x — A or - 2 
 1/ = 2 or - 4. 
 
 xciii. (Pnge 187.) 
 
 2. x* = 9 or 4 
 ?/ = 4 or 0. 
 
 5. :« = 50 or -5 
 i/ = 5 or - 50. 
 
 XCiv. (Page 187.) 
 
 2. .T=13 or ~3 
 7/ = 3 or -- 13. 
 
 5. re =10 ov 2 
 2/ = 2 or ]0. 
 
 xcv. (Page 188.) 
 
 2. a- = 5 or 6 
 ?/ " ( (!■ 5. 
 
 5. .»=5 or - 3. 
 // = 3 or - 5. 
 
 3. .'• = 25 or 4 
 7/ = 4 or 25. 
 
 6. .t = lO0 or - ], 
 //=1 or - 100 
 
 3. .r = 20 or -6 
 y = Q or -20. 
 
 6. if = 40 or y 
 2/ = 9 or 40. 
 
 3. .<; = 10 or 2 
 ij-'l or 10. 
 
 6. £ =7 or 4 
 y = \ or -7. 
 
ANSlVEJiS. 
 
 377 
 
 I . X — 5 or 4 
 // — 4 01- f), 
 
 4 a: = 3 
 2/ = 4. 
 
 I. .f = 4 or -3 
 ?/ = 3 (1-4. 
 
 4. rc=±M 
 7/ = ± t. 
 
 7. r:=- ±2 
 2/ =±5. 
 
 TO. .r=: +2 
 
 13. .r=10 or 12 
 
 2/ 
 
 = 12 or 10. 
 
 xcvi. (Page 181).) 
 2. a; = 4 or 2 
 
 '// = 2or 4. 
 1 
 
 2/ = 2. 
 
 I I 
 
 3- 
 
 X 
 
 ~3 
 
 or 
 
 .> 
 
 
 V 
 
 _ 1 
 
 ~2 
 
 • •r 
 
 1 
 'A' 
 
 6. 
 
 V 
 
 I 
 6 
 
 1 
 
 
 
 xcvii. (Page I'Jl.) 
 
 2. i';=±f) 
 ?/=±3. 
 
 5. 0;=') or 3 
 i/ = 3 or 5. 
 
 8. x = 6 
 II. a^=±7 
 
 3. r/;= + l() 
 7/= ±11. 
 
 6. x = 
 
 a or - 
 
 7/ = 2 or 
 ?/=±l. 
 
 95 
 
 2S 
 33 
 
 12. u; = 3 01 
 
 '?/ = 2 or 
 
 11 
 6 
 
 6- 
 
 14. ro = 4 or 
 2/ = 9 or 
 
 S.") 
 
 19 
 
 x=±\) (.r ±1l' 
 7/= ±12 or ±9. 
 
 xcviii. (Page 11»3.) 
 
 I. 72. 
 6. 29, 13. 
 
 2. 224. 
 
 7. 30. 
 
 3. 1«. 
 
 8. lo"; 
 
 4. no, 15. 
 
 9. 75. 
 
 5. 85, 76. 
 10. 20, 0. 
 
 II. 18,1. 12. 17,15. 13. 12,4. 14. 1290. 15. 5(> 
 
 1 
 
 16. 2601. 
 
 17. 6, 4. 18. 12, 5. 19. 12, 7. 20. 1, 2, 3. 
 
 21. 7,8. 22. 15, 16. 23. 10, 11, 1 
 
 >4. 12. 
 
 1$. 10. 
 
 26. £-2, 5s. 27. 12. 28, 0. 29. 75. 30. 5 aiK 
 
 1 71 
 
 lours. 
 
 31. 101 yds. and 100 yds. 32. O: 
 
 34. 10 yds., 2 yds. 35. 37. 
 
 33. 03 It., 45 ft. 
 
 36. 100 
 
 37- 1976. 
 
 I 
 
378 
 
 answers: 
 
 
 m 
 
 xcix. (Page 11)9.) 
 
 I. .x = 3 
 
 2. :'• = 5 3. X = 00, 71,52... down to 14 
 ,/^2. . ?/ = 3. ;(/r^0,13,26 up to 52. 
 
 4. ;/• - 7, 2 5. .r = 3, 8, 1 3 . . . 6. .r - 91 , 7G, 61 . . . down to 1. 
 7/-l,4. 7/-7,21,3r)... 7/ = 2. 13,24 up to GcS. 
 
 7. .r = 0,7,14,21,28 8. .r = 20,3y... 9. ,r-4(),49... 
 
 i/ = 44,33, 22, 11,0' ';/-cl,7... 7/=13,33... 
 
 10. .'• = 4,11 ....ip to 123 II. r'=2 12. .'=92,83. ...2 
 
 ^/=r53, 5>'...do\vnto2. i/ = (). ?/=l, 8. ..71. 
 
 13. ^nul? 14. iT'^'»hi 15- 3 ways, viz. 12, 7, 2; 2, «.](». 
 
 16. 7. 
 
 17. 12,57, 102... 18. 3. 19. 2. 
 
 21. 19 o.xen, 1 sheep and 80 hens. There is but one other 
 
 solution, that is, in the case where he bought no oxen, 
 and no hens, and 100 sheep. 
 
 22. A gives L' 11 sixpences, and B gives A 2 fourpenny pieces. 
 
 23. 2, 100, 27. 24. 3. 
 
 25. yl gives 6 sovereigns and receives 28 dollars. 
 
 26. 22,3; 16,9; 10, 15; 4,21. 27. 5. 28. 56,44. 
 29. 82, 18; 47,53; 12,88. 30. 301. 
 
 C. (Page 20.").) 
 
 (1) I. r^ 
 
 m 
 
 -1 .'••* +./>. 
 
 3- 
 
 4 r, 4 
 
 4- 
 
 I. 
 
 ,,-1 + ,^, 2 + ?,V-^^.3„-^ 
 
 
 3. 
 
 4- 
 
 4 < 
 3 ;» 
 
 1-1 
 
 « r 
 
 2. :i'^y + :r^!l-' +X' jl' . 
 
 (3 » I . 
 
 1 
 
 1 
 
 c,^«v' '"^ /^.r^^'^3.'' «• 
 
 1 _\ 1 
 
 ;;^y'«+3..7/'' + 5y^»- 
 
.down to 14 
 .... lip to 52. 
 
 .down to 1. 
 .... up to 68. 
 
 '; = 4(»,49... 
 /= 13,3:3... 
 
 = 92, 83... .2 
 = 1, 8... 71. 
 
 ,7, 2; 2; 6, 10. 
 
 19. 2. . 
 
 it ono otlier 
 ,dit no oxen, 
 
 enny pieces. 
 
 28. 56,44. 
 
 
 
 + <l 
 
 J 2 
 
 cy ' 
 
 + 4//-\ 
 
 1 
 
 1 
 
 i.'7/ ' ^ 'Of'- 
 
 ANSWERS. 
 
 379 
 
 4 
 
 3- ,-^-< + 
 
 3 1 
 
 - 1 i+ .. • 
 
 1 
 
 + 
 
 1 
 
 • 3.x ^u~^z a-'b 'h- 
 
 + 
 
 1 
 
 '-'■il,-K.' 
 
 a'% 
 
 (4) I. 2 4/.^-i-3 4/(,.//^)+i^ 
 
 •'■it 
 
 shi- . 3 ^hf 
 
 -^'11 
 
 
 1 1_ 1 
 
 2. "... "To. .> + •)• 
 •7 J. ■'')/■- "'' 
 
 Ci. (Page 20G.) 
 
 2. o 
 
 ,4m 
 
 81//'". 
 
 I. K^'' + ;rY'' + ?A- 
 
 3. :/;"■' -l-4f('V' + 16fd 4. tC-'^ + ^,rc^ -Ir'^ -Vr\ 
 
 5. 2(r " + %C"l'' - Aa^'c' - crh - ?/'+' + 'Ihr' 4- iC'c^ + h"c- - 2r'-+-. 
 
 _jr(.i -/' + ,.. 
 9. a;*" + ^c"" -f 3,i;2'' + 2,/:'' 4- 1 . I o. ro"'' - 2r"' + 3.r" - 2.c-'' + 1 . 
 
 cii. (P<i^e 207.) 
 I. iK^"* + re-"'?/"' + .r'"/r"' + ?/'"•. 
 
 ^,11 j.Sii I ,,S'i 
 
 2. ft;''"-a:^"'//"+:r"//:"-^"r +// • 
 
 5. rc^'+3a-'' k9x2' + -^7x'-i-81. 
 
 6. (r'"~2(r'V' + 4,r'. 7. 2 -.»■" + 3.c'^ 
 
 8. 4//"<'" - 5/r'". , 
 
 9. a""-t .V"' + 3cr-i-l, 10. a"' + 6'' + c', 
 
 J' 
 
 )ii*i 
 
 M 
 
 I 
 
 1*1 
 
38o 
 
 ANSWERS. 
 
 ciii. (Page 208.) 
 4. n + h f c - V>,r^r''i-\ 5. 10,,' - 1 l.,'l'//^ 4- r-.l//^^ 21? 
 
 I. r-;i(;^ + 3---l. 
 
 ?/• 
 
 4 1 2 
 
 6. iii — n. 7. 7/r" -|-4^/^w=* -t l()'/. 
 
 
 -27/>. 
 9. r'-3 + 2a-^'-' +rA 
 II. re-' +2.';-'7/'' +^/''''. 
 
 13. ,o-4.7:^ + l(),/--12.i + 0. 
 
 4 :; 2 1 
 
 1 4. 4.'" + 1 2,'-" + 25,<;7 + 24,r" + 16. 
 
 2 1 ] 111.' 112 
 
 15. X'^ - 2,c' if 4- 2,/--'-=5 4- ;v"' - 2^^^^ + ^. 
 
 1 11 11 1 111 
 
 16. 'X- -V 4x^i/ - 2,':*;:;* + 4//^ - 4i/:^ +z'\ 
 
 2. 112 
 
 10. x'-^ - 2(r'.'- -{-(('-^ 
 
 1 1 
 
 12. (i* + 2al)^-\-h^. 
 
 civ. (Pncre 209.) 
 
 I. .4 + -;/^ 
 
 2. ((^-ik 
 
 2 ] 1 
 
 3. x'^ + x''^ii'-^ +//•'. 
 
 1 1 
 
 4. • a'* -«''/r' +^;"'. 
 
 .' 2 1 -1 .1 I i 1 -.i 5 
 
 .'J 1 1 ] 1 ."^ 
 
 7. .rl +:3,':-//i + i),';i^- + 27//-*. 
 
 8. 27a^ + 1 Hah^ + I2ah>^ + Hh\ 
 
 4 ". 1 iJ -^ } ■) 4 
 
 5 . ,'• •"' - .'• ■"' // •■' + , (.■ •"' y ■"' - X •' y -' + 1 1 •'• . 
 
 9. ai-A 
 
 A 3 J _ I 
 
 10. Ill ■' I- 3o( •'" + 9iit'' + 277/t"' + SI. 
 
 .\ 
 
 1 
 
 II. .<•- + 10. 12. .';•'+ 4. 
 
 ^ 1. I. 11 ;i V 11 
 
 1 4. ,'••■' - x'-^ir - r'-'':.''' -H if + z-^ - 'f::^. 
 
 I 
 
 15. ,f'*-9,r' U>. 
 
 17. ^)^-2jj^ + l. 18. ..;--j/^-A 
 
 1 
 
 13. -h + 2h^-h'". 
 
 \ 
 
 , I J 1 
 
 16. 7/(- f ///-*/<•* -f- » 2. 
 
 1 1^ 
 
 19. a.-'^+i/". 
 
 ' , . -'. 
 
 r. 
 
 rr'" 
 
 4- 
 
 .-»;* - 
 
 7- 
 
 14 
 
 9- 
 
 Ax- 
 
" 9 
 
 /V2b/. 
 
 1 i; 
 
 1 3 
 •15 4 
 
 A 
 
 4 > 
 
 
 ANSWERS. 
 
 3»i 
 
 3. .t;"* — x~^. 
 
 CV. (Page 210.) 
 
 r. a-'^-h-K 2. x-« -&-'». 
 
 4. .r* + l4-x-*. 5. a-*-6-^. 6. rt-''' + 2rt-'r-i-ft-- }-<;-2. 
 
 g_ 4,,; .-. _ ,,;-4 1 3,,;-H + 2,ir^ 4- ,»:-' -f 1 . 
 ^ , 7.-:-:^ 107.'-- , 5r-i 7 
 
 I. ,x-;o \ 
 
 cvi. (Page 211.) 
 2. a + i"'. 
 
 4. f,■^■l-cM-» + c•-(^Hc(^3+c^-i 
 
 2 + 
 
 3. HI- - m?i~^ + )< -. 
 
 5. ;n/-» + .t--V 
 
 7. .«V--2 + ^:-¥- 
 
 9. <i-'&-2-l + «-^62. 
 
 10. a 
 
 -•2_ft-i6-i _ « -1(3-1 + 6 - - 6-'c-i 4-C-2. 
 
 cvii. (Page 211.) 
 
 r, 
 
 1 I 
 
 ^'■^r + 2?/. 
 
 3. JJ ^'"-^ . 
 
 , 22 , 421 _2 10 .. 1 
 
 5. "'^'^- 3'^"-.i2" r ^■ 
 
 4. 
 
 l.Vift-t 12 
 
 •2't 
 
 i_„i 
 
 6. .7;". 
 
 7. ^■"-//"- 
 
 .■1 r; 1 '.t 1, •„' 
 
 8. a--\-2a-h'' -2aV)--h •> 
 
 1 
 
 -1- 1 
 
 16. 2a^"' + 2tr^"-4'( c"-3(< ()-;j6''^'-^G/;c'". 17. c\ 
 
 13. a;"', 
 
 If 
 
 ill 
 
 i'' a 
 
 
3^2 
 
 ANSWERS, 
 
 19. tc^ + x'^+l. 
 
 20. a"'+" + 2rt'"+"-^ . hcx^ - a"'+"-2 6v;- - a"'+"-* c'-x''. 
 
 22. a'"-^ 
 
 21. xf''''-^^-ifp-^\ 
 
 23. oi'^-if". 
 
 ^'^' "' 144. 
 
 26. x + 3x^-2x2-7:c^ + 2x~^. 
 
 25. x'"" - x"iJ"-^>"' - J"'-^"'ij"' + If 
 
 4- 
 
 I i 
 
 a i 
 
 cviii. (Page 215.) 
 
 3. 4/(5832), 4/(2500). 
 
 4mHlC)n vui/,)ni 
 
 2. ^4/(1024), 1^/8. 
 
 5. "7«", "\//^"'- 
 
 6. 4/(a-5 + 2ah + 6'^), 4^(a3 - 'da% -t 3«6- - 6=0- 
 
 Cix. (Page 217.) 
 I. 2^6. 2. 5V2. 3- 2a Va. 
 
 5. 4zs^2y^)' 6. 10v'(10«). 
 
 4. 5aHy/{bd). 
 7. 12(;V5. 
 
 8. 42V(11*). 
 
 II. {it + x) . sjff'- 
 
 , 14. (3c'^-2/).x/(7j/). 
 
 16. 2xy . 4/(20.r|/). 
 
 18. 7a-'65^(46). 
 
 5.;; 
 
 10. rt 
 
 2 
 
 /a 
 
 9- ^^-V 3 > ■' 
 
 1 2. (.0 - y) fjx. 1 3. 5(rt - 6) . V-- 
 
 15. 3a'-^ ;/(26-). 
 
 17. 3m%''4/(4»). 
 
 1 9. (.0 + ij) . ^x. 20. (a - 6) . ^a. 
 
 I. x/(48). 
 
 5 7^1 
 
 - ^7- 
 
 9. V(»''''^ - *^''^)' 
 
 ex. (Page 217.) 
 2. >/(63). 3. ^/(1125). 
 
 6. s'i^Mi), 7- V(48«v:). 
 
 <'/' 
 
 4. v'(96). 
 
 5. ^/(3(^•'a;). 
 
 •"• (cl-Ii)^- "• C;-i-J' 
 
 20. 
 
ANSIVERS. 
 
 3»3 
 
 y 
 
 ip 
 
 If 
 
 i^/8. 
 
 la 
 h' 
 
 cxi. (rage 218.) 
 
 The numbers are here an'anged in order, the highest on the 
 left hand. 
 
 I. V3, 4^4. 2. VIO, ;^/15. 3. 3x^2,2^/3. 
 
 6. 2 v^87, 3 V33. 7. 3 4/7, 4 s% 2 V22. 
 
 8. 5^18,3^/19,34/82. 9. 54/2,24/14,3^3. 
 
 lo- 2^^^, 3^/3, -V4. 
 
 I. 29v/3. 
 
 4. 134/2. 
 
 8. 48^2. 
 
 12. 2V(70). 
 
 CXii. (Page 219.) 
 2. 30^10 +1^4^/2. 3. (a'-^ + fiHc'Ov/^^. 
 
 16. 2. 
 
 ^°- Vi 
 
 17. 
 
 5. 334/2. 
 
 9. 44/2. 
 
 13. 100. 
 3 
 
 6. V6. 
 10. 0. 
 14. 3a6. 
 
 7. 5^/3. 
 II. 4^3. 
 
 15. 2a6 4/(126). 
 
 18. yl 19. yl- 
 
 
 I 
 
 5 
 
 9 
 
 13 
 16 
 
 cxiii. (Page 220.) 
 
 sj{x;])), 2. sfixij-lf). 3. 3: + //. 
 
 18.C. 6. 56(x'+l). 7. 90 V0«''* - '4 
 
 -a;. 10. \-x. II. -12x'. 
 
 4. v/(x2-7/). 
 
 8. 2a;v/3. 
 1 2. 6a. 
 
 - ^{x^-'Jx). 14. 6V(-<^''^ + 74 15. 8(rt''2-l). 
 
 -(itr+12a-18. 
 
 cxiv. (Page 221.) 
 
 1 , x + 9 V'^ +14. 2. cc - 2 v/'-' - 15. 3. a. 
 
 4. a -53. 5. 3.'j + 5 vAo - 28. 6. 67; -54. 7. 6. 
 
 8. V(9'^- + 3x) + V(6-^'- - 3x) - V(6>^"^ - ./; - 1) - 2.C + 1. 
 
9- \^("-'') + \^('<-'^ - '«'^) - V(^*^ - f^-^^) -ii + -'■• 
 
 lO. I:'-f-«+ v'('^'*^ + »^''^)- 
 
 12. 2r, + 2 >s/{cix). 
 
 14. 2.»j + 1 1 + 2 V(a;2 + 1 la: + 24). 
 
 16. 2.y-6 + 2v/(a:^-6;/:). 
 
 18. 2x-2s/ic"-i/). 
 
 20. x*'^ + H-2VC'^"5-u). 
 
 II. 3: - ?/ -{- ;j 4- 2 Vx"'-!' 
 
 13. 432 + 42 VC'^- - 9) + ,.•■-. 
 
 15. 2..;-4 + 2x/(.c''2-4,o). 
 
 17. 4'; + 9-12 V-^' 
 
 19. a;- + 2u;-l-2 V(-^'*-a-). 
 
 cxv. (Page 222.) 
 
 f. ( Vc+ Jd){y/c- ^fd). 2. ((■+ x^'?)(o- VO- 
 
 3. ( Vc + rO ( s^c - d). 4. (1 + x^'/) (1 - V?/). 
 
 I. 
 
 CM 
 
 3^/( 
 6. 7//-H 
 
 rf'-^f? 
 
 9- /, 
 
 II. 
 
 14 
 
 (<- 
 
 X 
 
 ■ 2 
 
 16. a-W'' 
 
 5- 
 
 (1+ ^/3.ro)(l- A/3.a;). 6. ( v^o.'/t + 1)( v^5.??t- 1). 
 
 18. S + 
 
 7. 
 
 |2rt+ ^fC^x)\\2a- x/(3:c)|. 8. }3 + 2V(2».;| j3-2v/(2h)|. 
 I V(ll).H + 4!|V(ll).^^-4 . 10. {v + 2^r)(p-2^'7-). 
 
 21. - 4/ 
 
 11. 
 
 {sJl)+ V3.!z)(ViJ- \^3.5). 12. |a"' + P|}rt"'-P}. 
 
 
 13- 
 
 '*t-f. 14. "-'-^-t^^. 15. 24^-17,/2. 
 a^-b a-b ^ ^ 
 
 I. V7 + 
 
 16. 
 19- 
 
 2+^2. 17.3 + 2^3. 18.3-2^/2. 
 
 ft + .c + 2 ^/ (((,';) l+.'- + 2^.(; 
 
 a - X 1 - i/j 
 
 • 
 
 5. v^l<^- 
 9. 3 V7 - 
 
 21. 
 
 rt+ J(a- — .c^) •' ,/ i ix 
 ^ ^ ^ 22. m- - v/(''''^ - 1 ). 
 
 X ^ V / 
 
 - 
 
 23. 
 
 i 
 
 2a^~l + 2asf{ct^-^)' 24. ^^rj-^^-^ -. 
 
 I. 49. 
 
 '■ 
 
 cxvi. (Page 224.) 
 
 7. 27. 
 
 I 1. 
 
 19. 2. 11. 3- H-26^/(-l). 4. Of 4^3. 
 
 13. 12. 
 
 5- 
 
 2/) 4- 2 ^/(«^) - 12t(. 6. ((-' + a. 7. //' - a'K 
 
 
 8. 
 
 a~ + li'i 9. t-. 10. (;>v'(-i)_(.-W(-n_ 
 
 17. 3. 
 
 ! 
 
 • ■ 
 
 ■Is.y 
 
Jxz. 
 
 
 '^-9)+, 
 
 '*"'• 
 
 .i^-Ay). 
 
 
 t 
 
 X. 
 
 
 W(.^=*- 
 
 ^•). 
 
 d). 
 
 5.m- 1). 
 1-2v/(2h);. 
 
 ft"' - &^ } . 
 
 :+l7V2. 
 
 - 2 ^2. 
 
 1). 
 
 \. ") f 4 V3. 
 
 ANSWERS. 
 
 385 
 
 cxvii. (Page 224.) 
 
 :»-• + 1/ 
 
 a- + // 
 
 3^/(i//j' 3. 2^/(;^y 
 
 5. :';2- s/2.(rr + rf2. 
 
 X ., , . Q ^ / o 2(6 V' - 'Ih Ja 
 
 6. '/»-4- -v''2.?/?>j+7r. 7. 2xJx. 8. — >- . . ^ . 
 
 11. 
 
 i) ,.'2 
 
 (<- 
 
 10. «V2_2+ .;,., 
 
 a-s' - 
 
 X - 1 
 
 9. ,^-+n/-2..^/,^. 
 
 12. V(l-^'). 
 
 '4- 2"'\/(i«-'> ^- 2.-2,/(.2-«2). 
 
 16. a'-^ft?r. 17. - 1 + r)(r-(2 - f(2) + a{\Oa' a^ - 5) ^/( - I ). 
 
 18. H + 7,</3. 19. 4^/(3c4 20. a- 4/(3/'). 
 
 22. 0)/t-lO).V7. 23. 0. 
 
 1 
 
 21. 4^(-4h'') 
 
 cxviii. (Page 228.) 
 
 I. VV+ v''3. 2. VU+ V-5. 3- n/7- V2. 4. 7-3 Vr>. 
 5. ^no- ^^3. 6. 2Vr) -3^2. 7- 2^3- ^2. 8. 3^11-2. 
 9.3^7-2^3. io.3V7--2v/(3. ii.^(VlO-2). 12.3^5-2^3. 
 
 cxix. (Page 229.) 
 
 I. 49. 
 
 7. 27. 
 
 13. 12. 
 
 17. 3. 
 
 •IS.A.I 
 
 2. 81. 
 
 ■2 ^.5 
 
 14. 
 
 18. 10. 
 
 {a - hr 
 
 ,.2 
 
 19. 
 
 8. 5. 
 
 27. 
 
 6. 250. 
 
 153. 11. 
 
 (). 
 
 12. 3(). 
 
 15. 5. 
 
 
 16. G. 
 
 3(i 
 
 
 2 JJ 
 
 
386 
 
 ANSWERS. 
 
 I. 9. 
 
 7. 0, 
 
 -8 
 
 cxx. (Pag(3 2:n.) 
 
 2. 25. 3. 49. 4. 121. 
 
 8. 
 
 >/'. + 4V'* 
 
 
 6. 8,0. 
 10. 5. 
 
 I. 25. 
 
 6. ^^e^ 
 
 cxxi. (Page 231.) 
 2. 25. 3. 9. 4. 64. 
 
 7. a. 
 
 8. ^^orO. 
 
 9. 64. 
 
 36 
 5- -5- 
 
 10. 100. 
 
 I. 16, 1. 
 5. 5 
 
 cxxii. (Page 232.) 
 2. 81, 25. 
 
 3- ^'^l' 
 
 .3 
 
 4. 10, -13. 
 12252. 
 
 ,-. 6. -4, -32. 7. 9, -:i^. v.. ^.„ ^^ 
 
 9. 49. 10. V29. II. 4, -21. 12. 1 or^^. 13. ±24. 
 
 14. 5 or 221. 15. 5orj2|. i^- ^ ^^ ^- '7- ^j]- 18. 25. 
 19. ±9^2. 
 
 20. ± v^65 or ± /v/5. 
 
 21. 2a. 
 
 '>'> 
 
 26. 
 
 -2a. 
 
 1276 
 
 81 • 
 
 23. s ^^1" - ^ 
 
 24. M 
 
 25. 
 
 36 
 
 27. -5 . 
 
 G- "" 4- -'' 12" 
 
 28. ±5 o1'±3 V2. 29. ±14. 
 
 10 
 
 5 
 
 ^o. 6or--y. 31. 1. 32. 4. 33- 2orO. 34. 
 
 or 
 
 9a 
 16" 
 
 I. 2,5. 
 
 _7 5 
 5- 2' 3' 
 
 cxxiii. (Page 235.) 
 2. 3, -7. 3. -9, -2. 
 
 4. 5rt. fi& 
 
 6. . 
 
 227 
 
 83 
 
 19 ' 14' 
 
 Am \\n 
 7. 5 , -^ - 
 
 1 1. 
 
 I . r 
 
6. 8,0. 
 
 10. o. 
 
 36 
 5- -5- 
 
 lo. 100. 
 
 ANSWERS. 
 
 3fi7 
 
 g. - 2rt, - 3a and 3t6, 4a. 
 
 2a -h h- 3rt 
 
 1 1. 
 
 ac 
 
 9. ± 2, rt. 
 
 A e 
 
 c c 
 
 10. 
 
 0.6. 
 
 cxxv. (Page 239.) 
 
 ,. x2-llx + 30 = 0. 2. ..^ + ..:-20 = 0. 3- a;^ + 0:. + 14 = 0. 
 4. 6x^-7x + 2 = 0. 5. a^^-58^-35 = 0. 6. .^-3 = 0. 
 
 7. x'^-2mx4-m'^-7t'^ = 0. 8. ^'- "-^^-'^ + ^ = ^' 
 
 9. ^^^^-1=^- 
 
 } \ 
 
 4. 
 
 10, - 
 
 13. 
 
 8, 
 
 12252. 
 529 
 
 
 
 13. ±24. 
 
 25 
 
 . 18. 25. 
 
 3(j 
 21. 2ft. 
 
 25 
 
 • 12' 
 
 29. ±14. 
 
 14. Oor-. 
 
 4. 6rt. fi/i 
 m ll?i 
 
 5 ' T • 
 
 CXXVi. (Page 240.) 
 ,. (.-2.)0.-3)(.-6). 2. (.r-l)(.x-2)(x-4). 
 
 3. (;c-lO)(.x-M)C>^ + 4). 4.4(.i^ + l)(-'^+ -4--A•'■^~X">'• 
 5. (;^ + 2)(x + l)(6..--7). 
 
 6. (.c + 1/ + ^) {''^' + v' + ^' - ^y - ^^ - ''/^)- 
 
 7. (rt-6-c)(rt2 + 62 + c''2 + a6 + ac-6c). 
 
 8. (a:-l)(x + 3)(3x-7). 9- (x - 1) (x - 4) (2x + 5). 
 10. (a + l)(3x + 7)(5.T;-3). 
 
 cxxvii. (Page 242.) 
 4. 1 or "^> - 4. 
 
 au /5 2"/ 5 
 
 5- VrW'6- 
 
 6. 25 or . 
 
 I ■ 
 1\" 
 
 7- --9±v ^-cy-i-J- 
 
 9. 1 orl ±2x/15. 
 
 10. 3 or - a 
 
 1 5± vM.329 
 
 or 
 
 t '.i 
 
388 
 
 AiVSlVERS. 
 
 I 
 
 h «! 
 
 II. a + 2, or - - , or -^ V _y. 
 
 ... 0, or «, or "-±^«l-i«2±ii>. 
 
 cxxviii. (Page 245.) 
 I. 6 : 7, 7 : 9, 2 : 3. 2. The second is the greater. 
 
 3. Tlie second is the grcntor. 
 ad - bo 
 
 c-d ' 
 
 5. 10:9 ()i<) : 10. 
 
 CXXix. (Page 246.) 
 
 I. 2:3. 2. h:n. 3. b + d:a-c. 4. ±v^6-l:l. 
 
 5. 13 : 1, or, -1:1. 6. ± ;^f(m- + 4n') -m : 2. 7. 6, S. 
 8. 12,14. 9. 35,65. 10. 13,11. 11. 4:1. 12. 1:5. 
 
 1. 
 
 15' 
 
 cxxx. 
 
 8 
 9' 
 
 (Page 247.) 
 
 3. --"^. 
 x + y 
 
 a -h + r 
 a—h — c 
 
 ,2 
 
 m' - mn + 71'^ 
 m'^ + mn + n'^' 
 
 07-4)^' 
 
 6. .X = 4 or 0. 
 
 II. ;c=-30, ?/ = 20. 
 
 cxxxii. (Page 255.) 
 
 8. 440 yds. and 352 yds. per minute. 
 
 13- 
 
 16. 50, 75 and 80 yards. 
 
 19. 1 miles per hour. 
 o 
 
 21. 160 quarters. £2. 
 
 '24. i;20. 25. 90 : 79. 
 
 h^ 
 
 9 
 
 d' 
 
 ^5. 41. 
 
 17- 
 
 120, 160, 200 yards. 
 
 
 20. 1:7. 
 
 22. £80. 23. £60. 
 
 26. 45 miles and 30 miles. 
 
 I. 
 
 I. 
 
greater. 
 
 /6-1 : 1. 
 
 7- 6,S. 
 
 12. 1 : 5. 
 
 ,-h-c 
 
 limite. 
 
 9 
 
 41' 
 
 yards. 
 
 3C> miles. 
 
 
 ANSWERS, 
 
 389 
 
 4. 16^. 
 
 CXXXiii. (Page 262.) 
 «5. 5. 6. 12. 
 
 7 3^ 
 
 8. 
 
 5" 
 
 9. AozC 
 
 <t 
 
 13. :c- 
 
 108 
 ^3 • 
 
 10. 5. II. A = lB. 12. 64.o2 = 9.v\ • 
 
 14. 4x3 = 27?/2. 18. i/ = 3 + 2x + a;2. j^. i8ft. 
 
 I. 50. 
 
 cxxxiv. 
 
 2. 200. 
 
 6. 40. 
 
 (Page 266.) 
 3. 10' 
 
 4. -32-. 
 
 5. -\' 
 
 9. a;2 + ?/2-2(7(-2).r?/. 
 
 7. 117. 8. 0. 
 
 3071 - 26n - 2a + 6 * 
 
 10. 
 
 a + 6 
 
 I. 5050. 
 
 5. 24. 
 
 7n2 - 5n 
 ^ 2 • 
 
 cxxxv. (Page 268.) 
 
 2. 2550. 3. 820. 
 
 n.(u+ 1) 
 
 6. -31^;. 
 
 6 
 
 2 
 
 8. 
 
 4. 30. 
 3n''^ - 7J 
 '2" • 
 
 71-1 
 10. -g-. 
 
 I. -6. 
 5. -2. 
 
 cxxxvi. 
 
 a; 
 ^* "25* 
 
 6. -l| 
 
 (Page 269.) 
 
 1 
 
 3. g- 
 
 4. - 
 
 8* 
 
 cxxxvli. (Page 269.) 
 
 1. (i) -46. (2) 36-2. 
 
 2. 155. 3. 112. 4. 888. 
 
 (3)f. 
 
 (4) 4-4. 
 5. 100. 
 
 i 
 
390 
 
 AA^SlVEIiS. 
 
 6. 6433^-. 
 
 9 
 
 7. £135. As. 
 
 8. (i) 355, 7175. 
 
 (3) 161 +81x. 3321 + 1681:c. 
 
 (5) 4 ^'4 
 
 9. (i) 126, 63252. 
 
 (3) 45, - 1570-5a;. 
 (5) 71, 4899(1 -w). 
 
 (2) -156^2, -3116rj'''. 
 (4) 119^, 2357^. 
 
 (2) 25, 2250. 
 (4) 99,-1163^. 
 (6) 65, 05X + 8190. 
 
 cxxxviii. (Page 271.) 
 
 I. 6, 9, 12, 15. 
 
 5 
 
 5 1 
 3- ^12^ 1(5' 14- 
 
 12 2 1 
 
 7 13 2 11 
 ^' 15' 30' 5' 30' 
 
 II. 
 
 I 
 
 I. 
 
 2. 
 
 3. 
 
 CXXXix. (Page 272.) 
 
 3m + n m + n 971 + 3n 
 ~T ' 2""' ~4~' 
 
 5m + 3 bm + 1 5m - 1 bm - 3 
 5n''^+l 5?i2.f2 57i'-^ + 3 5?i2 + 4 
 
 5 
 
 5 ' 5' 
 
 2x^ + y^ 2 Sa;'-^ — ?/2 
 
 2 
 
 # I. 64. 
 5. 13122. 
 
 cxl. (Page 275.) 
 
 2. 78732. 
 
 3. 327680. 
 
 6. 16384. 
 
 ►7 
 
 ^' 2048" 
 
 1 
 96' 
 
AN^IVERS, 
 
 391 
 
 cxli. (l^ige 276.) 
 
 a 
 
 I. 65534. 
 
 ft (.c» - 1) 
 
 7. 7(2"-l). 
 
 2. 3«4. 
 
 (x2« - 1) 
 
 5" (rt+.'-)-Mr^^->»)' 
 
 8. - 425. 
 
 43 
 9()' 
 
 cxlii. (Page 278.) 
 
 I. 2. 
 6. -3. 
 
 2. 
 
 3" 
 
 27 
 
 3- 8' 
 
 4 
 4. 3. 
 
 ^ Q^ 8 2-. 9. 85.,. 
 
 7. 8-r. s. ^4' V 3 
 
 11 
 
 II. 
 
 a' 
 
 a -I) 
 49 
 
 12. Q. 
 
 13- 
 
 a;-^ 
 
 0; + ]/ 
 
 ^5- 90' 
 
 16. 
 
 46 
 55' 
 
 I. 9,27,81. 
 
 3 9 27 81 
 4- 4' 8' 16* 32* 
 
 cxliii. (Page 279.) 
 2. 4, 16, 64, 256. 
 
 I. (I) 558. 
 
 . 169 
 (5) --2 • 
 
 (9)1- 
 
 3157 
 
 (^2) ---80"- 
 
 # 
 
 5. 42. 
 
 CXliV. (Page 279.) 
 (2) 800. 
 
 (3) -K 
 
 ^^) 486- 
 (10) -84. 
 
 (7) - 
 
 
 
 1189 
 
 6. S"-!. 
 
 10. 
 
 5. 4 
 
 16x5 
 
 8x'-2+^l' 
 
 8^ 
 ^■^- '99' 
 
 3. 2,4,8. 
 
 , . 16 
 (4) -9 . 
 
 (8) 13^. 
 
 9999 V3 
 
 <^\') ~TVio+iW5' 
 
 6. ac—lr. 
 
 7. ±1- 
 
 4w 
 
 hi 
 
 :l:'(i 
 
9- 4. 
 
 lo. 10. 
 
 13- 4. 
 
 i6. 49, 1. T^ ol ^. r,l 
 
 J 7. 3 J, 6. 8.;. 
 
 14. 642. 
 18. GO. 
 
 4 3 2 1 
 ^ r>' 5' 5' 5 
 
 0. - 
 
 1 5> 3 4 
 
 ' ^ ) 
 
 5' 6' o 
 
 f^' 
 
 22. 3,, 7, 11, 15, 10. 
 
 23. 139. 
 
 23- 5, 15, 45, 135, 405. 
 26. 10 per cent. • 
 
 I. 8, 12. 
 
 ,1111 
 ^* 6' 9' 12' 15- 
 
 r 3 3 
 
 ^- 4' 2' ""' 
 
 cxlv. (Page 285.) 
 
 15 30 5 30 
 ' T' 13' 2" Il- 
 
 ls JB 4 
 
 29' 11' 5' 
 
 5. -2, 00, 2, 1, i. 
 
 3 
 
 3 
 
 4' 
 
 6 3 () :] G 3 
 
 5' 4' 11' 7' 17' 10' 
 
 7- r,. . 
 
 3y.'//H-2.'; ' 3//// { 4*: - :^,,7 
 
 ' • '? 
 
 a-'// (// + 1 ) 
 
 1 
 
 4' 
 
 1 111 
 
 10. 104, 234. 
 
 2?<.'' + 3// • 
 
 ' ' ' •"> '> 5 15 5 
 
 ^' 4' G'"' 31' 21' J 7' 2' 3' "4- 
 
 J •). _, ,i, b. 
 
 I. 132. 
 Ill 
 
 cxlvi. (Pao-e 200.) 
 2. 33GO. 3. lj(!280. 
 
 5. 
 
 8 
 
 6. 40320. 
 
 ;. 3G28S00. 
 
 S. 125. 
 
 10. G. 
 
 J^- 4. 12. 120. 
 
 14. 1^520, G720, 5040, 1GG32O0, 34G50. 
 
 4- 6720. 
 9. 2520. 
 
 13. liiGO. 
 
 cxlvii. (Page 295.) 
 
 I. 3921225. o (; - 1.,,. 
 
 - ^- 3- l-'(>. 4. 116280. 
 
 5- ^'' ^'- J-- 7. 8IGOOO. 8. 3353011200. 
 
 9. 7. 10. G3. 11. 52. 12. 123200. jj. 37699.. 5^3(^,^ 
 
 ■j*?^*^ 
 
I * 
 
 A XS I TEA'S. 
 
 393 
 
 [4. 642. 
 GO. 
 
 ^s 405. 
 
 cxlviii. (Page ;3UU.) 
 
 I . '/ ' + 4(r',<; + {Sa-x- 4- An:c^ •(■ .'»;*. 
 
 3. a" + 7a*^6 + 2i(f^/r' -i- 3.")^^ '/;•'■ -1- :3r)frV;' + •lld'-J,'' \. 'Jah'' + /'^. 
 
 4. .0'^ + 8.1-"// + 2Sx'''ij'' + :>ac'u"' + 7< yrhj ' + r)U,/;''//'' + 2s.'-'//'' 
 
 - + 8.';//i -\ ij\ 
 
 5 . ()25 + 2()0(Vfc + 2400a''^ + 1 28()(('' + 25(j(6*. 
 
 6. (ti" + 5(t«6c + 1 Oft"6"-^t'- + 1 Oc< »6\'' + ba-h\:^ + U'd\ 
 
 CXlix. (Pn-e 301.) 
 
 1 . n'' - 6^<\<; -H 1 ")(«"»,>;- - 2( )^r'vr + 1 ')((-,'.•■' - (vix;' + .»;". 
 
 2. //-■ _ 7//V -}- 2 1 /)''c- - 3:)/>«f" h 3r)f>\-^ - 2 1 6-c''' f TAr" - c^. 
 
 3. :52,c' - 240,/;'// + 720x-7/- - lOHO.o'-y/'' + 810,///^ ^ 2 13//''. 
 
 4. 1 - 1 ( ).(• + 4( )./;- - 8( Kc^ -I 8< Vu^ - 32.//'. 
 
 5. 1 - 1 ().'; + 45,/;'-'- 120x3 + 210,/;'- 252,';' + 21( ».'•''- 12(U-'' 
 
 6. (t -' ' - 8rt-'6'-^ + 28(6^*^// ' - 50^1 •'•/)" + 70rt 1 -//« - .5(k/'V;'" 
 
 + 28a"/>'--8a-'6'»r6'". 
 
 Cl. (Page 302.) 
 
 1 . ,v^ + Vm^h ~ 3«-V ■\- 1 2/(//- - I 'luhr + 3^?/;- 1 8//' - 1 -Ih'c + 66c- - c\ 
 
 2. l-6,<:-f-21.'''-44.'-"' fC:}'' n t-.-'f- 27.''''. 
 
 3. ./:•' - 3,/'" + 6,.' - 7.'," + (),';^ - '3x^ f ./;•'. 
 
 4. 27.K + 54,0'' -(- 63.'-' + 44,/;- -f- 2 1 ,>•- + 6, 
 
 )./'^>l. 
 
 v3 . 
 
 •' 1 
 
 5. a;'* + 3x"'-o+ ., ,. 
 
 :i :< 
 
 L 1. 
 
 1 1 
 
 .1 I 
 
 I I. 
 
 6. « ^ + M - c^ + 3(6-6-* + 3//-i//-' - 3/< -r i - :5//^rt + 3//^ 
 
 + 36 
 
 hK'^-[\,L^ib^C^, 
 
394 
 
 ANSWERS. 
 
 Cli. (Page 303,) 
 I. 33().(;7, 2. 495rt'%^■ 3. - 161 700ft''' //^ 
 
 4. Il)2192a«6«c^rf8, 5. 12870a«i^. 
 
 6. 70 Ai 7. - 92378tt"'69 and 92378a''6i". 
 
 8. l716aV and 171G(t".c^ 
 
 Clii. (Page 311.) 
 
 
 ^' ^ "^ 3 " 9 ^' Sf 
 3. rt^+ ^ 
 
 -. 4 
 
 S If 
 
 3a-5 9rt"' 81a-' 243ft '^ 
 
 t 1 •> 1 ■? '^ 1 ^ 
 
 4. 1 + a; - gX- -1- ^.^"^ - -^rt;*. 
 
 '4 1 1 _5 5 -0 
 
 5. ft-* + ft *:c - -ft •*»:''' + --ft -^ . ,>A 
 
 1 4 -M 2 -"4 4 -'M 
 
 6. a ■• ^,.a -^VA - . (t w /;2 + - . (,, -o.^,t 
 
 ::io 
 
 125 
 
 ■r- a;'* 
 
 ''• ^ 2 8 Hi 128" 
 
 _ - 7 ., 14 , 14 „ 
 8. l-,>'*-+,»'^'-u,"'". 
 
 9' 1-4 
 
 3 9 81 
 
 9,<; -i7x;;-_135 
 32 128 
 
 ^3 
 
 ,0. ^' ->:!/ + 6^ + 5473- 
 
 I. 
 
 3- 
 6. 
 
 I. 
 
ANSJFFRS. 
 
 395 
 
 II. 1 
 
 12. 
 
 5 
 
 fl 
 
 
 30 
 
 /2\:-: 2 /3\.l 1 3/3\.^ _4 .. 
 
 cliii. (Page 312.) 
 I . l-2a + 3<r - 4a^ + 5a ^. 2 . 1 + 3.^ + O.c- + 27,>:'' + 8 UA 
 
 J) .. •> 
 
 
 4. 14.,,+ --+^-+^^.. 
 
 5 . a-'" t- 1 ' hr'- c + 60(r"jc'' + 280rt-^''.'r^ + 1 1 20(r'\r\ 
 
 1 6.':'^ 21,';"^ 5(i,': 
 
 6. „ f- . + 
 a- \ 
 
 a' 
 
 
 cliv. (Page 313.) 
 
 '^28 16 128' 
 
 3.0- liW 35x" 31 :),/;*< 
 - ^ * 2 8 16 128 • 
 
 28 
 
 1 ,r:^ 2,<;«_ 14,r'' 
 
 1 _ ,r2 3,,;'» _ ho^ 
 
 clv. (Pi-e 311.) 
 
 , 7.6... (O-;-) , 
 ^- 1.2. ..(/--l)"^ • 
 
 ^ '^ 1 . 2 ... ^r- 1) 
 
 3' I- '^ • 1.2. ..(,■-!) •■'- • 
 

 396 AA'SIFEI^S. 
 
 1.2. 5. ..(3.-7) (_''-Y' J, 
 ^' i:273...(r-i) "V 3a/ ' 
 
 7,.9.ll...(2r + 3) , 
 9- 172.3... (/•-!) " • 
 
 (7^ 3.7.11 ...(4r- 5) /^Y'-'' 
 '°- 4--i' 1.2.3. ..(r-1) *Va/ " 
 
 "• 2"" • • ^^* 1.2.3...r •^•-•^• 
 
 ].3.r....(2r-l) , 5 1 
 
 ''■ 1.2.3...r •^^•'^- '^- !«• 1,- 
 
 m.(m+l) (w + 8) („+„ ,g 
 
 15. — -^ 2...... 9- 
 
 (l-5»0(l-4m) (l-w) !,-« 
 
 '^' 1.2......f5)»*' " •" 
 
 Clvi, (Page 315.) 
 
 1. 3-14137.... 2. r9r)2()4..,. 
 
 3. 3-04084.... 4. 1-98734.... 
 
 Clvii. (Pago 319.) * 
 
 I. 101.5032. 2. 1 00703 U. 3. 80451. 
 
 4. 31134. 5. 51117344. 6. 143322 1«. 
 
 7. 314r)(i and rcmaiiulcr 2. S. .52225(5 and remainder ]. 
 
 9. 4112. 10. 2437. 
 
r . :r' 
 
 r-l 
 
 ANSlVEIiS. 
 
 clviii. (Page 321.) 
 
 I. 5221. 2. J 2232. 3. 2139«j. 
 
 5. UlOUlOOllll. 6. Utee. 
 
 8. 211021. 9. -6^2. 10. 814. 
 
 12. 123130. 13. 16430335. 
 
 5^ 
 
 4. 104300. 
 7 6500445. 
 II. 61415. 
 [4. '2.1 1. 
 
 CliX. (Page 327.) 
 I. -41. 2. •162355013. 
 
 4. 12232-20052. 5. Senary. 
 
 3. 25-1. 
 
 6. Octoiuiry. 
 
 Clx. (Page 33G.) 
 
 I. 
 
 1-2187180, 
 
 2. 7-7074922. 
 
 3. 2-4036784. 
 
 4. 
 
 4-740378. 
 
 5. 2-924059. 
 
 6. 3-724833. 
 
 7. 
 
 5-3790163. 
 
 8. 4()-578098. 
 
 9. 62-9905319 
 
 10. 
 
 2-1241803. 
 
 II. 3-738827. 
 
 12. 1-61514132 
 
 Clxi. (Page 339.) 
 
 1. 2-1072100; 2-0969100; 3'3979400. 
 
 2. 1-6989700; 3*6989700; 2-2922560. 
 
 3. -7781513; 1-431.3639; 1 '7323939 ; 2*7601226. 
 
 4. 1-7781513; 2-4771213; -0211893; 5-6354839. 
 
 5. 4-8750613; 1-4983106. 
 
 6. -3010300 ; 2-8061800 ; -291GO0O. 
 
 7. •6989700 ; 1-0969100; 3-391U733. 
 
 8. -2, 0, 2 : 1, 0, -1. 
 
 9. (1) 3. (2) 2. 10. j: 
 
 9 
 
 
 '</=, 
 
 »'^ m 
 
398 
 
 ANSWERS. 
 
 11. (a) •3010300; 1-3979400; 1-9201233; 1-9979588. (6)103. 
 
 12. (a) -6989700; •G020G00; 1-7118072; 1-9880618. 
 
 (6) 8. 
 
 13. 3-8821260; 1-4093G94; 3'7455326. * 
 
 14. (i) a; = ^. 
 
 (2).'>J = 2. 
 
 ^"^^ ' loj' a + lo" IS 
 
 (4) ^' = 
 
 logc 
 
 «i lo<? rt + 2 lo(f 6* 
 
 ^^' 2 loi' c H- loff 6 - 3 Iog: a 
 
 (6) ^'= 
 
 l0f,'C 
 
 log a + ?/i log 6 + 3 log c 
 
 clxii. 
 
 I. 17-6 years. 
 
 3. 7 "2725 years nearly. 
 6. 12 years nearly. 
 
 (Page 343.) 
 
 2. 23-4 years. 
 
 4. 22-5 years nearly. 
 
 7. 11-724 rears. 
 
{h) 103. 
 
 T6g'6" 
 
 APPENDIX. 
 
 -♦•- 
 
 The following papers are from those set at the Aratrieulation 
 Examinations of Toronto, Victoria, and MeOill LTniversi- 
 t^eSi fcnd at the Examinations for Second Class Provincial 
 Certificates for Ontario. 
 
 II 
 
 years, 
 nccarly. 
 ., Years. 
 
 UNIVERSITY OP TORONTO. 
 
 — .i.,— 
 
 Junior Matric, 1872. Pass. 
 
 1. Multiply ix^-^xy + y^hjljx^ + l xy - ?/'. 
 
 Divide a* - 816* by a ± .36 and {x + af - {y - bf 
 by x + a-y + b. 
 
 2. What quantity subtracted from x'^ + px + q will 
 make tl^e remainder exactly diviyible hj x — a? 
 
 Shew that 
 
 {a + b + cf- {a+ b + c) {a^ + ¥ + c' - ab -be- ca) 
 - 3abG = 3 (a + b) (b + c) (c + a). 
 
 3. Solve the following equations : 
 
 4£c — 7 3x — 5 
 (b) T r + 1 i.- = 20. 
 
 ^X—1 ' U'-— 9 
 
 ^'aj — 3 X — 4 X — 5 X — 6* 
 
 (^) x + 
 
 V + .1 
 
 ^ i« + 2 n 
 
 1^ 3 + y- --^-g. 
 
 4. In a certain constituency are 1,300 voters, 
 ind two candidates, A and B. A is elected by a 
 
n 
 
 APPENDIX, 
 
 cortain niajority. But the election having heen de- 
 clared void, in llie second contest [A and B being 
 again the cat) di« lutes), B is elected by a majority of 
 10 more tlian /I'.s majority in the first election ; find 
 the number o>' votes polled for each in the second 
 election; having given that, the number of votes 
 polled for B in tiio first case : number polled in the 
 second case J I 43 : •' 4. 
 
 Junior Matric, 1872. Pass and Honor, 
 
 1. Multiply X -^^ y + z^ ~ 2yi z\ + 2z^ a^ - ^x^yh by 
 
 X + y + z^- + 2yh zi — 2z^ x^ — 2x^ ?/*, and 
 divide a^ + Sb' + 27 c^—Uabc by a'' + ib' ^- 9 c»— 
 2ab — 3ac — 6bc. 
 
 2. Investigate a rule for finding the //. C. D. of 
 two algebraical expressions. 
 
 if x + c be the //. C. D. of a^ + px + q, tfnd a;' + 
 p x + q, show that 
 
 (q-q'Y-p {q-q) (p -p) + q (p-pj =^0. 
 
 3. Shew how to find the square root of a binomial, 
 one 01 whose terms is ratiojial and the other a quad- 
 ratic surd. What is the condition that the result may 
 be more simple than the indicated square root of the 
 given binomial 1 Does the reasoning apply if one of 
 the terms is imaginaiy 1 Show that *y/ ~ im^ = y/m 
 
 + y/ -m. 
 
 4. Shew how to solve the quadratic equation ace' + 
 bx + G = o, and discuss the results of giving different 
 values to the coefficients. 
 
 If the roots of the above equation be as ^ to ^ 
 
 6'' (p + qy 
 
 show that — = • 
 
 ac 
 
APPENDIX. 
 
 Ui 
 
 5. Solve the equations 
 
 (a) 7r+ v^.c^+3a;-a- 111- . 
 
 XT/ +y^- iO 0, 
 
 (c) 
 
 ai^ + 6 X f 
 
 03 V 6 tc + 2 
 
 a? + Qx+ 4: x^ -h 6 x + S 
 
 a^ + 6 cc 4- 8^ 
 
 .^■2 + G~£c + rO' 
 
 re' 4- 6 re + i 
 a^ -h 6 x+Q 
 
 (d) Qx'~5x'-3Sx'-5x-^Q=^(^, 
 
 6. Shew how to find the sum of n terms of a geometrip 
 series. What is meant by tlie sum of an intiuit* 
 series 1 When can such a series be said to have » 
 sum 1 
 
 Sum to infinity the series i _}- 2?' + 3 r' -j- (fee. 
 and find the series of which the sum of n terms is 
 
 a 
 
 nq 
 
 1 
 
 a" — -~. 
 
 7. Find the condition that the equsitiong 
 
 ax + hy — cz = 0. 
 a^x -{- biy — CiZ = 0. 
 o-iX -\ h^y-CiiZ = 0. 
 
 may be satisfied by the same values of x, y, z. 
 
 8. A number of persons were engaged to do a ]:)iece 
 of work which would have occupied them m hours if 
 they had commenced at the same time ; instead of 
 doing so, they commenced at equal intervals, and thezi 
 continued to work till the whole was finished, tne 
 payments being proportional to the work done by 
 each ; the first comer received r times as much as tho 
 last : find the time occupied. 
 
 
APPENDIX, 
 
 vs. 
 
 I 
 
 i 
 
 Junior MalriCy 1872. Honor, 
 
 1. Tliero are three towns, A, B, and C ; the road 
 fiom B to ^ forming a right angle with that from B 
 to G. A person travels a certain distance from B 
 towards A, and then crosses by tlie nearest way to the 
 road leading from G to A, and finds himself three 
 miles from A and seven from C. Arriving at A, he 
 finds ho has gone farther by one-fourth of tlie distance 
 from B to C than he would have done had he not left 
 the direct road. Required the distance of B from A 
 and G. 
 
 2. If ay -f hx ex + az hz + aj 
 
 tlien will 
 
 a 
 
 a 
 
 z 
 
 3. 8()I \e the equations x"^ — v/.^ - a*, y^ — zx^ 6', «' — 
 
 4. Jf a, h, and c be positive quantities, shew that 
 
 a« (b+c) ^h' (c + a) + c^ (a + b) > 6abc. 
 
 5. Find the values of x and y from the equations 
 „ . 5?/ + 3 ^ 
 
 a;V5a;-i-v/ (7/-l) = 24. 
 
 6. A steamer made the trip from St. John to Boston 
 via Yarmoutli in 33 hours ; on her return she made 
 two miles an liour less between Boston and Yarmouth, 
 but resumed her former speed between tlie latter place 
 and St. Jolm, thereby making tlie entire return pas- 
 frtge in -^^ of the time she would have required had 
 lier diminished s})eed lasted throughout ; had she 
 made her usual time between Boston and Yai-mouth, 
 and two miles an hour less between Yarmouth and 
 
APPEND! A. 
 
 St. Jolui, her return trip would liave l)oon ncuUi in 
 44- of the time slie would liavci tMkon had tin- whole 
 of her return trip Ixmmi niado at the diniinislKMl rate. 
 Find the distance betweon St. John and Yannouth 
 ■ind between the latter place and Ijo.ston. 
 
 Junior Matrlc, Honor. ^ 
 Senior MiUric, Pass. J 
 
 1. Solve the following equations: 
 
 1S71 
 
 - 'Zx>j + 2/ 
 
 'Xf/ + y ~ 03. 
 
 (a) .... I ^. , 
 
 -,, ( 4x— 3x>/ = 171. 
 
 (h) . . . . j ^ "^ 
 
 (o) 
 
 Zy-ixy- ir)(). 
 /Ill 
 
 -^+- + :^^ 19. 
 
 j or ic?/ ir 
 
 u u 
 
 And fiiul one solution of tlie emiationii ; 
 
 W •••• \a;' + V'a; = 2/. 
 
 2. Find a number whose cube exceeds six times the 
 next greater number by three. 
 
 3. Explain the meaning of the terms Highest com- 
 mon measure and Lowest common multiple as applied 
 to algebraical quantities, and prove the rule for finding 
 the Highest common measure of two quantities. 
 
 4. Reduce to their lowest t(n-ms the following 
 fractions : 
 
 
 i 99 >y + 117x'^ - - 257a;^ — .325.^ — 50 
 '\ 'Sx^'+^x'^ix—lO. 
 
 j x' -i-jOx' \- :)r)x' -I r^Ox + 24: 
 
 '" \x' +'n? + 1 19.<r + 342x + 3G0 '' 
 
APPENDIX. 
 
 5. Find tlift sum of n terniH of tlie shiiea — A, ^, — 
 |, ifec, and (Jm! x'Mi U'xxw uf ilu; series 
 
 a; -f 1 
 
 X - 
 
 3 — a; 
 
 1' .r— r X r'^^' 
 
 6. Find tlie relations between tlie roots and co- 
 efticients of the equation a.x' +jL>X' + g-^O. 
 
 Solve the equation 
 
 7. A cask contains 15 gallon^' of a mixture of wine 
 and water, which is poured into a second cask con- 
 taining wine and water in the proportion of two of the 
 former to one of the hitter, and in the resulting mixture 
 the wine and water are found to be equal. Had the 
 quantity in the second cask originally been only one- 
 half of what it was, the resulting mixture would have 
 been in tl\e propoi'tion of seven of wine to eight of 
 water. Find the quantity in the second cask. 
 
 8. What rate per cent, per annum, payable half- 
 yearly, is equivalent to ten per cent, per annum, pay- 
 able yearly. 
 
 9. A is engaged to do a piece of work and is to 
 receive $3 for every day he works, but is to forfeit 
 one dollar for the first day he is absent, 1 ^ for the 
 second, three for the third, and so on. Sixteen days 
 elapse before he finishes the work and he receives $26. 
 Find the number of days he is absent. 
 
 Change the enunciation of this pi-oblem so as to 
 apply to the negative solution. 
 
 Junior Ma trie, 187G. Pdss. 
 1. Explain the use of negative and fractional in- 
 
 dices in Algelua. 
 
 Multiul 
 
 PO' 
 
 v 
 
 ^ i»v 
 
 (f 
 
 ' an( 
 
 Itl 
 
 le 
 
 12 la 
 
 product by *'ij 
 
 I J 
 
APPENDIX. 
 
 ▼H 
 
 SunpHfy - , writing the factors all in on<3 
 
 lino. 
 
 2. M ultiply toL,'('t li^r ct^ + ax 4- x\ a + x, a^ - ax H- .r\ 
 a — x, and divide IIk; product by a^~j?. 
 
 3. Divide 1 by i - 1j' v .c^ to six terms, and ;,Mve 
 the reinaindci". Also divide 27.'/-0^'- ^^ by '.xc^ \- 
 
 4. Multiply o. \- b l»y « +6 
 
 (!)• 
 
 5. Solve the equations : 
 
 _3a; + 4 _ 7a;-3 ^_2c- 16 
 ~5 "2 4 
 
 03 (y + 2^) = 2'lr, 
 
 (2). ^2/(i + a;) = 45, 
 « (ic + 2/) - 49. 
 
 Junior Matric, 1876. Honor. 
 
 1. An oarsman finds that during the first half of 
 the time of rowing over any course he rows at the 
 rate of five miles an hour, and during the second 
 half, a.t the rate of four and a half miles. His course 
 is up and down a stream which flows at the rate of 
 three miles an hour, and he finds that by going down 
 the stream first, and up afterwards, it takes him one 
 hour longer to go over the course tli.in by going first 
 up and then down. Find the length of the course. 
 
 2. Shew that if o?, b\ c- be in A. P., then will 6 + c, 
 • + «, « 4- 6 be in //.P. 
 
 Also, if a, b, c be in A. P., then will 
 be 
 
 I C((j ah 
 
 a-\- -^ , 6 f- -.^, c¥ 
 
 b ^ c G I- a a + 6 
 
 be in //./'. 
 
VIU 
 
 APPENDIX. 
 
 £ Tf ,9 rr r( + 6 + 6', then ' 
 
 ^{as + he) (hs + ac) (cs + ab) -- (s - a) (s -b)(s- c) 
 
 III',, na , 
 
 ■L 11 Ui + a^-i- +a„ -,-^y , then 
 
 (•^- - ^*i)'+ + (^- - ^g^ - «;-^+%-+ +a^,. 
 
 •'. if the fniotion ^, — wiien reduced to a re- 
 
 2n + 1 
 
 I'fU'iid, contains 2n figures, shew how to infer the hist 
 
 n digits after obtaining the first 7i. 
 
 Find the value of j\- by dividing to 8 digits, 
 6. Solve the equations 
 
 x-i/ + z = S, 
 
 xi^ + xz-2+^z, 
 
 Junior Matrlc, 1876. Honor. 
 
 1. Shew that the metliod of finding the square 
 root of a nunilx^r is analagous to that of finding the 
 square root of an algelmdc quantity. 
 
 Fencing of given length is placed in the form of 
 a rectangle, so as to include the greatest possible area, 
 wliich is Ibund to be 10 acres. The shape of the 
 lield is then altered, but still renuiins a rectangle, and 
 it is found tiiat with 102 yards more fencing, the 
 same area as before nuiy be enclose^. Find the sides 
 of the latter rectaufde. 
 
 2. Prove the rule for finding the Lowest Common 
 Multiple of two compound algebraic quantities. 
 
 Find the L.C.M. of a? - b"" + c' + Zahc and a;\b + o) 
 - lr(c + a) + c^ (a + b)+ abc. 
 
 3. Tf a, 6' l»e tlie roots of the e([uation x^ + px + q^ 
 0. show thiit the <Mpi:ition may lu- thrown into the 
 form {x - a) (x - fS) -- 0. 
 
APPENDIX. 
 
 LX 
 
 + 7- 
 
 3 + \/2 is a root of the eijuatioii .f^ - 5.t;^ + 2^' + ;« 
 : find the other roots. 
 
 4. (1) Shew how to extract the square root of a 
 
 binomial, one of whose terms is rational, 
 and the other a quadratic surd. 
 
 (2) Find a factor which will rationah'ze x^ - v/i. 
 
 5. a, h are the iirst two terms of an //. P., what is 
 the /Uh term ] 
 
 If a, I, c be in //. P., shew that 
 b\a - cf - 2c\b - ay + 2a\c - bf. 
 
 6. A and B are to race from M to N and back. A 
 moves at the rate of 10 miles an hour, and gets a start 
 of 20 minutes. On A's returninyf from N, he meets 
 B moving towards it, and one mile from it ; but A is 
 overtaken by B when one mile from M. Find the 
 distance from M to N. 
 
 7. Solve the equations 
 
 (1). ar' + 8-2^^+llxfU. 
 / X 51 
 
 \ X 12 xy 
 
 i 1 
 
 Second Class Ceriijicates, 1873. 
 
 1. Multiply .^. - + 1 ijy , + - - 1. 
 b a •^ /' ^ 
 
 . 2. Shew tliat 
 
 b a 
 a'' - 3ab -\- 2b' co" ~7(fb + 121^ 
 
 a - 26 
 can be reduced to the foiui 36. 
 
 a - 36 
 
 // 
 
iMHnilHBUU 
 
 APPENDIX, 
 
 3. Tierliice to its loudest ttn-ms tlio fmction, 
 
 x' 
 
 
 i 
 -0 
 
 x' + ^ - 
 
 1 
 "9 
 
 4. (a) Prove that af' ~ t/* is divisible by x-y with- 
 out remainder, wJioii in is any positive integer. 
 
 (/>) Is there a remainder wlien a;'""- 100 \sk 
 li\ ided by a; - 1 '/ If so, write it down. 
 
 5. Given ax v hy =1^ 
 
 X y 1 
 
 pnd — + , =~^' 
 a ao 
 
 Find the difference between x and y. 
 (5. Given3--^il^-2(— i)i K^"^) 
 
 8(ic-l) 3(a;+l) 
 
 Find a; in terms of m. 
 
 0. 
 
 X 
 
 7. Given - 
 2/ 
 
 2 . 7.7; flT) 
 
 o. Find the value of ^ ^r 
 
 S. Given 
 
 2 
 
 nnd 
 
 x — y 
 6' 
 
 5^ 
 
 a:; +:y 
 10 
 
 aj -v/ aj + y 
 
 I, 
 = 3. 
 
 Find a; and y. 
 
 9. There is a-»niiinbcr of two digits. By inverting 
 the digits we obtain a nnniber which is le^ss by 8 than 
 (hree times the original number; but if we increase 
 each of the digits of the original number by unity, 
 and invert the digits thus augmented, a number is 
 obtained which exceeds the original number by 29. 
 Find tlie number. 
 
 10. A stud(Mit takes . certain number of minutea 
 to walk from his residence to the Normal School. 
 Were the distance j^th of a mile g]'eat(!r, ho would * 
 need to increase his pace (number of miles per hour) 
 
APPENDIX. 
 
 xi 
 
 Vjy ^ of a mile in the hour, in oi-der to reacli the 
 school in the same time. Find how much he would 
 have to diminish his ])ace in ordei' still to reach the 
 school in exactly tlie same time, if the distance were 
 -g^ of a mile less than it is. 
 
 1 
 
 Secotid Glass Gertijlcates, 1875. 
 
 1. Find the continued product of the expressions, 
 a -\-\) + c, G + a -h, b + c- a, a + b - c. 
 
 a^ + a^b a{a-b) 2ab 
 
 2. Simplify - ;,, - ,3 - r-, yi — ~2 — Ti' 
 
 ^ - a^b -P b {a + b) cv' - b^ 
 
 3. Find the Lowest Common Multiple of TiX" - 2.x' - 1 
 and 4.//' - 2.«''^ - S.'K + 1. 
 
 4. Find the value of x from the equation, ax — 
 «'■' — obx ijbx — 5rr bx -\- 4' a 
 
 2a~ 
 
 -aU^ - bx + 
 
 a '^a 4 
 
 5. ^So!ve the simultaneous equations, 
 
 (I b 
 
 — + - =r m, 
 X y 
 
 c (I 
 
 - + - -- n. 
 X n 
 
 6. In the immediately preceding question, if a 
 |)upil should say that, when nb — <md, and be -ad, the 
 values of x and y o))tained in the ordinary method, 
 have the form ", and that he does not know how to 
 interpret such a result, what would you reply % 
 
 7. Two travellei's set out on a journey, one with 
 $100, the other with |?48 ; they meet with robljeis, 
 who take from the iirst twire as much as they take 
 IVom the second ; and what remains with the tiist is 
 3 times that whijh remains with the be«;ond. How 
 Uiu< '?. moiMriV did each travellci' lost; '/ 
 
i 
 
 xn 
 
 APPENDIX, 
 
 8. A and B labor togetlier on a piece of work for 
 Iwo clays ; and tlicn B linishcK the work by himself 
 in 8 days ; but A, \vith half of the assistance that B 
 could render, would liave finished the work in 6 days. 
 In what time could each of them do the whole work 
 alone % 
 
 9. P and Q are travelling along the same road in 
 the same direction. At noon P, who goes at the rate 
 of m miles an hour, is at a point A ; while Q, who 
 goes at the rate of n miles in the hour, is at a i)oint 
 J3, two miles in advance of A. ^yhen are they to- 
 gether ] 
 
 Has the answer a meaning when m — n is nega- 
 tive % Has it a meaning when m = n1 If so, 
 state what interj)retation it must receive in these 
 cases. 
 
 10. P is a number of two digits, x being the left 
 hand digit and y the right. By inverting the digits, 
 the number Q is obtained. Prove that 11 (x + y) 
 (P-Q)^-9(a;-v/)(P + Q). 
 
 Second Class Certijicates, 1876. 
 
 1. Divide (1 + m) x^ — (m + n) xy (x — y) — (n — 1) ?/' 
 by x^ — xy + y*. 
 
 Shew that (a + a^B + hf-- {a — a^h^ + hf is ex- 
 actly divisible by 2a^b^. 
 
 2. Resolve into fMx'tora x* + 2.*;// (.-c' — y") — ?/^, 
 
 a\b — c) + h-{c — a) + 'c\a — b), aiid 2bx' + 
 bx^- — X — 1. 
 
 3. If x^ ■\- px^ \ qx V r is exactly divisible by2;' + 
 mx + w, then nq — n^ - rm, 
 
 4. Prove that if m be a common measure of p and 
 
1 
 
 ork for 
 liimself 
 that B 
 6 days, 
 le work 
 
 road in 
 lie rate 
 Q, who 
 
 a point 
 ;liey to- 
 
 IS nega- 
 
 If so, 
 
 n these 
 
 the left 
 digits, 
 (x + y) 
 
 -1)?/ 
 is ex- 
 
 25a;* + 
 ;) and 
 
 APPENDIX, 
 
 xiu 
 
 q, it will also measure the difl'erence of any miiltiplea 
 of jD and q. 
 
 Find the G. C. INF. of x*~p:r' 4 (<?— l)a." + px— 
 q and rw* — qx^ -f (;j — 1 )x^ + qx~j> and of 1 + 
 
 x^ + X + x'^ and 2^ + %^ + 3a;- + 3a;^" 
 
 5. Prove the rule for multiplication of fructions. 
 
 x'—{y—zY y'^—iz—xf ■z'—ix—y)^ 
 Simplify ^% — V X 7 ^2 o >: / --vT-'^-i 
 
 and 
 
 a 
 
 a' 
 
 2a^--b^—ab'' 
 
 6. What is the distinction between an iderdity and 
 an equation ? If a? — a =-y + b, prove a; — b = y + a. 
 
 Solve the equations (2 + a;) {7n — 3) - — 4 — 2mx, 
 
 . 16a;— 13 40a;— 43 32a^-30 2Ga>— 24 
 and - , ~^^~ + - ^ — ,-- = -„— ->r + 
 
 4a;— 3 
 
 8a;— 9 
 
 8a^-7 
 
 4x — 5 
 
 7. What are simultaneous equations ? Ex])lain why 
 there must bo given as many independent equations 
 as there are unknoivn quantities involved. If tl:'erti 
 is a greater number of equations than unknown quan- 
 tities, what is the inference 'i 
 
 Eliminate x and u from the e(juations ax + by 
 = c, ax + b'y = c, a"x + b"y -- c", 
 
 8. Solve the equations — 
 
 (1) Vn + x+ '^\/ a — X'=m 
 
 (2) 3a; + 2/ + 2;=--13 
 3v/ + z+x=^ 1 5 
 3z + x + y- 17 
 
 9. A person has two kiiKls of foreign money ; ifc 
 takes a pieces of the iirwt kind to make one £, and b 
 pieces of the second kind: lie is offeied one £ for r 
 j)ieces, how many pieces of each kind must he take ? 
 
 A' 
 
XIV 
 
 APPENDIX. 
 
 10. A pei'son starts to walk to a railway station 
 four and a-lialf miles otF, intending to arrive at a 
 certain time ; but after walking a mile and a-lialf he 
 is detained twenty minutes, in consequence of which 
 he is obliged to walk a mile and a- half an hour faster 
 in order to reach the station at the a])poiiited time. 
 Find at what pace he started. 
 
 11. («.) It -J- = ^ then will ^^r:j:^. 
 
 a«c' 
 
 h" d "'^"h' + d'~W(P 
 
 (jb) Find by Horner's raetiiod of division the 
 value of 
 a;^ 290a;H 279a;"— 2892u;^— 586a>— 312 when 
 a; = _289. 
 (c) Shew without actual multiplication that 
 
 (a + b-\- cf—{a + b-^c) (a^'—ab + b''—bc + c'— «c) 
 — 3aic-3(rt + ii) (6 + c) (c + a). - 
 
 I' 
 
 u-iitmmmm^mm 
 
 ii ii 
 
 
 'I?:- 
 
station 
 e at a 
 lialf he 
 which 
 v faster 
 d time. 
 
 ion the 
 
 1 2 when 
 
 that 
 h c' — <ic) 
 
 McGILL UNIVERSITY. 
 — 4. — 
 
 First Year Exhibitions, 1873. 
 
 1. The difference between the first and second of 
 four numbers in geometrical progression is 12, and 
 the difference between the 3rd and 4th is 300 ; find 
 them. 
 
 2. Find two numbers whose difference is 8, and 
 the harmonical mean between them li. 
 
 3. Prove the general formula for finding the sum 
 of an arithmetical series. 
 
 4. The differences between the hypotenuse and the 
 two sides of a right-angled triangle i\re 3 and 6 
 respectively ; find the sides. 
 
 6. Sdlve the equations 
 
 , a; + y = 1 ; 
 X a; + 1 13 
 
 X + y = ^1/ 
 
 + 
 
 x+l X 
 
 x + y + z = ^, x + y^z-'{ ; x- 3 ^^y + z 
 x+ 4: 3.C + 8 
 
 ————— -r J. ,. — . 
 
 3.X' + 5 
 
 2ic + 3 
 
 6. A cistern can be filled by two ])ipcs in 24' and 
 30' respectively, and emptied by a third in 20' ; in 
 what time would it be filled, if all three were running 
 together. 
 
 7. Shew tliat 
 
 a^ + h^- c^ (a + h + c) (a-i-b-c) 
 
 1 
 
 I 
 
 / 
 
 ^ 
 
■MH 
 
 mil 
 
 XVI 
 
 APPENDIX. 
 
 8. Prove the rule for finding the greatest common 
 measure of two quantities. 
 
 I) 
 
 First Year Exhibitions, 1874. 
 
 1. The sum of 15 terms of an arithinotic series is 
 GOO, and the common difference is 5 ; find the first 
 term. 
 
 2. Find the hist term and the sum to 7 terms of 
 the scries 
 
 1-4 + 1G-&C. 
 
 3. Find tlie aritlimetical, geometric, and harmonic 
 means between 3| and IJ. 
 
 4. The difference between the hypotenuse and each 
 of the two sides of a right-angled triangle is 3 and 6 
 respectively; find the sides. 
 
 5. The sum of the two digits of a certain number 
 is six timc.tj their difference, and the number itself 
 exceeds six times their sum by 3 ; find it, 
 
 6. Solve the equations : — 
 
 3.^. _7 4.C-10 
 
 X 
 
 CC + 5 
 
 1 
 
 X 
 
 ^(2/-2)^5; 4y-H«'' + 10) = 3. 
 
 U2x + 1 8.cj^5 
 OOJ + 1 x—\ 
 
 7. A man could reap a field by himself in 20 hours, 
 but with his son's help for 6 hours, he coulcfdo it in 
 16 hours ; how long would the son bo in reaping the 
 Held by himself] 
 
 8. Find the value in its simplest form of 
 
 X + y 2a; 
 
 y 
 
 + 
 
 1 "X 
 
 xy — X 
 
 x + y a?y-f 
 
ommoD 
 
 series ig 
 ihe lii'st 
 
 «rin3 of 
 
 armonic 
 
 ,nd e.ach 
 3 and 6 
 
 number 
 r itself 
 
 ) hours, 
 do it in 
 )ing tlie 
 
 xvu 
 
 APPENDIX, 
 
 9. Find the greatest common measure of 
 307^ + S.c'^ - 15a: 4- 9 and 3u;^ + Zx" - 21a;' — - 9^, 
 
 First Year ExhlbitloiiSy 187G. 
 1. Solve the equations 
 
 12 
 
 a 
 
 
 u: y X 
 
 --r - = 1 - - 
 
 a b c 
 
 V X y 
 
 - f - = 1 H- - . 
 a b c 
 
 2. Reduce to its .simj»lesh form the expression :- 
 
 7 Vsl + 3 yYa + ^' 2— 5 ^l28. 
 
 3. Find the greatest common measure of 
 
 2x' -{-x' — Sx + d and 7a;» — 1 2^+5. 
 m^ + n* 
 
 4. Simi)Iify 
 
 n 
 
 m 
 1 
 
 m' 
 
 n' 
 
 m-^ + ^^^ 
 
 5. A number consists of two digits, of which the 
 left is twice the right, and tlie sum of the digits is 
 one-seventh of the number itself. Find tlie number. 
 
 6. Solve the following : — 
 
 x 1/ X z y z 
 
 - + - = +1, - + - =2, - + -=3; 
 
 a b (10 be 
 
 1 1 
 
 - + -=2, a;+2/=:2. 
 
 X y 
 
 7. Find the sum of n terms of the scries 1, 3, 5, 
 7, itc. 
 
 ((«.) Shew that the reciprocals of the first four 
 terms, and also of any consecutive four terms, are in 
 harmonioal proportion. 
 
 mi. 
 
 i 
 
 ' if \ 
 
UNIVERSITY OP VICTORIA COLLEG-E. 
 
 J i 
 
 Matriculation, 1873. 
 
 1. What is the " dimeuRion " of a term 1 When U 
 an expression said to be " homogeneous " "J 
 
 2. Remove the b^'ackets from, and simplify the 
 following expression : — 
 
 (26*— 3c + 4tZ)-;5(i — (m + 3a)| + |5rt — (— 4 
 ^d)\—\?>a — {ia — bd — i)\. 
 
 3. Prove the '^ Rule of Signs" in Multiplication. 
 
 2 1 ^ 2 
 
 4. Multiply a — '— by a; + — 
 
 X' 
 
 a X 
 
 5. Divide" ads^ + hx^ + ex + d hy x — r, 
 
 6. Divide 1 by 1 + ». 
 
 7. Find the Greatest C( mnaon Measure of 6a* — 
 a^x —I2x and Oa" r 12aV — 6aV— Sx^. 
 
 r.x — a^ 
 
 8. From 3a 
 ffi — a 
 
 2z 
 
 ar*— 1 
 
 subtract 2a — x — 
 
 a; + 1 
 
 9. Given 
 
 j 8 9 ( 
 
 { 
 
 X 
 
 9 
 
 y 
 
 8 
 
 -.+-=- 43 
 
 ) 
 
 to find X and i/. 
 
 10. Divide the nunxber a into four such parts that 
 the second shall exceed the first by m,, the third shall 
 exceed the second oy n, and the fourth shall exceed 
 the third by p. 
 
 11. A sum of nioiicy put out at simple interest 
 
APPENDIX. 
 
 xiz 
 
 EGE. 
 
 tVhen la 
 
 lify the 
 _(-4 
 
 lation. 
 
 amounts in m months to a dollars, and in n months 
 to h dollars. Re(|uired tlie sum and rate per cent. 
 
 12. Given a*' + ah --■ bx^, to fin»l the values of x. 
 
 13. Divide the number 49 into two sucli parts Hiat 
 the quotient ot the i^'rcater divided ))y the less niay 
 be to the quotient of the less divided b^ the greater, 
 as f to a. 
 
 14. !*Jivide the number 100 into two PAich j)arts that 
 their product ma^ be ec^ual to the ditierence of their 
 squares. 
 
 (x'-\ xy - 56, \ 
 16. Given -| - to find \ alues of .x-aud?/, 
 
 (ajy+22/'--GO,j 
 
 16. A farmer bought a numbei of sheep for $80, 
 and if he had bought four more for the same money, 
 he would have paid $1 less for each. How many did 
 he buy 1 
 
 f 6a* — 
 
 — X — 
 
 arts that 
 ird shall 
 1 exceed 
 
 interest 
 
 Matriculation^ 1874. 
 
 1. Find the Greatest Common Measure of 2 J' — 
 lOaJ' + %a% and 9i** — ZaW + ^a^h" — 9rt'6, and de- 
 monstrate the rait. 
 
 2. Add together a — a; + 
 
 a' + aj' o «* — oa? 
 , 3«— , 
 
 a + ic a + a; 
 
 2a;— 
 
 3(i2_2.c-. 
 
 a 
 
 X 
 
 , and — 4a 
 
 a^ + X' 
 
 a 
 
 'X 
 
 ,.'i 
 
 1 
 
 X 
 
 X 
 
 m 
 
 i + a; 
 
 3. Divide --— + ^---~ by - 
 1+03 1 — a; 1 
 
 and reduce. 
 
 4. Given I {x — a) — l'() {±c — ob) — \ {a ~ x) 
 - 10a4 116 to find x. 
 
 5. A sum of monev was divided among thi-ce per- 
 Suns, A, B, and 0, ah follows : tlie share of A 
 e-^cetdod 4 of the sha.-es of h a^»d hy $120 ; the 
 
 ^1 
 
 m 
 
 
 /\ 
 
'!! 
 
 ■• t- 
 
 
 . 
 
 
 XX 
 
 APPENDIX. 
 
 share of B, f of tbo shares of A and C by $120; 
 and tlie shiii-e of C, \ of tlie shares of A and B by 
 $120. \V luit was eac/i poison's share ] 
 
 6. Given | "^ \ t, + ^f^ ^^ = «» I to 6nd =» and^. 
 
 7. Shew thiit a quadratic efpiation of one unknown 
 quantity cannot liave more than two roots. 
 
 2v/.i; + 2 4— v/iB 
 
 8. Given 
 
 \ ^ y/ X y/x 
 
 '; to find tlie value of x. 
 
 9. The e is a stack of hay whose len<,'th is to its 
 breadth as 5 to 4, and wliose hei'dit is to its bieadtb 
 as 7 to 8. It is wo.tb as m.inv cents per cubic foot 
 as it is feet in broacHh ; and the whole is worth at 
 that rate 221 times as many cvnts as there are squai'e 
 feet on the bottom. Find the dimensions of the stack. 
 
 10- Given 
 
 j^-V.^.5 I 
 
 ]?3 
 
 2xy _ 
 
 y 
 
 y/xij 
 
 ) 
 
 to find X and y. 
 
 11. In attemi)ting to arrange a number of counters 
 in the form of a square it was found there wej'« seven 
 over, and when the side of the square was increased 
 by one, there was a deficiency of 8 to complete the 
 square. Eind the number of counters. 
 
 12. Keduce to its simplest form 
 
 t. — (^ — ^f + h^ — ip 
 
 So^ (h + cY — a'' 
 
 (^ + c)« _ h^ {a + by 
 
 13. A and B can do a piece of work in 12 daysj 
 in how many days couhl each do it alone, if it would 
 take A 10 days longer tiian B l 
 
 14. Given 
 
 X 
 
 y ^ 
 X — ;y = 4 
 
 Z V)—- li 
 
 , j^ 4- ;v* + 
 
 to find 
 
 aj, y, «, 
 
 and w. 
 
 + w;2 = 62j' 
 
! by $120; 
 . and B by 
 
 lid X and^. 
 le unknown 
 
 value of X. 
 
 ;h is to its 
 its lueadth 
 oiibic foot 
 is worth at 
 ! art! squai'e 
 )f tlie stack. 
 
 X and y. 
 
 of counters 
 
 wer« seven 
 
 ,s increased 
 
 )mplete the 
 
 a — by 
 
 y — a'' 
 
 in 12 days; 
 if it would 
 
 to find 
 
 X, y, z, 
 and w. 
 
 APPENDIX. 
 
 xjd 
 
 15. Pind tlio last t<!rni, and the sum of 50 terms, 
 of the series 2, 4, G, 8, ifcc. 
 
 16. Write down tho expansion of ^ -^ — > 
 
 17. How many it'lftfttinc snaias may be rung on 
 ten different; bells, supposing all tho combinations to 
 produce different uote« * 
 
 ^M 
 
4 
 
 'I ', 
 
 i 
 
 ANSWEHg 
 
 y^ 
 
 Junior Mairic, 1872. Pass, 
 
 1. f x-» - ( fVr^^y - W + y'') ; (a^ + 9^.^) {a + 36) ; 
 
 (:c + ft)2 + (a; + «) (yy - i) + (y _ b)\ 2. a^ + ap + 9 
 
 3. W, li; (^), ]^; (c), 4i; (./), 1, 1. 4. 640, 660. 
 
 Junior McUriv , 1872. Atw and Honor, 
 1. {.i+(,.i-yi)|V|^i-(:.i-^J)r = 
 
 I ^ - (.£> ^ 'nhf { ; ,, .|. 26 + 3c.. 2. We have 
 c^ — ;;r;^9' = and c^ -- p'c + q' ^0, from 
 
 which to elini.nato r. 
 4. If /3 be one root, --i^ aYi + ^-Y ' ./3''^, 
 
 and, eliiiiinating 'v, _^ A^. . 
 
 rtc pr/ 
 
 6. (a), 1, - 7, i(— 3rhv^277) ; (6), 3, 2, ; — 3,- 2 
 7 5 7 5 
 
 I. 
 
 n/6 
 
 /6 
 
 -f. |2. (r/), Dividf. thionrrh by x' and ]»ut _y for 
 x+ - , and .*.y/^ — 2 for ic"+ - , then u = 
 
 10 
 — or 
 
 3 
 
 T) 
 
 and x-3 J, — J or — 2. 
 
 
'i^ + aq) + q 
 S40, 660. 
 
 OT. 
 
 We hare 
 - 0, from 
 
 1 
 
 ANSWERS. 
 
 xxiii 
 
 « 1 a^— 1 \ 'P , P + 9 P+29 I 
 
 6. ; Va +« +a .+ .... ^ 
 
 (1 — ry a — 1 J ) 
 
 7. a (biC^ — biCi) + CLiifi-/^" ^^s) + ^aC^^i — V) =0. 
 8. 
 
 1+r 
 
 Junior Matrlc, 1872. Ilouor. 
 
 1. 8 and 6 miles. 2. Each of the first set of 
 
 fractions may be shewn equal to 
 
 - 2/ 
 
 2rt6c » or 2ahG f> 
 
 or 2abc 
 
 z 
 
 G 
 
 a- 
 
 ^b^-c' 
 
 _, v/hich are therefore equal. 
 
 3. Multiplying the equations successively by y, z, z 
 and z, X, y, we obtain c'^x + cJ^y + 1/^ = 0, 
 
 l^x + (^y + (^z - ; thence -- 
 
 X 
 
 y 
 
 a' - b'c 
 
 c'a^ 
 
 z , ±a(a*-b^c'') 
 
 c* - a'b*' y/\(a'-b'cy-{b'-c^a'){a'-a'b')]' 
 
 4. a« + 6^>2a/;,.-.c(rt'^ + //)>2«Z>c, etc. 
 
 5. 3^0; -2, -5;- 3, 6; ~ 8, I. 6. 90 and 240 mU 
 
 
 -3,-2 
 
 [.),-3 
 
 put y for 
 len y a 
 
 %funior Mafric, Ifoiior. \ 
 h^emor Ma trie. , I ass. ) 
 
 1. (a), From first x = 2y or ?/, and then solutions a»"e 
 
 3, i; -3,— .|j_n/2I^ v/2T; — v/2r,-~ V2i: 
 
 (/;), T^j(41±v/769), i-(-37±i/761)). (6-), J, 4; 
 _i _i. 1 i-_i. _i. A/), 4, 18. 2. 3. 
 
 , , , 33a;»+61a;+lO ^,^ a;'^ + 3a;+2 
 
 <• (")' ^T2 ^ ^^^' Tr+TTrTTso- 
 
XXIV 
 
 ANSWERS. 
 
 _ X (Z—x) 
 
 6. 05 - 2 and a; + 5 are_ factors, and roots are, 2, - 5, 
 J (-3=t^/35). 7. 7 J gals. 
 
 '^^ 4.88 per cent. 9. 4 days. 
 
 lie receives $3 every day the work continues ; 
 he returns notliing the first day he is idle, 
 $1 the second, and so on, and the number of 
 days he works is IG. 
 
 Junior Matric., 1876. Pass. 
 
 I 2. m-n jn—2 —1, « « « . . 
 
 i. a , a be d. 2. a^-7?) a^+i?, 
 
 3. 1 + 2iK + 3ar^ + 4a^ + 5a;* + 6ic^ + ...... ; rem. 7a:*- 
 
 Qx\ ^x^ — Qx+\. 
 
 5. (1), 2. (2), 2, 5, 7; or -2,-5,-7. 
 
 1 
 
 Junior Matric, 1876. Uonw. 
 
 1. 35 mis. 2. (2), These quantities are in //. P. if 
 
 -^; j-,&c., are in il.P., f.c., if a, b, c 
 
 ao + aG + bo * ' 
 
 are in y(. P. 
 
 5. It may be shewn that the remainder at the nth 
 
 decimal place is 2/i ; hence if the nth digit be 
 increased by unity, and the whole subtracted 
 from 1, the remainder is the remaining part 
 of the period. 
 
 6. ««=4,a; = 2or-3^=3or-2;2;=--l,a;x.2±v^ror, 
 
 3/« -2*^/10. 
 
ANSWERS. 
 
 ZZY 
 
 Junior Matric, 1876. Honor,, ^ 
 
 1. 121 and 400 yards. 
 
 2. (a — b + c) (ab + bc + ca) (a^ + b'^ + c* + ab + bc — ca). 
 
 3. Irrational roots go in pairs/. 3--^/ 2" is a root; 
 
 and other roots ave | (-—1 rhv^ Ha). 
 
 4. £C- + oj'y^ + x^f/'-^ + xy + x^y^ + 2/^* 
 ab 
 
 5. 
 
 6. 3 nils. 
 
 6 + (n— 1) (a— 6) 
 7. (1), Plainly x -f 2 divides both sides, and roots 
 are — 2, 24- /t"- (2), oj = 3, y = 4 or ;|- ; .r =• 
 — 3, 2/ - — 4 or 
 
 i- 
 
 Second Glass Certificates, 1873. 
 2. (a-6)-(a-46) = 36. 
 
 3 _(^+w-{^y-^i^h 
 
 4. (i),-99. 
 
 5. (a -b) {x-y) = ; .*. if a be not = ?>, x -y = ; 
 
 if a - b, X - y may have any value. 
 
 6. ^4 -lo* '^' h pi'ovided x bo not= - 2^ ; 
 
 then fi-action becomes § and is indetei mluate. 
 
 11 
 
 x-y ' a; + 7 * ^ ^ 
 
 9. 13. 
 
 1 0. -J of a mile j)er hour, 
 
XXVI 
 
 ANSWERS. 
 
 Second Glass Certificates, IST^ 
 1. 2(a'6« + 6V + cV) - (a' + b' + c'). 
 
 3. (3a; + 1) (iic" - 2x' -3x+ 1). 4. 
 
 oa 
 
 2. 1- 
 
 a + h 
 
 2a{2b^ - 5) 
 
 ~4a - 36~' 
 
 5. x= 
 
 he 
 
 ad he 
 
 ad 
 
 nb — md mc — na 
 6. X and y are indeterminate : therp ia but one 
 equation. 7. $88, $44. 8. 14 days, 11| days. 
 
 /in ^^^^* '^^ — ** negative means that they 
 
 were together 
 
 they are never together. 
 10. Each side equals 99(ar^ —- 2/2). 
 
 hrs. before noon. wi=«, 
 
 Second Class Certificates, 1876. 
 
 1. {\-fm)x-{\-'n)y.' 2. {x + yf {x-y)] (a-b) 
 (b-c) (c-a); (Saj^-l) {5x' + x+l), 
 
 3. Let the other factor be a; + a; multiply and equate 
 co-efficients ; eliminating a, nq ~n^- rm; other 
 condition is ;;?^ - vm = r. 4. a; - 1 ; 1 + x-i. 
 
 5. i^-^y-^) {^-y + ^) fa + g-a;) . J _ 
 
 (x + y + zf ' a-b 
 
 6. -§; 1. 
 
 7. a'(^'c " f>c') + b"{ac - a'c) + c>'6 - «6') = 0. 
 
 8. (1,) Cube, and 3(n4-a;)i (/i - a;)4 (wi) -= ?>i" ~ 2n, 
 
 q a (c -- ?>) b (a - c) 
 a — 6 a - 
 
 10. 3 miles an hour. 
 
oa 
 
 a + h' 
 W - 5) 
 i-'6h" 
 
 ANSWERS* 
 
 xxvii 
 
 U. (a), See §359. (J), 2,000. (c), Substitute suc- 
 cessively ~b, —c, -a for a, b, c, in the left 
 hand side, and it appears that a-\-h, b + c, 
 c + a are factors, and .*. expression is of form 
 N{a + 5) (6 4- c) {c + a)i putting a - 6 = c = 1, 
 
 we get iV"= 3. 
 
 but one 
 ll|days. 
 
 that they 
 
 )n. wi=«, 
 
 First Year Exhibitions, 1873. 
 
 1.3,15,75,375. 2. 9andl,or J^and-ll. 4.9,12. 
 5. (a), 4,-3; -3,4. (6),2,-3. (c),4,-5,r,. (cZ},-|. 
 
 0. 40 . 7. = ^: — -— — - . 
 
 2ao 
 
 ; («-6) 
 
 id equate 
 m'y other 
 
 ; 1 + x-i. 
 
 /\'n9< Fear ExJdbitions, 1!^74. 
 
 1. 5. 2. (— 4)^ 3277. 3. 2^\; 2J; 2/^. 
 
 4. 9, 12. 5. 75. 
 
 6. (a),3,2;— 2,— 3. (6), 7 or— If (o),5,3. {d),U. 
 
 7. 30 hours. 
 
 8. 
 
 ?/ 
 
 a; + v/ 
 
 9. 3{x-h'd). 
 
 First Vear ExJdbitions^ 1876. 
 
 m* - 2n, 
 
 1 
 
 1 
 
 b 
 
 1 1 
 
 -I 4 3 a 
 
 1. _„,-.,-.__. J 
 
 r' 1 
 
 1 1 
 
 ft c 
 
 1 
 
 ..2 
 
 1 — .12y2. 3. aj— 1. 4. ??i. 
 5. 21, 42, 63, or 84. G. o, b, 2r. -, 1, 1. 
 
 .1 
 
 I 
 
 7. n' 
 
XXVIU 
 
 ANSWERS. 
 
 B' ( 
 
 2. 
 
 5. 
 
 8. 
 
 10. 
 11. 
 
 12. 
 11. 
 15. 
 IG. 
 
 Matriculation, 1873. 
 
 \\a — 3c — TmI -\- m. 4. — ax. 
 
 ax^ -|. {ar -f 6) aj + (a?' + hr -\- c) -V 
 ar^ + br _j- cr + tZ 
 a; — r 
 
 l—x + x' — x'+,,.,. 7. 3«' + 'U-», 
 
 (a — x) {x'' — .2) 
 
 x'^ — l 
 
 1: (f* — 3y/i — 2/» — p), SiG. 
 mb — na 1200 (f^* — b) 
 
 9. 144, 216. 
 
 \ 
 
 iiij — M 
 J 
 
 itib — na 
 ±l^ab. 13. 28, 21. 
 
 50(v/5 —1), 50(3 — 1/5). 
 x^ ±10, 1/ ^ =F 10 ; it; ~- zfc 4n/ 2, 2/ = ± 3/2 . 
 16. 
 
 Matriculation, 1874. 
 
 1. t* — 5. 2. 
 
 4. _5,t_36. 
 
 6. 2, 4 ; 4, 2. 
 
 <). 20, 16, lift. 
 
 12. 1. 
 
 14. 6, 2, 4i, U, or 
 
 15. 100, 2550. 
 
 4(7,' + a^x — 2ax^ + a? 
 ,7r — a'^ 
 5. 600, 480, 360. 
 
 8. 4 or 9i. 
 
 10. 40, 10; 10, 40. 
 
 13. 30 and 20 days. 
 
 -2, — 6,— lj.~4i. 
 
 X t 
 
 11. 56. 
 
 Kr^-l 
 
 16. ft.T __ 7aj» + 21a;' — 35,c + 35.^ 
 — o:'\ 17. 1023. 
 
 21a;-' + 7a5 
 
 -» 
 
56. 
 
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 MASON'S (JHADUATED SERIES OF ENGLISH GRAMMARS. 
 
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 PKCtMINEST FEATURKS ! 
 
 The book is divided into five parts as follows : 
 
 PART I. 
 
 Contains the words in common use in daily life together with abbrevia- 
 tions, forms, etc. If a boy has to leave school eaily, h£ should at least 
 know how to spell the words of common occurrence in connection with his 
 business. 
 
 PART II. 
 
 Gives words liable to be spelled incoiTectly because the same sounds are 
 spelled in various ways in them, 
 
 PART HI. 
 
 Contains words pronounced alike but sjicllod diflFerently with different 
 meanings. 
 
 PART IV. 
 
 Contains a large collection of the most difficult words in common use, 
 and is intended to supply material for a general review, and for spelling 
 matches and tests. 
 
 PART V. 
 
 Contains literary selections which are intended to be memorhed and re- 
 cited as well as used for dictation lessons and lessons in moral.^. 
 
 DICTATION LESSONS. 
 
 All the lessons are suitable for dictation lessons on the slate or in dicta- 
 tion book. 
 
 REVIEWS. 
 
 These will b;; found throughout the book. 
 
 An excellent compendium. Al^'x. McRai\Pi'in. Acad'ii,Di(]hy,N.S. 
 I regard it as a necessity and an excellent compendium of the subject 
 of which it treats. Its natural and judicious arrangement well accords 
 with its title. Pupils instnicted in its principles, under the care of diligent 
 teachers, cannot fail to become correct spellers. It great value will, doubt- 
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 Supply a want long felt. John Johnston, I.P.S., Belleville. 
 
 The hints for teaching spelling arc excellent. I have shown it to a num- 
 ber of experienced teachers, and they all think it is the best and most i)rac- 
 tical work on spelling and dictation ever presented to the public. It will 
 supply a want long felt by teachers. 
 
 Admirably adapted. Colin W. Roscor, r.P.S., Wolf ri lie, N. S. 
 
 The arrangement and grading of the different classes of words I rcganl 
 as excellent. Much benefit must arise from committing to memory the 
 " Literary Selection.s." The work is admiraljly adapted to our public 
 schools, and I shHU recommend it as the best I have seen. 
 
• J- (iaflc S: QTo'g. ^ciu €bufatiottal oSEorks. 
 
 TEXT HOOKS ON ENGLISH GRAMMAR. 
 
 BY MASON AND MACMILLAN. 
 
 Revised Ed. Miller's Language Lessons. 
 
 Adajited as an introductory Text Book to Mason's Grammar. By J. 
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 to Hif;h Schools, and teaches Grammar and Comjiosition sinmltaneously. 
 Sixth Edition, 200th thousand. 
 
 Price, 
 
 25 Cents. 
 
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 New and improved edition. With copious and car«fully graded exercises. 
 243 pages. 
 
 Price, 60 Cents. 
 
 Mason's Advanced Grammar. 
 
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 ters on the analysis of difficult sentences is of itself sufficient to place ths 
 work far beyond any EnglishGrammar l)efore the Canadian public."— Alex. 
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 English Grammar Exercises. 
 
 By C. P. Mason. Reprinted from Common School Edition. 
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a J. ©age ^ Co'0. HeU) €b«attional MoxU. 
 
 WORKS FOR TEACHERS AND STUDENTS, BY JAS. L. HUGHES. 
 
 Examination Primer in Canadian History. 
 
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 ronto. A T'liiiior for Stiideiitu i)rcpaiiii;; for Examination. Price, 25C 
 
 Mistakes in Teaching. 
 
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 ADOPTKD BY STATK UN1VKR8ITY OF IOWA, AS A.S ELEMENTARY WORK FO!l USK 
 
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 This work discusses in a terse manner over one hundred of the mistakes 
 commonly made hy untrained or inex)ieneneed Teachers, li isdesig-ned to 
 warn younft- Teachers of the errors tliey are liable to make, and to help the 
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 be preventinff their hi<jfher success. 
 
 The niistakes arc arranjj^cd under the following heads : 
 
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 The work contains : The Squad Drill proscribed for Public Schools in On- 
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 I. — An Introduction, embracing Definitions of Literary Terms, Classifica- 
 tions of Prose and Poetry, F'i<,nires of Speech, CJontemix»rary Writers 
 (with their works) of Athlison, G(Jldsmith and Cow^Hjr. 
 
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 with numerous references to Mason's Grammar. 
 
 [iii*^arv 
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 iglaiid. 
 .Songs, 
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 The Deserted Village, The Task, and Ad- 
 dison's Sir Roger De Coverley. 
 
 In one volume. Interleaved. With remarks on Analysis of Sentences ; 
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 Lives of the Authors. By Walter McLeod, F.R.G.S., F.C.P., Francis Storr, 
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 With IntroiUictlon, Notes, Maps and Glossary. Interleaved. Ry Kdwar 1 
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 (Copy) Edvcation Office, Halifax, N. S., Nov. 17, S19,. 
 
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 Read extract frovi letter from the fo.tnous American Poet, John 
 
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 iving each 
 
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