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Hontion : MACMILLAN AND CO. i8;9 [The Right of Translation is reserved.] Camliritige: PRINTED BY C. J. CLAY, M.A., AT THE UNIVEKSITT PUESS. y s^ ci e S S *-• \ PEEFACE. This work contains all the propositions which are usually included in elementary treatises on algebra, and a large number of examples for exercise. My chief object has been to render the work easily intelligible. Students should be encouraged to examine carefully the language of the book they are using, so that they may ascertain its meaning or be able to point out exactly where their difficulties arise. The language, therefore, ought to be simple -and precise j and it is essential that apparent conciseness should not be gained at the expense of clearness. In attempting, however, to render the work easily intelligible, I trust I have neither impaired the accuracy of the demonstrations nor contracted the limits of the subject ; on the contrary, I think it will be found that in both these respects I have advanced beyond the line traced out by previous elementary writers. The present treatise is divided into a large number of chapters, each chapter being, as far as possible, complete in itself. Thus the student is not perplexed by attempting to master too much at once ; and if he should not succeed in fully comprehending any chapter, he will not be precluded from going on to the next, reserving the difficulties for future consideration : the latter point is of especial importance to those students who are without the aid of a teacher. 984604 Vi niEFACE. The onler of succession of the several chapters is to some extent arhitraiy, because the position which any one of them shoukl occupy must depend partly upon its difficulty and partly upon its importance. But, since each chapter is nearly independ- ent, it will be in the power of the teacher to abandon the order laid down in the book and to adopt another at his discretion. The examples have been selected with a view to illustrate every part of the subject, and, as the number of them is more than two thousand, I trust they will supply ample exercise for tlie student. Complicated and difficult problems have teen excluded, because they consume time and energy which may be spent more profitably on other branches of mathematics. Each set of examples has been carefully arranged, commencing with some which are very simple and proceeding gradually to others which are less obvious; those sets which are entitled Miscellaneous Examples, together with a few in each of the other sets, may be omitted by the student who is reading the subject for the first time. The answers to the examples, with hints for the solution of some in which assistance may be needed, are given at the end of the book. I will now give some account of the sources from which the present treatise has been derived. Dr Wood's Algebra has been so long current that it has become public property, and it is so well known to teachers that an elementary writer would naturally desire to make use of it to some extent. The first edition of that work appeared in 1795, and the tenth in 1835 ; the tenth edition was the last issued in Dr Wood's life-time. The chapters on Surds, Ratio, and Proportion, in my Algebra are almost entirely taken from Dr AVood's Algebra. I have also frequently used Dr Wood's ex- amples either in my text or in my collections of examples. Moreover, in the statement of rules in the elementary part of my book I have often followed Dr Wood, as, for example, in the PKEFACE. Vll Kule for Long Division ; the statement of such rules must be ahnost identical in all works on Algebra. I should have been glad to have had the advantage of Dr Wood's authority to a greater extent, but the requirements of the present state of mathematical instruction rendered this impossible. The tenth edition of Dr "Wood's Algebra contains less than half the matter of the present work, and half of it is devoted to subjects which are now usually studied in distinct treatises, namely. Arithmetic, the Theory of Equations, the application of Algebra to Geometry, and portions of the Summation of Series ; the larger part of the remainder, from its brevity and incompleteness, is now unsuitable to the wants of students. Thus, on the whole, a very small number of pages comprises all that I have been able to retain of Dr Wood's Algebra. For additional matter I have chiefly had recourse to the Treatise on Arithmetic and Algebra in the Library of Useful Knowledge, and the works of Bourdon, Lefebure de Fourcy, Mayer and Choquet, and Schlomilch ; I have also studied with great ad^^antage the Algebra of Professor De Morgan and other works of the same author which bear upon the subject of Algebra. I have also occasionally consulted the edition of Wood's Algebra published by Mr, Lund in 1841, Hind's Algebra, 1841, Colenso's Algebra, 1849, and Goodwin's Elementary Course of Mathematics, 1853. Althouofh I have not hesitated to use the materials which were available in preceding authors, yet much of the present work is peculiar to it ; and I believe it will be found that my Algebra contains more that is new to elementary works, and more that is original, than any of the popular English works of similar plan. Originality however in an elementary work is rarely an advan- tage ; and in publishing the first edition of my Algebra I felt some apprehension that I had deviated too far from the ordinary methods. I have had great satisfaction in receiving from eminent viii PREFACE. tetichers favourable opinions of the work generally and also of those parts which arc peculiar to it. The present edition has been carefully revised, and two new cliapters have been added. Three hundred miscellaneous examples have also been supplied ; these are arranged in sets, each set con- taining ten examples ; the fii*st hundred relate to the first twenty chapters of the book, the second hundred extend to the end of the fortieth chapter, and the last hundred relate to the whole book. I have to return my thanks to many able mathematicians who have favoured me with suggestions which have been of service to me ; the improvements which have been effected in the work will, I trust, render it still more useful in education, and still more worthy of the approbation which it has received. I have drawn up a treatise on the Theory/ of Equations to form a sequel to the Algebra; and the student is refen-ed to that treatise as a suitable continuation of the present work. I. TODHUNTER St John's College, October, 1870. CONTENTS. PAGE I. Definitions and Explanations of Signs .... 1 II. Change of the order of Terms. Reduction of Like Terms. Addition, Subtraction, Use of Brackets ... 7 III. Multiplication . .14 IV. Division 23 V. Negative Quantities 33 VI. Greatest Common Measure 45 YLl. Least Common Multiple 54 VIII. Fractions 57 IX. Equations of the First Degree 70 X. Problems which lead to Simple Equations with one Un- known Quantity ........ 80 XI. Simultaneous Equations of the First Degree with two Un- known Quantities 88 XII. Simultaneous Equations of the First Degree with more than two Unknown Quantities ...... 94 XIII. Problems which lead to Simple Equations with more than one Unknown Quantity ....... 98 XrV. Discussion of some Problems which lead to Simple Equa- tions 106 XV, Anomalous forms which occm- in the solution of Simple Equations 117 XVI. Involution 129 XVII. Evolution 132 XVin. Theory of Indices . • 149 XIX. Surds 157 XX. Quadratic Equations 169 XXI. Equations which may be solved like Quadratics . . . 175 XXII. Theory of Quadratic Equations and Quadratic Expressions 189 XXIII. Simultaneous Equations involving Quadratics . . . 197 XXIV. Problems which lead to Quadratic Equations . . . 204 CONTENTS. XXV. Imaginary Expressions XXVI. Batio XXVII. Proportion XXVIII. Variation XXIX. Scales of Notation XXX. Ai-ithmetical Progression . XXXI. Geometrical Progression XXXII. Harmonical Progression , XXXIII. Mathematical Induction XXXIV. Permutations and Combinations XXXV. Binomial Theorem. Positive Integi-al Exponent XXXVI. Binomial Theorem. Any Exponent XXXVII. Multinomial Theorem XXXVIII. Logarithms ...".... XXXIX. Exponential and Logarithmic Series . XL. Convergence and Divergence of Series XLI. Interest ....... XLII. Equation of Pajonents XLIII. Annuities ....... XLIV. Continued Fractions XLV. Beduction of a Quadratic Surd to a continued Fraction XL VI. Indeterminate Equations of the First Degree . XL VII. Indeterminate Equations of a Degree higher than the first XLVIII. Partial Fractions and Indeterminate Coefficients XLIX. Becurring Series . . .... L. Summation of Series LI. Inequalities ..... LII. Theory of Numbers .... LIII. ProbabiUty LIV. Miscellaneous Equations . LV. Miscellaneous Problems • , LVI. Convergence and Divergence of Series "" LVn. Continued Fractions . LYIII. Miscellaneous Theorems . Miscellaneous Examples ANSWERS ALGEBEA. I. DEFINITIONS AND EXPLANATIONS Of SIG^JJ. ' ' - 1. The method of reasoning about numbers, by means of letters which are employed to represent the numbers, and signs which are employed to represent their relations, is called Algebra. 2. Letters of the alphabet are used to represent numbers, which may be either known numbers, or numbers which have to be found and which are therefore called unknown numbers. It is usual to represent known numbers by the first letters of the alphabet, as a, b, c, and unknown numbers by the last letters, as X, y, z; this is not however a necessary xule, and so need not be strictly obeyed. Numbers may be either whole or fractional. The word quan- tity is frequently used as synonymous with number. The word integer is often used instead of whole number. 3. The sign + signifies that the number to which it is prefixed is to be added. Thus a + b signifies that the number represented by b is to be added to the number represented by a. If a repre- sent 9, and b represent 3, then a + b represents 12. The sign + is called the plus sign, and a + b is read thus " a plus b." Similarly a + b + c signifies that we are to add b to a, and then add c to the result. 4. The sign - signifies that the number to which it is prefixed is to be subtracted. Thus a — b signifies that the number repre- sented by b is to be subtracted from the number represented by a. If a represent 9, and b represent 3, then a — b represents 6. The sign — is called the minus sign, and a — 6 is read thus " a minus b.' T. A. 1 2 DEFINITIONS AND EXPLANATIONS OF SIGNS. Similarly a — h — c signifies tliat we are to subtract h from a, and then subtract c from the result ; a + h — c signifies that we are to add h to a, and then subtract c from the result ; a — h-^-c signi- fies that we are to subtract h from a and then add c to the result. 5. The sign x signifies that the numbers between which it stands are *to -be multiplied together. Thus axh signifies that the number repi'eSellied by a is to be multiplied by the number repre- ss j\ti3cl' by J). 'If- « re][>resent 9, and h represent 3, then axh repre- sents 27. The sign x is called the sign of muUi2?licatio7i, and a xh is read thus " a into b." Similarly a xhxc denotes the product of the numbers represented by a, h and c. It should be observed that the sign of multiplication is often omitted for the sake of brevity ; thus ah is used instead of axh, and has the same meaning ; so abc is used for axh x c. Sometimes a point is used instead of the sign x ; thus a . 6 is used for axh or ah. But the point is here superfluous, because, as we have said, ah is used instead of « x &. Nor is the point, nor the sign x , necessary between a number expressed in the ordinary way by a figure and a number represented by a letter ; so that, for example, Za is used instead of 3 x a, and has the same meaning. The sign of multiplication must not be omitted when numbers are expressed by figures in the ordinary way. Thus 45 cannot be used to express the product of 4 and 5, because a diflferent mean- ing has already been appropriated to 4 5, namely forty-five. We must therefore express the product of 4 and 5 thus 4x5, or thus 4.5. To prevent any confusion between the point thus used as a sign of multiplication and the point as used in the notation for decimal fractions, it is advisable to write the latter higher up ; thus 4*5 may be kept to denote 4 f ■^. 6. The sign -f- signifies that the number which precedes it is to be divided by the number which follows it. Thus a-^h sig- nifies that the number represented by a is to be divided by the number represented by h. If a represent 9, and h represent 3, tlien a -^h represents 3. The sign ^ is called the sign of division^ and a H- 6 is read thus " a hy b." There is also another way of DEFINITIONS AND EXPLANATIONS OP SIGNS. 3 denoting that one number is to be divided by another ; the divi- dend is placed over the divisor with a line between them. Thus - b is used instead of a-^b and has the same meaninjr. 7. The sign = signifies that the numbers between which it is placed are equal. Thus a = b signifies that the number repre- sented by a is equal to the number represented by h, that is, a and b represent the same number. The sign = is called the sign of equality, and a = 6 is read thus " a equals b " or " a is equal to b." 8. The difference of two numbers is sometimes denoted by the sign ~ ; thus a~h denotes the difference of the numbers denoted by a and h, and is equal to a — 6 or to 6 — a, according as a is greater than b or less than b. 9. The sign > denotes greater than, and the sign < denotes less than; thus a > 6 denotes that the number represented by a is greater than the number represented by b, and bed. 4. ac + ^icd — 2eb. 5. abc + ibd-\-ec- -fd. 6. a' + b' + c'+/\ 7. cd 4:be cd b "^ 3a 24 • 8. c'-4:c' + 3c-6. 9. b' + c' 2c -3a' 10. d'-c' d' -\-dc + o'' 11. ^/(276)- 4/(2^) + J(2c). 12. J{3bc)+'J{9cd)^l/{2e'). EXAIVrPLES. I. 7 13. Find the value of (9 -y) (x + l) + {x + 5) {y + 7)-n2, when re = 3 and y = 5. 14. Find the value of x J{x^ - 8^/) + y J{x' + Sy), when x = 5 and y = S. 15. Find the value of a J{x^ -Za)-\-x J{x^ + 3a), when a; = 5 and a = 8. 1 6. Find the value oi a + h J{x + y)-{a — l) ^(x — y), when a = 10, b = 8, x=12, and 3/ = 4. 17. If a = 16, 6 = 10, x = 5 and y = 1, find the value of (6 - x) {J a + h) + J{{a -h){x^y)}; and of {a - y) {J{2hx) + x^] + ;^/{(a - cc) (6 + 2/)}. 18. Tf a = 2, 6 = 3, a; = 6 and 2/ = 5, find the value of U{{a + 6)^2/} + y{(« + «') (y - 2a)} + 4/{(y - &)' «.}. IL CHANGE OF THE ORDEll OF TERMS. 'EEDUCTION OF LIKE TERMS. ADDITION, SUBTRACTION, USE OF BRACKETS. 25. When the terms of an expression are connected by the sign + it is indifierent in what order they are written ; thus a + 6 and h-\- a give the same result, namely the sum of the numbers which are denoted by a and h. We may express this fact algebraically thus : a + h = h+ a. Similarly a-irh + c = a-{- c + h = h + a+c = h-¥ c + a = c-\-a-\-h = c + h + a. 26. When an expression consists of some terms preceded by the sign + and some terms preceded by the sign — , we may write the former terms first in any order we please, and the latter terms after them in any order we please. This appears from the same considerations as before. Thus, for example, a-\-h — c — e = a-irh — e — c = h-^a — c — e = h-\-a — e — c. 8 CHANGE OF THE ORDER OF TERMS. 27. In some cases it is obvious that we may vary the order of terms still fui-ther, by mixing up the terms preceded by the sign — with those preceded by the sign + . Thus, for example, if a represent 10, 6 represent 6, and c represent 5, then a + h — c = a — c + h = h — c ■¥ a. If however a represent 2, h represent 6, and c represent 5, then the expression a — c + h presents a difficulty because we are thus apparently required to take a greater number from a less, namely 5 from 2. It will be convenient to agree that such an ex- pression as a — c + 6 when c is greater than a shall he understood to mean the same thing as a + b — c. At present we shall i\ever use such an expression except when c is less than a + b, so that a + b — c presents no difficulty. Similarly we shall consider - b + a to mean the same thing as a— 6. We shall recur to this point hereafter in Chapter Y. 28. Tlius the numerical value of an expression remains the same whatever may be the order of the terms which compose it. This, as we have seen, follows, partly from our notions of addition and subtraction, and partly from an agreement as to the meaning we ascribe to an expression when our ordinary arithmetical notions are not strictly applicable. Such an agreement is called in Algebra a convention, and conventional is the corresponding adjective. 29. "We shall frequently, as in Article 2Q, have to distinguish the terms of an expression which are preceded by the sign + from the terms which are preceded by the sign — , and thus the follow- ing definition is adopted : The terms in an expression which are preceded by no sign or which are preceded by the sign + are called positive terms ; the terms which are preceded by the sign - are called negative terms. This definition is introduced merely for the sake of brevity, and no meaning is to be given to the words positive and negative beyond what is expressed in the definition. The student will notice that terms preceded by no sign are treated as if they were preceded by the sign + . ADDITION. 9 30. Sometimes an expression includes several like terms; in this case the expression admits of simplification. For example, consider the expression Aia% - Sa^c + 9ac^ — 2a^b + 7a^c — 66' ; this may be written 4a'6 - 2a'b + Id'c - 3a'c + 9ac' - 65' (Art. 28). Now 4a'6 - ^a^h is the same thing as ^a^h, and la^c — 3a^c is the same thing as 4a*c. Thus the expression may be put in the simpler form 2a^b + ia^c + ^ac^ — 66'. ADDITION. 31. The addition of algebraical expressions is performed by writing the terms in succession each preceded by its proper sign. For suppose we have to add c — c? + e to a — b] this is the same thing as adding c + e — d to a — b (Art. 28). Now if we add c + e to a — b we obtain a—b + c + e; we have however thus added d too much, and must consequently subtract d. Hence we obtain a — b-¥c-\-e — d, which is the same as a — b + c — d + e', thus the result agrees with the rule above given. The result is called the sum. We may write our result thus : a — b-\-{c — d + e) = a — b + c — d+e. 32. When the terms of the expressions which are to be added are all unlike, the sum obtained by the inile does not admit of simplification. But when like terms occur in the ex- pressions, we may simplify as in Art. 30. Hence we have the following rules : When like terms have the same sign their sum is found by taking the sum of the coefficients with that sign and annexing the common letters. Example; add 5a - 36 and 4a - 76 ; the sum is 9a- 106. For the 5a and the 4a together make 9 a, and the 36 and 76 together make 106. Again; add 4a*c-106(ie, Ga^c-9bde and lla'c-36cfe. The sum is 21a^c-22bde. 10 SUBTRACTION. When like terms occur with different signs their sum is found by taking the difference of the sum of the j)Ositive and the sum of the negative coefficients witJi tJie sign of the greater sum and an- nexing the common letters as before. Example ; add 7a — % and 5b — ia. Tlie sum is 3a — 46. Again ; add together 3a^ + 46c - e* + 10, 5a* + 66c + 26^-15 and 4a*- 96c- 106^ + 21. The sum is 12a' + be -de' +16. SUBTRACTION. 33. Suppose we have to take 6 + c from a. Then as each of the numbers 6 and c is to be taken from a the result is denoted by a — b-c. That is a— (b + c) = a — b — c. We enclose the term 6 + c in brackets, because both the num- bers 6 and c are to be taken from a. Similarly a + d—{b + c+e) = a + d — b—c — e. Next suppose we have to take 6 — c from a. If we take b from a we obtain a — b; but we have thus taken too much from a, for we are required to take, not 6 but, 6 diminished by c. Hence we must increase the result by c ; thus a — [h — c) = a — b + c. Similarly, suppose we have to take b — c — d+e from a. This is the same thing as taking b + e — c — d from a. Take away 6 + c from a and the result is a — b — e ; then add c + d, because we were to take away, not b + e but, b + e diminished by c + d ', thus a— (b — c — d + e)==a — b — e + c + d = a-b + c + d — e. ■ 34. From considering these cases we arrive at the following rule for subtraction : Change the sign of every term in the expres- sion to be subtracted, and then add it to the other expression. Here as before, we suppose for shortness, that where there is no sign before a term, + is to be understood. BRACKETS. H For example ; take a — h from "da + h. Za-¥h-{a-h) = ?>a-\-h-a + h = 2a + 2b. Again ; take 5a* + 4a6 — Qxy from 11a' + 3a6 — 4a?y. 11a* + 3a6 - 4ic?/ - (5a* + 4a5 - Qxy) = 11a* + 3a& — ixy - 5a* — 4a& + ^xy = 6a* — a6 + '2xy. BRACKETS. 35. On account of the frequent occurrence of brackets in algebraical investigations, it is advisable to call the attention of the student explicitly to the laws respecting their use. These laws have already been established, and we have only to give them a verbal enunciation. When an exjyression within hrackets is preceded by the sign + the brackets may be removed. Thus a-b + {c-d+e) = a-b + c-d + e, (Art. 31). And consequently any number of terms in an expression may be enclosed by brackets, and the sign + placed before the ivhole. Thus a — b + c—d + e may be written in the following ways : a-b + c-¥ {- d + e), a-d + {c + e-b), a + (- c? + c + e - 5), and so on. When an expression within brackets is preceded by the sign — the brackets may be removed if the sign of every term within tlie brackets be changed, namely + to — and — to +. Thus a-{b-c-d + e) = a-b-\-c + d-e, (Art. 34). And consequently any number of term^s in an expression may he enclosed by brackets and the sign — placed before the whole, provided the sign of every term, within the hrackets be changed. Thus a-b + c + d — e may be written in the following ways : a - 6 + c - (- c? + e), a-ib-c-d + e), a + c — {b-d+ e), and so on. 12 BRACKETS. 36. Expressions may occur with more than one pair of brackets; these may be removed in succession by the preceding rules hegiyining with the inside pair. Thus, for example, a + {b + {G-d)} = a + {b + c-d] = a + h + c-df a + {b- (c-d)} = a+{b-c + d] -a + b-c + df a-{b + {c-d)} = a~{b + c-d] = a-b-c + d, a-{b-{c-d)] = a-{b-c + d} = a-b + c-d. Similarly, a-[b-{c-{d~ e)]] = a - [b -{c -d + e]] — a — [b — c + d — e] =a — b + c—d + e. It will be seen in these examples that, to prevent confusion between various pairs of brackets, we use brackets of different shapes; we might distinguish by using brackets of the same shape but of different sizes. A vinculum is equivalent to a bracket; see Art. 11. Thus, for example. a-[b-{c-{d^e-/)}] = a-[b-{c-{d-e+f)}] = a-[b-{G-d-\-e -/}] =a-[6-c + (i-e +/] = a-b + c-d + e -/. In like manner more than one pair of brackets may be intro- duced. Thus, for example, a-b + c-d + e = a-{b-c + d-e}=a-{b-{c-d + e)}. 37. The beginner is recommended always to remove brackets in the order shewn in the preceding Article ; namely, by removing first the innermost pair, next the innermost pair of all which re- main, and so on. We may however vary the order ; but if we remove a pair of brackets including another bracketed expression within it, we must make no change in the sign of the included ex- pression. In fact such an included expression counts as a single term. Thus, for example, EXAMPLES. II. 13 a + {b + (c - d)} = a + h + (c ~d) = a + h + c - df a + {b-(c-d)]=a + b-{c-d) = a + b-c + df a~{b + {c-d)} = a-b-{c-d) = a-b-c + d, a~ {b-{c-d)} = a-b + {c-d) = a~b + c-d. Also, a-[b-{c-{d- e)}] = a-b + {c- (d-e)} = a-b + c-{d — e) = a-b + c-d + e. And in like manner, a — [b — {c— (d-e —/)}^ = a-b + {c-(d-e -/)} =a-b + c-(d-e -f) — a—h + c — d + e —f= a—b + c — d+e —f. EXAMPLES. 1. Add together 4a - 56 + 3c - M, a + b-4c + 5d, 3a - 76 + 6c + id and a+ ib -G~7d. 2. Add together x^ + 2x' - 3aj + 1, 2x^ - 3x' + 4:X-2, 3x' + 4x'+5 and 4ic' - 3a;' - 5a; + 9. 3. Add together a;* - 3xt/ + y' + x+y-l, 2x* + ixy - Zif -2x-2y + 3, 3x' — 5x1/ — 4j/' + Sx + iy -2 and 6a;* + 1 Oxy + 5y^ -k-x + y, 4. Add together a' - 2ax^ + a^x, x^ + 3aa;' and 2x^ - ax*. 5. Add together 4a6 - x', 3x' - 2ab and 2ax + 2bx. 6. From 5a - 36 + 4c - 7c? take 2a - 26 + 3c - d. 7. From X* + 4a;' - 2a;' + 7a; - 1 take x* + 2a;' - 2x* + 6a; - 1. 8. Subtract a' — ax + x' from. 3a' — 2aa; + a;'. 9. Subtract a - 6 — 2 (c - c?) from 2 {a-b)-G+ d. 10. Subtract {a — b) x — (b - c) y from (a + 6) a; + (6 + c) y. 11. Remove the brackets from a - {6 - (c — d)]. 12. Kemove the brackets fr-om a — {(b - c) — d]. 13. Remove the brackets from a + 26 — 6a — {36 — {Qa — 66)} 14. Remove the brackets from 7a— {3a — [4a — (5a — 2a)]}. 14 EXAMPLES. II. 1 5. Also from 3a-[a + b-{a + b + c-{a + b+c + d)}]. 16. Also from 2x-[3ij-{^x-{5y-(Jx)}]. 17. Also from a-[2b + {3c - 3a - (a + b)} + 2a-{b + 3c)]. 18. Also from a -[5b -{a- (3c - 36) + 2c -{a -2b- c)]]. 19. If a = 2, 6 = 3, a: = 6 and y = 5, find the value of a + 2x- {b + y -[a-x-(b- Sy)]}. 20. Simplify 4.x'-2x' + x+l-{3x'-x'-x-7)-{x^-ix' + 2x + S). III. MULTIPLICATION. 38. We have already stated that the product of the numbers denoted by any letters may be denoted by writing those letters in succession without any sign between them ; thus abed denotes the product of the numbers denoted by a, b, c and d. We suppose the student to know from Arithmetic, that the product of any num- ber of factors is the same in whatever order the factors m^ay be taken ; thus abc = acb = bca, and so on. 39. Suppose we have to form the product of ia, 5b, and 3c ; this product may be written at full thus : 4xax5x5x3xc, or 4 X 5 X 3 X abc, that is 60 abc. And thus we may deduce the following rule for the multiplication of simple terms : multiple/ together the numerical coefficients and put the letters after the product. 40. The notation adopted to represent the powers of a num- ber, (Art. 17), will enable us to prove the following rule: the powers of a number are multiplied by adding the exponents, for a^ y. a^ = a xa xay. a -Ka^a^ = a^^'^ ] and similarly any other case may be established. Thus if m and n are any whole numbers, a"* x a" = a"*"^". MULTIPLICATION. 15 41. We may if we please indicate the product of the same powei-s of different letters by writing the letters witliin brackets, and placing the index over the whole. Thus a'^ x b' = (aby j this is obvious since (aby = ab x ab — a x a x b \ b. Similai'ly, a' X 6^ X c^ = (abcy. Thus a" X 6"= (aby j a" x 6"* x c"= {abcy; and so on for any number of factors. 42. Suppose it required to multiply a + b by c. The pro- duct of a and c is denoted by ac, and the product of b and c is denoted by be ; hence the product of a+b and c is denoted by ac + be. For it follows, as in Arithmetic, from our notion of multiplication, that to multiply any quantity by a number we have only to multiply all the parts of that quantity by the number and add the results. Thus (a + b) c = ac + be. 43. Suppose it requii-ed to multiply a — b by c. Here the product of a and c must be diminished by the product of b and c. Thus (a — b)c = ac — be. 44. Suppose it required to multiply a + b hj c + d. It follows, as in Arithmetic, from our notions of multiplication, that if a quantity is to be multiplied by any number, we may separate the multiplier into parts the sum of which is equal to the multiplier, and take the product of the quantity by each part, and add these partial products to form the complete product. Thus {a + b){c + d) = {a + b)c + {a + b)d; also (a+b)c = ac+ be, and (a + b) d = ad + bd j thus (a +b) (c -hd) = ac + bc + ad + bd. 45. Suppose it required to multiply a — b by c + d. Here the product of a and c + d must be diminished by the product of b and e + d. Thus (a-b) {e + d) = a (e + d) - b (c + d) = ae + ad- (be + bd) = ae + ad — be — bd. 16 MULTIPLICATION. 46. Suppose it required to multiply a + b by c-d. Here the product of a + b and c must be diminished by the product oi a + b and d. Thus {a + b) {c - d) = {a + b) c - {a + b) d = ac + bc- (ad + bd) = ac+bc-ad— bd. 47. Suppose it required to multiply a — b by c — d. Here the product of a — b and c must be diminished by the product of a ~b and d. Thus (a - 5) (c - c?) = (a - 6) c - (a - 6) c^ = ac — bc — (ad — 6xy - y' by 3a; - 2y. 1 2. Multiply a;' — a;*^^ + xy* -y^ hy x + y. 1 3. Multiply a; + 22/ - 3;2; by a; - 2y + 3z. 14. Multiply 2a;' + 3xy + iy" by 3a;' - ixy + y\ 15. Multiply a;' + a;y + 2/' by a;* + a;« + «'. 16. Multiply a' + 5' + c' + 6c + ca - a6 by a 4- 6 - c. 17. Multiply a;'-a;2/ + 2/' + a; + y+l bya; + 2/-l. 1 8. Multiply a;^ + 4a;' + 5a; - 24 by a;' - 4a; + 1 1 . 1 9. Multiply a;' - 4a;' + 1 la; - 24 by a;= + 4a; + 5. 20. Multiply a;' - 2a;' + 3a; - 4 by 4a;' + 3a;' + 2a; + 1. 21. Multiply a;* + 2a;' + a;' - 4a; - 1 1 by a;' - 2a; + 3. 22. Multiply a;=-5a;' + 13aj'-a;'-a;+ 2 by a;'-2a;-2. 23. Multiply a" - 2a' + 3a* - 2a + 1 by a* + 2a' + 3a' + 2a + 1. 24. Multiply together a — x, a + x, and a' + a;'. 25. Multiply together a;— 3, a;— 1, a;+l, and a; + 3. 26. Multiply together a;'- a; + 1, a;' + a; + 1, and a;* — a;' + 1. 27. Multiply x* — aa;' + hx'—cx+d by x* + aa^ - hx' + cx-d. 28. Shew that (x + a)* = x* + ia?a + ^x^a^ + 4a;a' + a*. 29. Shew that a; (a; + 1) (a; + 2) (a; + 3) + 1 = (a;' + 3a; + 1)'. 30. Multiply together a + x, h + x, and c + x. 31. Multiply together a; — a, x—h, x — c, and x — d. 32. Multiply together a+b — c, a + c — h, h + c — a, and a+h + c. 33. Simplify (a + h) {h + c)-{c + d) {d + a)-{a + c) {b - d). 34. Simplify {a + h + c + dy + (a-b-c + dy + {a-b + c~dy + {a + b-c- dy. EXAMPLES. III. 28 35. Prove that {x + y + zf- {x^ + y^ + z^)= 3{i/ + z) {z +x) (x + y), 36. Simplify {a -^h ■¥ cf - a{h + c - a) - h {a + c-h)-c{a+b-c). 37. Simplify {x - yf + {x + yf + 3 {x- yY {x+y) + 3{x+7/y (x-y). 38. Simplify {a*+b'+cy-(a+b+c){a+b-c){a+c-b)(b+c-a). 39. Simplify (a' + b'+cy+{a+b+c){a + b-c){a + c-b){b+c-a), 40. Prove that «« + 2/' + (a; + yf = 2{x' + xy + y')* + Sx'^y' {x + yf {x^ + xy + y^). 41. Prove that 4icy (a;' + y"") = {x' + xy + y'Y - (o;^ -xy + yy. 42. Prove that ixy {x' - y') = (x' + xy- yy - (x" -xy - yy. 43. Multiply together (a* - 3a; + 2)* and a;'' + 6a; + 1. 44. Multiply x^ + a^ - ax {x^ + a?) by a;' + a^ - ax {x + a). 45. Multiply {a + bf by {a - b)\ 46. If5 = a4-6 + c, prove that s {s - 2b) {s -2c)+s{s- 2c) {s - 2a) + 5 (5 - 2a) (s - 2b) = (s - 2a) {s - 2b) {s - 2c) + ^alc. IV. DIVISION". 68. Division, as in Ai'ithmetic, is the inverse of Multipli- cation. In Multiplication we determine the product arising from two given factors ; in Division we have the product and one of the factors given, and our object is to determine the other factor. The factor to be determined is called the quotient. 59. Since the product of the numbers denoted by a and b is denoted by ab, the quotient of ab divided by a is 6 ; thus ab-T-a=b; and also ab-7-b = a. Similarly, we have abc-i-a = bc, abc-i-b = ac, abc-i-c = ab ', and also abc-hbc = a, abc-r-ac = b, abc-^ab = c. These results may also be wiitten thus : ahc J abc abc , — = be. -7- = ac, — = ao : a b c ' abc abc , ahc be ac ab 24 DIVISION. GO. Suppose we require the quotient of QOabc divided by 3c. Since GOahc = 20ah x 3c we have Q0abc-^3c = 20ab. Similarly, 60a6c-i-4a= 156c ; 60abc-r-5ah=12c ; and so on. Thus we may deduce the following rule for dividing one simj^le term by another : If the numerical coeflcient and tlie literal product of the divisor he found in the dividend, the other part of the dividend is the quotient. 61. If the numerical coefficient and the literal product of the divisor be not found in the dividend, we can only indicate the division by the notation we have appropriated for that purpose. Thus if 5 « is to be divided by 2c, the quotient can only be indi- cated by 5a-r-2c, or by -^. In some cases we may however simplify the expression for the quotient by a principle already used in arithmetic. Thus if 15a'b is to be divided by 66c, the quotient is denoted by ■ . Here the dividend = 36 x 5a', and the divisor = 36 x 2c ; thus in the same way as in Arithmetic we may remove the factor 36, which occurs in both dividend and divisor, and denote the quotient by — . 62. One power of any number is divided by another power of the same number by subtracting the index of the latter power from the index of the former. Thus a^-^a* = axaxaxa'x a-r-a x a = axaxa = a^ = a*~*. Similarly any other case may be established. Hence if m and n be any whole numbers, and m greater a*" than n, we have a^'-^a" or — - = a"^", a* —^ 63. Again, suppose we have such an expression as — j . We a' X 1 may wiite it thus -^ -^ ; then, as in Aii;. 61, we may remove DIVISION. 25 a? 1 the common factor a'. Tims we obtain — = — . Similarly any other case may be established. Hence if m and n be any whole numbers, and m less than n, we have a -^0!" or — = . or- a"-"* a' 64. Suppose such an expression as p- to occur ; this may be written thus ( t ) • ^^ ^^ ( T ) means t ^ r, and t ^ r = T2 > as we know from Arithmetic, and as will be shewn in Chapter viil. Similarly any other case may be established. Hence if 72, be any whole number — - = ( - Q5. When the dividend contains more than one term, and the divisor contains only one term, we Tnust divide each term of tJie dividend by the divisor, and then collect the 2'>(^^f'i<^l quotients to ob- tain the complete quotient. Thus, —7 — = a - c ; for (a — c)b = ab- cb. b ab* — ahc + abd ab b-c + d ; for (h-c + d) ab = ab^ - abc + abd. In the first example we see that corresponding to the tenn ab in the dividend and to the divisor b there is the term a in the quotient ; and corresponding to the tenn — cb in the dividend and to the divisor b there is the term — c in the quotient. We have already stated in Art. 49, that the following results are admitted for the present, subject to future explanation : b X ~ c = — be, —b X — c = bc. Similarly, the following results may be admitted : — bcy be , — = 0, - — = — 6. -c — c 26 DIVISION. Thus in Division as in Multiplication, the sign of the quotient is deduced from the signs of the dividend and divisor by the rule, like signs produce + , and unlike signs produce — . 66. When the divisor as well as the dividend contains more than one tenn, we must perform the operation of algebraical division in the same way as the operation called Long Division in Arithmetic. The following rule may be given : Arrange both dividend and divisor according to the powers of some common letter, either both according to ascending powers, or both according to descending powers. Find how often the first term of the divisor is contained in the first term of the dividend, and write down this result for the first term of the quotient ; multiply tJie whole divisor by this term, and subtract the product from the dividend. Bring down as many terms of the dividend as the case may require, and repeat the operation till all the terms are brought down. Example. Divide a'-2a& + 6^ by a — b. The operation may be arranged thus : a-bja'-^ab + b' [a-b a? — oh -ab + b' -ab + b' The reason for the rule is, that the whole dividend may be divided into as many parts as may be convenient, and the com- plete quotient is found by taking the sum of all the partial quo- tients. Thus, in the example, a' - 2ab + 6^ is really divided by the process into two parts, namely, a^ — ab and —ab + b^, and each of these parts is divided by a — b ; thus we obtain the complete quotient a -b. 67. It may happen, as in Arithmetic, that the division can- not be exactly performed. Thus, for example, if we divide a' — 2ab + 26' by a-b, we shall obtain as before a — b in the DIVISION. 27 quotient, and there will then he a remainder h'. This result is expressed in a manner similar to that used in Arithmetic ; we say =za — o A f ; that is, there is a complete quotient a-0 a—b b* a-h and a fractional part v . To the consideration of alge- braical fractions we shall return in Chapter viii. 68. The following examples are important : x-a)x^-a^ (a' + xa-\-a^ x-a)x*-a* (x^ + x^a + xa^ + a' aj* — flj'a x^a - c? x^a - a* x^a — xa* xa' — a 4 3 ^2^« ~^3 xa —a' /v.«3 -4 The student may also easily verify the following statements : a'-a* x*-a* 3 ^ j 3 = x—a\ = x — xa + xa — a : ac + a x + a x^ -^a^ , - a;* + a' 4 - „ „ 3 . = x -xa + a : = x --xa + xa —xa +a\ x+a 'x+a Each of these examples of division furnishes an example of multiplication, as the product of the divisor and quotient must be equal to the dividend. Thus we have the following results which are worthy of notice : x' -a* = {x + a) {x - a), x^ — a^ = {x — a) (x' + xa + a'), a^ + a^ = {x + a) {x' -xa + a'), X* — a* = {x — a) (at' + x'a + xa' + a"), x* — a* = {x + a) (a^ - x'a + xa' - a*), a:* + a' = (ic + a) (a* - x*a + xW - xa^ + a*). 28 DIVISION. G9. It will be useful for the student to notice the following facts : x^ — a" is always divisible hj x — a whether the index n be an odd or even whole number. cc" - a" is di\dsible hj x + a if the index 7i be an even whole number. ic" + a" is divisible hj x + a if the index n be an odd whole number. It will be easy for the student to verify these statements in any particular case, and hereafter we shall give a general proof of them. See Chapter xxxiii. 70. By means of the results which have been obtained in the preceding Articles we may often resolve algebraical expres- sions into factors. Thus whatever A and B denote we have A'-B' = {A+B){A-B\ and the student will frequently have occasion to use this general result with various forms of A and B. For example, suppose A = a^, and B = ¥, so that A^ = a*, and B^ = b^ ; then we have and as a^ -b^ = (a + b) {a- b), we obtain a* - &* = (a' + 5') {a + b){a- h). Again, suppose A = a^, and B = b^, so that A^ = a^, and B^ = 6*; then we have a'-b' = {a' + b'){a'-b'); and, as in Art. 68, a' + b^ = {a + b){a'-ah + b'), (^-b' = {a-b){a' + ab + h'), so that a'-b' = {a + b) (a-b) {a' + ab + ¥) {a'-ab + b'). DIVISION. 29 Again, suppose A = a* and B = b\ so that A* = a^, and ^ = 6' ; then we have = {a' + b*) {a' + b') (a+h) (a-^b). Again, take the general result A^-E'={A-B) {A' + AB + B% and suppose A — a', and B = b^ ; thus we obtain a'-b'= {a' - b"') {a' + a'b' + b*) ; and by comparing this with the result just proved, a'-b' = {a + b) (a - b) (a' + ab + b') {a' -ab + b% we infer that (a^ + ah + b-) {a- - ab + b') = a* + a%^ + b*. This can be easily verified by the method of Art. 56. For (a' + a6 + 6') (a'-ab + b') = (a' + b'-hab) (a'+b'-ab) = {a' + by-a'b' = a*+2a'b' + b*-a'b' = a* + a'b' + b\ We may also in some cases obtain iiseful arithmetical applica- tions of our formulae. For example, (127)'- (123)' = (127 + 123) (127-123) = 250x4 = 1000; thus the value of (127)^- (123)^ is obtained more easily than it would be by squaring 127 and 123, and subtracting the second result from the first. The following additional examples are deserving of notice : (a* + abJ2 + b') {a' -abJ2 + b') = (a' + bj - {ab J 2)' = a*+2a'b' + b*-2a'b' 30 DIVISION. {a? + abJZ + 6*) {a* -abJS + l/)^ {a' + hj - {ah JSy = a'-a'b' + b\ a'+b' = {a' + b''){a*-a'b' + b^) = {a' + b') (a' + abJS + b') {a' -ah ^3 + b'). 71. The following are additional examples of Division. Divide 8a* - 22a'b + iSa'b' - dSab' + 246* by 2a' - dab + ib\ 2a» - 3ab + W) 8a* - 22a'6 + iZa%^ - 38a6' + 246* (4a*-5a6 + 66' 8a*-12a'6 + 16a^6» -10a'6 + 27a'6'-38a6' -10a36 + 15a''6*-20a6^ 12a'6'-18a6' + 246* 12a''6»-18a63 + 246* The quotient is 4a* - 5a6 + 66*. Divide a;^-(a + 64-c)ar* + (a6 + 6c + ac) a; - a6c bj cc - a. x-a\ a:^-(a + b + c)x^ + {ab + bc + ac) X- abc ix^ - (b + c)x + bc a? — ax^ — {b + c)x^-¥ (ab + bc + ac) x — (6 + c) £c' + (a6 + ac) x hex — a6c hex — abc The quotient is ic^ — (6 + c) a; + be. These two examples suggest the following statement : When the dividend and the divisor are homogeneous so also is the quo- tient ; the number of the dimensions of the quotient is equal to the excess of the number which expresses the dimensions of tlie di\ddend over the number which expresses the dimensions of the divisor. See Art. 57. EXA3IPLES. IV. 31 EXAMPLES OF DIVISION. 1. Divide a? + 1 by a; + 1. 2. Divide 27a;' + 81/ by Sx + 1y, 3. Divide a^ - 2ab' + b^hy a-b. 4. Divide a^ - 2a'b - 3ab' hja + b. 5. Divide Qix*^ - y^ by 2x - y, 6. Divide a* + 6* by a + b. 7. Divide a? — x^y + xy^ — 2/^ by £C - y. 8. Divide £c^ — 7ic — 6 by ic - 3. 9. Divide 32a;^ + y^ by 2x +y. 1 0. Divide x^ — x*y + o^y^ - cc^ + xy*" - y' hj a? - 'i^. 11. Divide x* -^x^— ix^ + 5ic - 3 by a' + 2a: - 3. 1 2. Divide a* + 2a'6' + 96* by a^' + 2ab + 36^ 1 3. Divide a' - b' by a^ + 2a'b + 2ab' + b\ 14. Divide 32a* + 5iab^ - Sib* by 2a +'36. 15. Divide a^ - 20;^ + 1 by £c^ - 2a: + 1. 16. Divide x' - 6x* + 9a;'- 4 by a;'- 1. 1 7. Divide a* + a'b - 8a'b' + 1 9ab^ - 1 56* by a' + Sab - 5b'. 18. Divide the product of ar^- 12a; + 16 and ar'-12a;-16 by a;'- 16. 1 9. Divide the product of a;^ — 2a; + 1 and a^ — 3x + 2 by a;3-3a;' + 3a;-l. 20. Divide the product of a;' — a; — 1, 2a;' +3, a;' + a; — 1, and a;- 4 by a;*- 3a;' + 1. 21. Divide the product of a^ + ax + x' and a^ + a^ by a* + a'a;' + x*. 22. Divide the product of a;* - 4af a + 6a;V - 4a;a' + a* and ic' + 2xa + a' by a;* - 2x^a + 2xa^ - a*. 23. Divide a^ + a'b + a'c - abc - b'c - be' by a' - be. 32 EXAMPLES. IV. 24. Divide ^7? + iahx"" - Q>a'J/x - Aa?h^ hjx+ 2ah. 25. Divide the product of x^ — 3x' + 3a: — 1, x^ — 2x+l and x-l by aj* - io^ + Gx^ - ix + 1. 26. Divide 6a* - a^b + 2a'b' + 1 3a¥ + 46* by 2a' - Zah + 46'. 27. Divide x^ + y^ + 3x7/ -1 hj x + y - I. 28. Divide a^ + 6'-c^+ 3a6c by a + 6-c. 29. Divide 2a'b - 5a'b' - Ua'b^ + 5a'b' - 2QaJ'b' + la'b' - 12ab' by a* — ia^b + a'b' — 3ab^. 30. Divide a'b' + 2abc' - aV - bV by ab + aG- be. 31. Divide the product of a + b — c, a—b + c, and b + c— a hy a'--b'-c' + 2bc. 32. Divide {a + b + c) {ab + bc + ca) — abc hj a + b. 33. Divide(a'-6cf + 86Vby a' + 6c. 34. Divide b{3(?-a^) + ax {x' - a') + a^ {x- a) by {a + b){x — a). 35. Divide x]^ -\-2y^z — xy'z-^xyz' — oi^y — 2y:^ + a?z — xz^ by y + Z — X. 36. Divide a' (b + c)- b' (a + c) + c^ {a + b) + abc hj a-b + c. 37. Divide {a-b)x^ + {b^-a')x+ab{a''-b') by {a-b)x+a'-b\ 38. Di^dde ax' — ab' + b"x — a^hj {x+b){a — x). 39. Divide (b — c)a^ + {c- a) b^ + {a — b) c^ by a' - ab-ac + be, 40. Divide {ax + byf + {ay — bxy + c^x^ + c^y^ by a^ + y\ 41. Divide a^6 — 6ar + a^aj — 2c^ by (5C + 6) (a - a:). 42. Resolve a^ — b^-c^ + d^ — 2 {ad — be) into two factors. 43. Divide b {a? + a^) + ax {a? — a") + a^{x + a) by {a+ b) {x + a). 44. Shew that {x^ — xy + ^ff + {xr + xy + y^)^ is divisible by 2x' + 2f, 45. Shew that {x + yf — xJ — y'^ is divisible by {o? + xy + 2/^)^ 46. If A = bc - 2y^, B = ca — q^, C = ab - r^, F =qr - op, Q = rp — bq, and R^pq — cr^ find the value of , , AB-R' QR-AP RP-BQ ^ PQ-CR -, — , , and . c p q T EXAMPLES. IV. 33 47. Resolve a^" - a;'" into five factors. 48. Kesolve 4a'6* - (a' + 6^ - c^f into four factors. 49. Resolve 4 (o^ + hcf - (a' - 6' - c'* + (/')' into four factors. 50. Shew that {ay - hx)^ + (J)z - cyY + {ex - azf + {ax + by + cz)' is divisible by a' + 6' + c' and by ic' + 3/' + «'. V. NEGATIVE QUANTITIES. 72. In Algebra we are sometimes led to a subtraction which cannot be performed because the number which should be subtracted is greater than that from which it is required to be subtracted. For instance, we have the folio ^dng relation : a — {b -h c) = a — h — c ; suppose that a = 7, 6=7 and c = 3 so that 6 + c = 10. Now the relation a — {b + c) = a — h — c tacitly sup- poses 5 + c to be less than a ; if we were to neglect this supposi- tion for a moment we should have 7 — 10 = 7 — 7 — 3; and as 7 — 7 is zero we might finally write 7 — 10 = — 3. 73. In writing such an equation as 7 — 10 = — 3 we may be understood to make the following statement : " it is impossible to take 10 from 7, but if 7 be taken fr-om 10 the remainder is 3." 74. It might at first sight seem to the student unlikely that such an expression as 7 — 10 should occur in practice j or that if it did occur it would only arise either from a mistake which could be instantly corrected, or from an operation being proposed which it was obviously impossible to perform, and which must therefore be abandoned. As he proceeds in the subject the student will find however that such expressions occur frequently ; it might happen that a-b appeared at the commencement of a long investi- gation, and that it was not easy to decide at once whether a were greater or less than b. Now the object of the present Chapter is to shew that in such a case we may proceed on the supposition that a is greater than b, and that if it should finally appear that a is less than b we shall still be able to make use of our investigation. T. A. 3 •34! NEGATIVE QUANTITIES. 75. Let us consider an illustration. Suppose a merchant to gain in one year a cei-tain number of pounds and to lose a certain nimibcr of pounds in the following year, what change has taken place in his capital 1 Let a denote the number of poiuids gained in the fii'st year, and b the number of pounds lost in the second. Then if a is greater than 6 the capital of the merchant has been increased by a — 6 pounds. If however h is greater than a the capital has been diminished hj b — a pounds. In this latter case a — b is the indication of what would be pronounced in Arithmetic to be an impossible subtraction ; but yet in Algebra it is found convenient to retain a — J as indicating the change of the capital, which we may do by means of an appropriate system of iiiterpre- tation. Thus, for example, if a = 400 and b — 500 the merchant's capital has suffered a diminution of 100 pounds ; the algebraist indicates this in symbols, thus 400 -500 = -100, and he may turn his symbols into words by saying that the merchant's capital has been increased by — 100 pounds. This . language is indeed far removed from the language of ordinary life, but if the algebraist understands it and uses it consistently and logically his deductions from it will be soiuid. 76. There are numerous instances like the preceding in which it is convenient for us to be able to represent not only the magnitude but also what may be called the quality or affection of the things about which we may be reasoning. In the preceding case a sum of money may be gained or it may be lost y in a ques- tion of chronology we may have to distinguish a date before a given epoch from a date after that epoch ] in a question of posi- tion we may have to distinguish a distance measured to the north of a cei'tain starting-point from a distance measured to the south of it ; and so on. These pairs of related magnitudes the algebraist distinguishes by means of the signs + and — . Thus if, as in the preceding Article, the things to be distinguished are gain and loss, he may denote by 100 or by + 100 a gain, and then he mil denote by — 1 00 a loss of the same extent. Or he may denote a loss by 1 00 NEGATIVE QUANTITIES. 35 or by + 100, and then he will denote by — 100 a gain of the same extent. There are two points to be noticed ; first, that when no sign is used + is to be understood ; secondly, the sign + may be ascribed to either of the two related magnitudes, and then the sign — will throughout the investigation in hand belong to the other magnitude. 77. In Arithmetic then we are concerned only with the numbers represented by the symbols 1, 2, 3, (fee, and intennediate fractions. In Algebra, besides these, we consider another set of symbols — 1, — 2, — 3, (fee, and inteiTnediate fractions. Symbols preceded by the sign — are called negative quantities, and symbols preceded by the sign + are called positive quantities. Symbols without a sign prefixed are considered to have 4- prefixed. The absolute value of any quantity is the number repre- sented by this quantity taken independently of the sign which precedes the number. 78. In the preceding Chapters we have given rules for the Addition, Subtraction, Multiplication, and Division of algebraical expressions. Those rules were based on arithmetical notions and were shewn to be true so long as the expressions represented such things as Arithmetic considers, that is positive quantities. Thus, when we introduced such an expression as « — Z) we supposed both a and 6 to be positive quantities and a to be greater than h. But as we wish hereafter to include negative quantities among the objects of our reasoning it becomes necessary to recur to the con- sideration of these primary operations. Kow it is found con- venient that the laws of the fundamental operations should be the same whether the symbols denote positive or negative quantities, and we shall therefore secure this convenience by means of suitable definitions. For it must be observed that we have a power over the definitions ; for example, multiplication of positive quantities is defined in Arithmetic, and we should naturally retain that defi- nition ; but midtiplication of negative quantities, or of a positive and a negative quaiUity has not hitherto been defined ; the temis are 3— :j 36 NEGATIVE QUANTITIES. at present destitute of meaning. It is therefore in our power to define them as we please provided we always adhere to our definition. 79. The student will remember that he is not in a position to judge of the convenience which we have intimated will follow from our keeping the fundamental laws of algebraical operation perma- nent, and giving a wider meaning to such conmion words as addition and multiplication in order to insure this permanence. He must at present confine himself to watching the accuracy of the deductions drawn from the definitions. As he proceeds he wHl see that Algebra gains largely in power and utility by the intro- duction of negative quantities and by the extension of the meaning of the fundamental operations. And he will find that although the symbols + and - are used apparently for two purposes, namely, according to the definitions in Arts. 3 and 4, and according to the convention in Art. 76, no contradiction nor confusion will ulti- mately arise from this circumstance. 80. Two quantities are said to be equal and may be con- nected by the sign = when they have the same numerical value and have the same sign. Thus they may have the same absolute value and yet not be equal ; for example, 7 and — 7 are of the same absolute value but they are not to be called equal. 81. In Arithmetic the object of addition is to find a number which alone is equal to the units and fractions contained in certain other numbers. This notion is not applicable to negative quan- tities ; that is, we have as yet no meaning for the phrase " add — 3 to 5," or " add — 3 to — 5." "We shall therefore give a meaning to the word add in such cases, and the meaning we propose is deter- mined by the following rules : To add two quantities of the same sign add the absolute values of the quantities and place the sign of the quantities before the sum. To add two quantities of different signs J subtract the less absolute value from the greater, and place before the remainder the sign of that quantity which has the greater absolute value. NEGATIVE QUANTITIES. 37 ^vlU, ty tke first rule, if we add 3 to 5 we obtain S ; if" we add - 3 to — 5 we obtaia — 8. By the second rule, if we add 3 to — 5 we obtain — 2 ; if we add — 3 to 5 we obtain 2. 82. It will be seen that the rules above given leave to the word add its common aritlimetical meaning so long as the things which are to be added are such as Arithmetic considers, namely, positive quantities, and merely assign a meaning to the word in those cases when as yet it had no meaning. The refader may perhaps object that no verbal definition is given of the word add but merely a rule for adding two quantities. We may reply that the practical use of a definition is to enable us to know that we use a word correctly and consistently when we do use it, and the rules above given will ensure this end in the present case. 83. The rules are not altogether arbitrary : that is, the stu- dent may easily see even at this stage of his progi-ess that they are likely to be advantageous. Thus, to take the numerical example given above, suppose a man to be entitled to receive 3 shillings from one person and 5 shillings from another, then he may be con- sidered to possess 8 shillings. But suppose him to owe 3 shillings to one person and 5 shillings to another ; then he owes altogether 8 shillings ; this may be considered to be an interpretation of the — 8 which arises from adding — 3 to — 5. Next, suppose that he has to receive 3 shillings and to pay 5 shillings ; then he owes altogether 2 shillings ; this may be considered to be an interpreta- tion of the — 2 which arises from adding 3 to — 5. Lastly, suppose that he has to receive 5 shillings and to pay 3 shillings, then he may be considered to possess 2 shillings ; this may be considered to be an interpretation of the 2 which arises from adding -3 to 5. 84. Thus in Algebra addition does not necessarily imply augmentation in an arithmetical sense ; nevertheless the word sum is used to denote the result. Sometimes when there might be an uncertainty on the point, the term algebraical sum is used to distinguish such a result from the arithmetical sum, which would 38 NEGATIVE QUANTITIES. be obtained by the arithmetical addition of the absolute values of the terms considered. 85. Suppose now we have to add the five quantities — 2, +5, ~ 13, — 4 and + 8. The sum of — 2 and +5 is + 3 ; the sum of + 3 and — 13 is — 10 ; the sum of — 10 and — 4 is — 14 ; the sum of — 14 and +8 is —6. Thus — 6 is the sum required. Or we may first calculate the sum of the negative quantities — 2, — 1 3 and — 4, and we thus get —19; then calculate the sum of the positive quantities + 5 and + 8, and we thus get +13. Thus the proposed sum becomes + 13 — 19, that is, — 6 as before. It mtH be easily seen on trial that the same result is obtained whatever be the order in which the terms are taken. That is, for example, -2-13 + 5 + 8-4, 8-13-2-4 + 5, and so on, all give — 6. 86. Next suppose we have to add two or more algebraical expressions ; for example, 2a — 3& + 4c and — a — 26 + c + 2d, We have for the sum 2a-3b + 4:c-a-2h +c + 2d. Then the like terms may be collected ; thus • 2a — a = a, — 36 — 25 = — 56, 4c + c = 5c ; and the S7im becomes a - 56 + 5c + 2d. Thus we may give the following rule for algebraical addition : Write the terms in the same line j)f needed hy their proper signs; collect like terms into one, and arrange tlie terms of the result in any order. 87. In arithmetical subtraction we have to take away one number, which is called the subtrahend, from another which is called the minuend, and the result is called the remainder. The remainder then may be defined as that number which must be added to tiie subtrahend to produce the minuend, and the object of subtraction is to find this remainder. NEGATIVE QUANTITIES. 39 We shall use the same definition in algebraical subtraction, that is, we say tliat in subtraction we have to find the quantity which must be added to the subtrahend to produce the minuend. From this definition we obtain the mle : Change the sign of every term in the subtrahend and add the result so obtained to the minu- end, and the result will be the remainder required. For it is obvious, that if to the expression thus fonned we add the subtrahend, giving to each teiTa its proper sign, all the terms of the subtrahend will disappear and leave the minuend ; which was required. 88. "We have still another point to notice. According to what has been laid down, the sum oi + a and — 6 is denoted by a — b', if we take — b from a, the result is a + b ; and the sum of — «, +6, and — c is — a + 6-c; and so on. But we have as yet supposed that the letters themselves stand for positive numbers ; for example, when we say that the sum of + a and —b is a — b, a may be 6, and b may be 10 ; but suppose that a is — 6, and 6 is — 10, do the iTiles adopted apply here? Since b is —10, — b or —(—10) will natui'ally be taken to mean 10, and +a or + (— 6) will be taken to mean — 6 ; and the sum of 10 and — 6 is -i. 89. Tlius if a be itself a negative quantity, we have assigned a meaning to + a and to — a ; and the meanings are these : let a = — a, so that a is a positive quantity, then + rt or + (— a) = — a, and —a or — (— a) = a. We said in the preceding Article that these meanings followed naturally from what had preceded ; it is however of little consequence whether we consider these meanings to follow thus, or whether we look upon them as new interpreta- tions ; the important point is to use them uniformly and con- sistently when once adopted. Since + (— a) = — a, and — (— a) = a, that is, + o, we may enun- ciate the same rule as formerly, namely, that like signs produce + and unlike signs — . 90. There are four cases to consider in multiplication. Let 40 NEGATIVE QUANTITIES. a and b denote any two numbers, then we have to consider + ax + 6, —ax + h, +ax—b, —ax — h. The fii'st case is that of common Arithmetic and needs no remark. The ordinary definition of multiplication may also be applied to the second case ; for suppose, for example, that 6 = 3, then — . 100. If equal quantities be multiplied by the same or by equal quantities, the products will be equal. Thus too i£ a = b then a" = h" and ^a = ^6. 101. If equal quantities be divided by the same or by equal quantities, the quotients will be equal. 102. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. 103. If a quantity be both multiplied and divided by another, its value will not be altered. 4r4« EXMIPLES. V. 104. It is important to draw the attention of tlie reader t<5 the fact, that these propositions are still ti*ue whether the quanti- ties spoken of are positive or negative, and when the terms addi-» tion, subtraction, multiplication, and division have their extended meanings. For example, if a = 5, and c = d, then ac = hd', this is ob^dous if all the letters denote positive quantities. Suppose however that c is a negative quantity, so that we may represent it by — y ; then d must be a negative quantity, and if we denote it by — S, we have y = S ; therefore ay = 6S j therefore — ay = — 68 3 and thus ac = bd, MlSCELLAKEOtJS EXAMPLES. 1. Shew that x^ ■¥ y^ -ir iz^ + 2xy + ^xz and 4 (oj 4- zY become identical when x and y each = a. 2 2. If a = 1, 6=0) ^ = 7 and y = 8, find the value of 5{a-h) lJ[{a + x)y^}-hJ{{a-\-x)y} + a. 5 1 9 3. Ifa = ;;, 6 = Qj oc = 5 and 2/ = ^ > ^^ the value of (10a + 206) ^{{x -h)y\- 3a U{y' {x - 6)} + 56. 4 10 4 4. If a = - , 6 = 2, ^ = -^ and y = -^, find the value of (a + 6) U{{x - 6) f] - a J{y {x - 6)} + x. 5. Substitute 3/ + 3 for x in x'^ — x^ + 2x^ — 3 and arrange the result. 6. Shew that {(a - 6)'' + (6 - cf + (c - a)''}' = 2 {(a - 6)* + (6 - c)* + (c - a)*}. 7. If 2s = a + 6 + c, shew that (s - of + (s - 6)' + (5 - c)' + s' = a' + 6' + c». 8. If 2s = a + 6 + c, shew that 2(s-a)(s-6) + 2(s-6)(s-c) + 2(s-c)(s-a) = 2s*-a«-6^-c». 9. If 2s = a + 6 + c, shew that 2 (s — a) (s - 6) (s - c) + a (s - 6) (s - c) + 6 (s - c) (s - a) + c (s — a) (s — 6) = a6<;. EXAMPLES. V. 45 10. Shew that (a + 6 + c)' - (6 + cY -{c + ay -{a + bf + a'' + h' + c^ = 6abc, 11. Shew that if a^ + a^ + . . . + a„ = ^ s, theu (s-a,)' + (s-a^)'+...+(5-aJ" = < + <+... + a„». 12. If 2s = a + 6 + c and 2o-' = «' + &'' + c^ shew that (a' - a') (o-' - 6^) + (cr« - b') {a' - c') + {a' - c^ (o-'' - a') =: 4s (« - a) {s -b){s- c). VI. GREATEST COMMON MEASURE. 105. In Arithmetic the greatest common measure of two or more whole numbers is the gi-eatest number which will divide each of them without remainder. The term is also used in Algebra, and its meaning in this subject will be understpod from the following definition of the greatest common measure of two or more alge- braical expressions: Let two or more algebraical expressions be arranged according to descending powers of some common letter ; then the factor of highest dimensions in that letter which divides each of these expressions without remainder is called their greatest common measure. 106. The term greatest common measure is not very appro- priate in Algebra, because the words greater and less are seldom applicable to algebraical expressions in which specific numerical values have not been assigned to the various letters which occur. It would be better to speak of the highest common divisor or of the highest common measure; but in confomiity with established usage we retain the term greatest common measure. The letters G. c. M. will often be used for shortness instead of this teim. When one expression divides two or more expressions without remainders we shall say that it is a common measure of them, or more briefly, that it is a measure of them. 46 GREATEST COMMON MEASURE. 107. The following is the rule for finding the g. c. m. of two algebraical expressions : Let A and B denote the two expressions j let them be arranged according to descending powers of some common letter, and suppose the index of the highest j^ower of that letter in A not less than the index of the highest power of that letter in B. Divide A by B ; then make the remainder a divisor and B the dividend. Again, make the new remainder a divisor and the preceding divisor the dividend. Proceed in this way until there is no remainder ; then the last divdsor is the G. C. M. required. 108. Example : find the G. c. m. of a^-Qx + S and 4a;' -21x'+l5x+ 20. x'-6x + 8J4:x^-2lx'+15x+20 i^ix + 3 4:x^-24:x' + 32x 3a;'- 17a; + 20 3a;' -18a; + 24 x— 4 a;-4^a:'-6a; + 8 (a;-2 a;' — 4a; -2a; + 8 -2a; + 8 Thus a; — 4 is the g. c. m. requii^ed. 109. The truth of the i-ule given in Ai't. 107 depends upon the following principles ; (1) If P divide A, then it will divide 7nA. For since P divides A, we may suppose A = aF, then mA = maF^ thus P divides mA. (2) If P divide A and B, then it will divide mA ± nB. For since P divides A and B, we may suppose A = aF, and B = hF, then mA :i=nB = {ma ± nh) F ; thus P divides mA ± nB, We can now prove the rule given in Art. 107. GREATEST COMMON MEASURE. 47 110. Let A and B denote the two expres- B) A (j? sions ; let them be an-anged according to de- pB scending powers of some common letter, and ~77] w / suppose the index of the highest power of that ri letter in A not less than the index of the highest power of that letter in B. Divide A y ^ v by B j let i) denote the quotient, and C the ^ remainder. Divide B hj C ; let q denote the quotient, and D the remainder. Divide C by D, and suppose that there is no remainder, and let r denote the quotient. Thus we have the following results : A=pB + C] B = qC + D; C = rD. We shall first shew that D i^ a common measure of A and B. D divides C, since G = rD ; hence (Art. 109) D divides qC and also qC + J) ; that is, D divides B. Again, since D divides B and C, it divides 2^^+ C ] that is, D divides A. Hence D divides A and B. We have thus shewn that D \s, a common measure of A and B] we shall next shew that it is their greatest common measure. By Art. 109 every expression which divides A and B divides A —pB^ that is, C ; thus every exj^ression which is a measure of A and ^ is a measui'e of B and C. Similarly every expression which is a measure of B and (7 is a measure of C and D. Thus every expression which is a measure of A and B divides D. But Uo expression higher than D can divide D. Thus D is the G. c. M. tequii-ed. 111. In the same manner as it is shewn in the preceding Ai"ticle that D measures A and B, it may be shewn that every expression which divides D also measures A and B. And it is shewn in the preceding Article that every expression which mea- sures A and B divides D. Thus every measure of A and B divides then- G. c. M. ; and every divisor of their g. c. m. measures A and B. 43 GREATEST COMMON MEASURE. 112. As an example of the process in Art. 110, suppose we hai-e to find tlie G. c. M. ofx' + dx + i and x^ + 4a;' + 5a; + 2. x' + 5x + i) a;* + 4a;' + 5a; + 2 (a; - 1 a;' + 5a;' + 4a; - a;* + a; + 2 - a;' — 5a; - 4 Qx-^-Q 6a; + 6^ a;* + 5a; + 4 (g + g x^ ■{■ X 4a; + 4 4a; + 4 THs example introduces a new point for consideration. The last divisor here is ^x-^-Q; this, according to tlie rule, must be the G. c. M. required. We see from the above process that when a;' + 5a; + 4 is divided by 6a; 4- 6 the quotient is ^ + ^ . If the other given expression, namely x^ + 4a;* + 5a; + 2, be divided by a; X 1 6a; + 6, it will be found that the quotient is -^ + ^ + ^ . It may at first appear to the student that 6a; + 6 cannot be a measure of the two given expressions, since the so-called quotients really contain fractions. But we see that in these quotients the letter of reference x does not appear in the denominator of any fraction although the coefficients of the powers of x are fractions. Such X 2 a;* a; 1 expressions as ^ + k and ^ + o "^ q ' therefore, may be said to be integral expressions so far as relates to x. Thus, in the example, when we say that 6a; + 6 is the G. c. M. of the two given expressions, we merely mean that no measure can be found which contains higher powers of x than 6a; +6. GREATEST COMMON MEASURE. 49 Other measures may be found which differ from this so far as resjDects numerical coefficients only. Thus 3a; + 3 and 2x + 2 will be found to be measures ; these are respectively the half and the third of 6x + 6, and the corresponding quotients when we divide the given expressions by these measures will be respectively twice and three times what they were before. Again, re + 1 is also a measure, and the corresponding quotients are cc + 4 and x^+3x+2 ; we may then conveniently take cc + 1 as the greatest common mea- sure, since the quotients are free from fractional coefficients. 113. In order to Avoid fractional coefficients in the quotients it is usual in performing the operations for finding the g. c. m. to reject certain factors which do not form part of the g. c. m. re- quired. Suppose we have to find the G. c .M. of ^ and B ; and at any stage of the process suppose we have the expressions X and H, one of which is to be a dividend and the other a divisor. Let H = mS, where m has no factor which K has ; then m may be re- jected : that is, instead of continuing the process with K and R we may continue it with A" and S. For by what has been already shewn we know that A and B have just the same common measures as JK and B have. Now any common measure of K and aS' is a common measure of K and B, and is therefore a common measure of A and B. And any common measure of K and i? is a common measure of K and ')nS. But m has no factor which IC has. Therefore any common measure of K and ^ is a common measure of X and S. Hence any common measure of A and ^ is a common mea- sure of K and S. Thus we see that A and B have just the same common mea- sures as K and JS have; and this is what we had to shew. 114. A factor of a certain kind may also be introduced at any stage of the process. Suppose we have to find the G. c. M. of A and B ; and at any stage of the process suppose we have the expressions K and R, one T. A. "^ 50 GREATEST COJSBION MEASURE. of wliich is to be a dividend and the other a divisor. Let L = nK, where n lias no factor which JR has ; then n may be introduced : that is, instead of continuing the process with JST and B we may continue it Avith L and Ji. For by what has been already shewn we know that A and B ha\'e just the same common measures as K and E have. Now any common measure of K and -ff is a common measure of L and B ; so that any common measure of A and ^ is a com- mon measui'e of L and B. And any common measure of L and -ff is a common mea- sure of nK and B. But n has no factor that B has. Therefore any common measure of L and J? is a common measure of K and Bf and is therefore a common measure of A and B, Thus we see that A and B have just the same common mea- sures as L and B have ; and this is what we had to shew. 115. We see then that certain factors may be removed from either a dividend or a divisor, or introduced into either : in practice w^e usually remove factors from divisors, and introduce factors into dividends ; and such factors are generally numerical factors. The reasoning of Arts. 113 and 114 shews that these operations may be performed at any stage of the process, for example at the begin- ning if we please. By means of such modifications of the process for finding the G. c. M., we may avoid the introduction of fi'actional coefficients. The following example will guide the student. Be- quired the g. c. m. of 3a;*-10a;^ + 15a; + 8 and a' - 2rc* - 6a^ + 4a^ + 1 3a; + 6. x'-^x'-^x^-^^x'+Ux + Q) Zx' -lOa^ -j-15a; + 8(3 3a;' - 603* - 18a;3 + 12a;' + 39a; + 18 6a;* + 8ar'- 12a;'- 24a;- 10 Before proceeding to the next division we may strike out the factor 2 from every term of the new divisor, and multiply every term of the new dividend by 3. Then continue the operation thus : GREATEST COMMON MEASURE. 51 3xU 4x^-(jx*-l2x-5J 3a;»- 6a;*- 18a;' + 12a;' + 39a; + 18 (^x 3a;'+ ix*- Qx'-12s(^-5x -lOx'~l2x' + 2ix'+i4:X+lS Hemove the factor 2 from every term of the last expression, and then multiply every t^i-m by 3. Thus we have - 15a;* - 18a;» + 36a;' + 66a; + 27. Proceed with the division 3a;* + 4a;'- 6a;'- 12a; -5^ - 15a;*- 18a;' + 36a;' + 66a; + 27 (- 5 - 15a;*- 20a;' + 30a;' + 60a; + 25 2a;^ + 6x' + 6a; + 2 Kemove the factor 2 and then continue the operation thus ; iB»+3a;' + 3a; + i; 3a;* + 4a;'- 6a;'-12a;-5 (3a;-5 3a;*-t-9a;'+ 9a;' + 3a; -5a;3-15a;'-15a;-5 -5a?-15x'-15x-5 Thus a;* + 3a;' + 3a; + 1 is the g. c. m. required. 116. Suppose the original expressions A and B to contain a common factor F, wliich is obvious on inspection ; let A z=aF, and B = hF, Then F will be a factor of the G. c. M. ; as is she\sTi in Art. 111. We may then find the g. c. m. of a and 5, and multiply it by Fy and the product will be the g. c. m. of A and B, 117. Similarly, if at any stage of the operation we perceive that a certain factor is common to the dividend and divisor, we may strike it out, and continue the operation with the remaining factors. The factor omitted must then be multiplied by the last divisor which is obtained by continuing the operation, and the product will be the required G. c. M. 118. Suppose, for example, that we require the G. c. M. of (a; - 1 )' (a; - 2) (a; - 3) and (a; - 1 )' (a; - 4) (a; - 5). Here the factor (a;— 1)' is common to both the proposed expressions, and is there- fore a factor of the G. c. ll. Moreover in this example (a;— 1)' forms the entire q. c. m. ; for no common measure can be found, except unity, of (a; - 2) (a; - 3) and (a; - 1) (a; - 4) (x - 5) which ai*e the 4—2 52 GREATEST COMMON MEASURE. remaining factors of the proposed expressions. Tlie last statement can be veiified by trial, but when the student is acquainted with the subject of the resolution of algebraical expressions into factors it will be obvious on inspection. Tlie resolution of algebraical expressions into factors is discussed in the Theory of Equations. 119. Next suppose we require the g. c. m. oi three algebraical expressions A, B, C. Find the g. c. m. of two of them, say A and B -y let i> denote this G. c. M. ; then the g. c. m. of D and C is the required G. c. M. of A, B and G. For by Art. Ill every measure of D and C is a measure of A, B and G ; and also every measure of A, B and (7 is a measure of D and G. Thus the G. c. m. of JD and G is the g. cm. of A, B and G. 120. In a similar manner we may find the g. cm. oi Jour algebraical expressions. Or we may find the g. c m. of two of the given expressions and also the g. c m. of the other two ; then the G. c M. of the two expressions thus found will be the g. c m. of the four given expressions. 121. Tlie definition and operations of the preceding Articles of this Chapter relate to polynomial expressions. The meaning of the term greatest common measure in the case of simple expressions will be seen from the following example : Eequired the G. c M. of 432a*6"iC3/, 270a%^x^z and dOa^ho^. "We find by Aiithmetic the G. c M. of the numerical coeffi- cients 432, 270, and 90 ; it is 18. After this number we write every letter which is common to the simple expressions, and we give to each letter respectively the least exponent which it has in the simple expressions. Thus we obtain ISa^bx, which will divide all the given simple expressions, and is called their greatest com- mon measure. EXAAIPLES, VI. 53 EXAMPLES OP THE GREATEST COMMON MEASURE. Find the g. c. m. in the following examples : 1. x'~3x+2 and x^-x-2. 2. x^ + 3x^ + ix + 12 and x^ + 4a;* + 4aj + 3. 3. x^ + x^ + x-3 and x^ + 3x^ + 5x + 3. 4. x^+l and x^ + mx^ + mx + 1. 5. 6x^ - lax' - 20a'x and dx^ + ax- id^, 6. x" - if' and x' — y^. 7. 3a;'- 13a;' + 23a; - 21 and 6a;* + a;' - 44a; + 21. 8. a;*-3a;'+2a;' + a;-l and a;' - a;' - 2a; + 2. 9. a;* - 7a;' + Sa;' + 28a; - 48 and a;' - 8a;' + 19a; - 14. 10. a;*-a;'+2a;' + a; + 3 and a;* + 2a;3 - a; - 2. 11. 4a;* + 9a;^ + 2a;' - 2a; - 4 and 3a;' + 5a;-' - a; + 2. 12. 2a;*-12a;'+19a;'-6a; + 9 and 4a;' -18a;'' + 19a;- 3. 1 3. ^x^ ^-0? -X and 4a;^ - ^^x?— 4a; + 3. 14. 12a;' - 15?/a; + 3?/' and 6a;' - 6ya;' + 2\fx - 2y\ 15. 2a;' -11a;' -9 and 4a;' + 11a;* + 81. 16. 2a* + 3a»a; - 9aV and 6a'a; - 1 7a'a;' + 1 4a V - 3aa;*. 17. 2a;3+ (2a -9) a;' -(9a + 6) a; + 27 and 2a;'- 13a; + 18. 18. aV-a26ary + a&'a;?/'-6y and 2a'6a;'y - a6'a;2/' - 6y. 19. x^-\-ax^- axy - if and x^ + 2d^y - a^x^ + xSf - 2axy'^ - y\ 20. a;' + 3a;*-8a;2-9a;-3 and a;' - 2a;*- ea;^^ 4^,2 4. 13^;+ 6. 21. 6a;' -4a;* -11a;'- 3a;-- 3a;- 1 and 4a;* + 2a;'- 18ar'+3a;-5. 22. X* - aa;' - aV - a'a; - 2a^ and 3a^ - lax" + 3a'a; - 2a'. 23. a;'-9ar'+26a;-24, a;^. iOar+ 31a;-30 and a;3-lla;'+38a;-40. 24. a;^-10a;«+9, a;* + 10a;' + 20a;'- 10a;- 21 and a;* + 4a;'-22ar'-4a; + 21. 54 LEAST COMMON MULTIPLE. YII. LEAST COMMON MULTIPLE. 122. In Arithmetic the least common multiple of two or more whole numbers is the least number which contains each of them exactly. The terai is also used in Algebra, and its meaning in this subject will be understood from the following definition of the least common multiple of two or more algebraical expressions : Let two or more algebraical expressions be arranged according to descend- ing powers of some common letter ; then the expression of lowest dimensions in that letter which is divisible by each of these expressions is their least common multiple. 123. The letters l. c. m. will often be used for shortness instead of the term least common multiple; the term itself is not very appropriate for the reason already given in Art. 106. Any expression which is divisible by another may be said to be a multiple of it. 124. We shall now shew how to find the L. c. M. of two algebraical expressions. Let A and B denote the two expres- sions, and D their greatest common measure. Suppose A = aD and £ = hD. Then from the nature of the greatest common measure, a and h have no common factor, and therefore their least common multiple is ah. Hence the expression of lowest dimensions which is divisible by aD and hD is ohD, AB And ahD = Ah = Ba = -rr , Hence we have the following rule for finding the l. c. m. of two algebraical expressions : find their g. c. m. ; divide either ex- pression by this G. c. m., and multiply the quotient by the other expression. Or thus : divide the product of the expressions by their G. c. M. LEAST COMMON MULTIPLE. 55 125. If M he the least common multiple of A and B, it is obvious that ev^ery multiple of M is a common multiple of A and B. 126. Every common muUijjle of two algebraical expressions is a multiple of their least common inultiiyle. Let A and B denote the two expressions, M their l. c. m. ; and let N denote any other common multiple. Suppose, if possible, that when N is divided by M there is a remainder R ; let q denote the quotient. Thus R=^ N — qM. Now A and B measure M and N, and therefore (Art. 109) they measure R. But ^ is of lower dimensions than M ; thus there is a common multiple of A and B of lower dimensions than their L. C. M. Tliis is absurd ; hence there can be no remainder R j that is, iV is a multiple of M. 127. Next suppose we require tlie l. c. m. of three algebraical expressions A, B, C. Find the l. c. m. of two of them, say A and B ; let M denote this l. c. m. ; then the L.-c. M. of M and C is the required l. c. it. of A, B and C. For every common multiple of M and (7 is a common multiple oi A, B and C (Art. 125). And every common multiple of ^ and ^ is a multiple of M (Art. 126); thus every common multiple of A, B and (7 is a common multiple of 3f and C, Therefore the L. c. M. of M and C is the l. c. m. of A^ B and G, 128. By resolving algebraical expressions into their compo- nent factoi-s, we may sometimes facilitate the process of deter- mining their G.c.M. or l,c.m:. For example, required the l.c.m. of X* — a' and ic* — a*. Since x* — a' = (a; — a) (oj + a) and x^ — a^ = {x--a) (re* + ax + a^, we infer that x — a is the g, c. m. of the two expressions ; conse- quently their L. c. M. is (aj + a) {x* — a'), that is, X* + ax' - a*x - a\ 56 EXAMPLES. VII. 129. The preceding articles of this Chapter relate to 'polyno- mial expressions. The meaning of the temi least common mul- tiple in the case of simple expressions will be seen from the following example : Requii-ed the L.C.M. of 432a^6^a;?/, ^lOa^^x^z and 90a^6cc'. We find by Arithmetic the L. C. M. of the numerical co- efficients 432, 270 and 90; it is 2160. After this number we write every letter which occurs in the simple expressions, and we give to each letter respectively the greatest exponent which it has in the simple expressions. Thus we obtain 2\^0a*h^o(?yz, which is divisible by all the given simple expressions, and is called their least common multiple. 130. The theories of the greatest common measure and of the least common multiple are not necessary for the subsequent Chap- ters of the present work, and any difficulties which the student may find in them may be postponed until he has read the Theory of Equations. The examples however attached to the preceding Chapter and to the present Chapter should be carefully worked, on account of the exercise which they affiDrd in all the funda- mental processes of Algebra. EXAMPLES OF THE LEAST COMMON MULTIPLE. Find the L. c. M. in the following examples : 1. 6a:'-(c-l and 2a;'' + 3a; - 2. 2. 9? -1 and x^ + x-2. 3. ar* - 9a;' + 23a; - 15 and a;' - 8a; + 7. 4. 3a;* -5x + 2 and 4a;^ - 4x'^ - a; + 1. 5. (a; + 1) {x' — 1) and a;^ - 1. 6. a^+ 2x'y - xif - lif and x^ - Ix^j - xif + 2^. 7. 2PJ-1, 4a;' -1 and 4ar^+l. 8. ar — a;, a:^ — 1 and a;^ + 1. 9. a;'-4rt', (a; + 2< and (a; -2a)? EXAMPLES. VII. 57 10. x'-Gx'+Ux-e, x^-9x'+26x-24: iincl a;'- 8a:'' + 19a;- 12. 11. a;'-9a;'+26a;-24,a;'-lCx-+31a;-30 and a;'-lla;'+38a;-40. 12. a;'-10a;'+9, a;'+10a;3+20a;'--10a;-21 and a;V4a;'-22a;'-4a;+21. 13. x'- 4a^, 0;'+ 2ax''+ ia'x + 8a^ and x^- 2ax^-\- ia^x - 8a\ 14. x^— {a + b)x + ah, x'— (b + c)x + bc and x' — {c + a) x + ca. 15. 2x^+{2a~Sb)x'-{2b' + 3ab)x-h3b' and 2a;^-(36-2c)a;-36c. 1 6. 6 {a^ - b') (a - bf, 9 (a' - b') {a - by and 1 2 (a' - b')\ YIII. FEACTIONS. 131. We propose to recall to the student's attention some propositions respecting fractions wliicli lie has already found in Arithmetic, and then to shew that these propositions hold uni- versally in Algebra. In the following Articles the letters repre- sent whole numbers, unless it is stated otherwise. 132. By the expression j- we indicate that a unit has been divided into b equal parts, and that a of such parts are taken. Here - is called 2^ fraction ; a is the numerator and b the denominator^ b so that the denominator indicates into how many equal parts the unit is to be divided, and the numerator indicates how many of those parts are to be taken. Every integer may be considered as a fraction with unity for p its denominator ; that is, ^^ = r . 133. Rule for multiplying a fraction l)y an integer. Either multiply the numerator by that integer^ or divide the denominator by that integer. 58 FRACTIONS. Let J- denote any fraction, and c any integer; then will =- X c = -r- . For in each of the fractions ^ and -j- the unit is divided into b equal parts, and c times &s many parts are taken ac . a . ac . , a in ^ as in 7 : hence -7- is c times =- . 00 This demonstrates the first form of the Kule. Again ; let 7- denote any fraction, and c any integer ; then will rj- X c — T • For in each of the fractions z- and 7- the same be be b number of parts is taken, but each part in - is c times as large as each part in — , because in y- the unit is divided into c times as be be many parts as in t ; hence r is c times 7- . This demonstrates the second form of the Rule. 1S4, Rule for dividing a fraction by an integer. Either mul- tiply the denominator by that integer^ or divide the numerator by that integer. -a i-iet Y denote any fraction, and c any integer ; then will Cb a „ fi . ., a Vc -T-c= V-. For ^ is c times ^, by Art. 133: and therefore b be -^ ' a . 1 . , ^a 7- is - tn of Y • be c This demonstrates the first form of the Rule. ac Again ; let — denote any fraction, and c any integer ; then ac a —, ac . .. a will -7— ^c = Y . For — is c times -r , by Art. 133; and there- 066 . a . \ ac fore Y 13 - th of Y- • b c b This demonstrates the second form of the Rule. FRACTIONS. 59 135. If any quantity be both multiplied and divided by the same number its value is not altered. Hence if the numerator and denominator of a fraction be multiplied by the same number the value of the fraction is not altered. For the fraction is multiplied by any number by multiplying its numerator by that number, and is divided by the same number by multiplying its denominator by that number. (Arts. 133 and 134.) Thus =- = -r- . And so also if the numerator and denominator of a 00 fraction be divided by the same number the value of the fraction is not altered. 136. Hence, an algebraical fraction may be reduced to an- other of equal value by dividing both numerator and denominator by any common measure ; when both numerator and denominator are divided by their g. c. m. the fraction is said to be reduced to its 6a;^ - 7a; - 20 lowest terms. For example, consider the fraction -r-^ — —- . ^ ix - 21 X + 5 Here the G. c. m. of the numerator and denominator will be found to be 2x — 5; hence, dividing both numerator and denominator by this we obtain ex'-7x-20 _ Sx + i 4:x^ - 21 X + 5 "" 2x^ + ox-\' 137. Since j-= — t (Art. 94) it is obvious that we may change the signs of the numerator and denominator of a fraction without altering the value of the fraction, 138. To reduce fractions to a common denominator: multi- ply the numerator of each fraction hy all the denominators except its own for the numerator corresponding to that fraction, and mid- dply all the denominators together for the common denominator. Thus, suppose t , ;, , and -r, to be the proposed fractions ; then, di J - o - ct adf c cbf , e ebd , , adf cbf . by Art. 13a, -= ^, -= _, and^= ^y, tlius j^^, ~, and 60 FRACTIONS. ebd are fractions of the same value respectively as the proposed hdf fractions, and having the common denominator hdf. 139. If the denominators liave any factors in common, we may proceed thus : find the l. c. m. of the denominators and use this as the common denominator ; then for the "new numerator cor- responding to each of the 'proposed fractions^ multijyly the numerator of tlmt fraction hy the quotient xohich is obtained by dividing tlie L. c. M, by the denominator of that fraction. Thus suppose, for example, that the proposed fractions are -^ — and — . Here the l.c.m. of the denominators is mxyz; mx my mz , a ayz b bxz , c cxy and — = — ^- , — = , and — = - mx mxyz^ my mxyz' mz mxyz 140. To add or subtract fractions, reduce them to a common denominator, then add or subtract the numerators and retain the common denominator. For example, - + ^ = —. — : this follows immediately from the '■boo meaning of a fraction. a c ad cb ad + ch ^ b'^d^ bd^Ur bd '' 1 1 a — b a^-b 2a ^ b a b ac b ac + b a-\- - = - + -=— + - = : c I c c c c a + b a-b _ 2{a'-b') (a + b)' (a - bf a-b'^ a + b" a'-b' '^ a'-b' '^ a' -b' _ 2a'-2b' + a' + 2ab + b' + a'-2ab + b' ^ ^dr FRACTIONS. 61 a c a — c b b b ' a c ad be ad — bc^ b~d^bd~bd^ bd ' a c + d a{c — d) b {c + d) _ac — ad — {be + bd) b ~ c-d^b{c-d) ~b{c-d)^ b(c-d) ac — ad—bc — bd " b{c-d) ' a + h a-h _ {a + by {a-bf _ {a + bY-{a-bf ^:r6~^T6~ a^-b^ a--b''~ a'-b^ _ a^ + 2ab + 5^ - (a" - 2ab + b") a^- b'' a^ + 2ab +b''-a^ + 2ab - b^ 4ab a--b^ d'-V 141. The iiile for the multiplication of two fractions is, mul- tiply the numerators for a new numerator^ and the denominators for a new denominator. ft ft The following is usually given for a proof. Let 7 and - bo two fractions which are to be multiplied together ; put t = ^> and -,— y\ therefore a = bx, and c = dy, therefore ac = bdxy ; ac divide by bd ; thus j— = xy. This process is satisfactory when x and y are really integers, though under a fractional foi-m, because then the word multiplica- tion has its common meaning. It is also satisfactory when 07ie of the two, x and y, is an integer, because we can speak of multiplying a fraction by an integer, as in Art. 133. But when both x and y are fractions we cannot speak of multiplying them together with- out defining what we mean by the term multiplication, for, ac- 62 FRACTIONS. cording to tlie ordinary meaning of this tenn, the multiplier must be a whole number. In fact the so-called rule for the multiplication of fractions is really a definition of what we find it convenient to understand by the multiplication of fractions. And this definition is so chosen that when one of the fractions we wish to multiply together is an integer in a fractional form, or when both are such, the result of the definition coincides w4th the consequences drawn from the or- dinary use of the word multiplication. 142. The following verbal definitions may shew more clearly the connection between the meaning of the word multiplication when applied to integers, and its meaning when appKed to frac- tions. When we multiply one integer a by another 6, we may describe the operation thus : what we did with unity to obtain b we must now do with a to obtain b times a. To obtain b from unity the unit is repeated b times ; therefore to obtain b times a the number a is repeated b times. Now let it be required to ft c multiply the fraction y- by -; adopting the same definition as Q above, we may say that, what we did with unity to obtain -r u)e must now do with - to obtain ^ times - . To obtain -., from unity b d b d "^ the unit is divided into d equal parts, and c of such parts are taken ; therefore, to obtain -. times r* the fraction 7- is divided into d d h b equal parts, and c such parts are taken. Now, by Art 134, if ^ be divided into d equal parts, each of them is ^ , and if e such parts hd' CLG be taken the result is t-, • bd The definition then of multiplication may be given thus : to obtain the product of the multiplier and multij^licand we treat the multiplicand in the same way as unity was treated to obtain the multiplier. FRACTIONS. 63 143. To multiply three or more fractions together, multiply all the numerators for the new numerator y and all the denominators for the new denominator. 144. Suppose we have to divide t by - . Here, by the , nature of division, we have to find a quantity such that if it be c a multiplied by -, the product shall be ^ • This is the meaning of division applied to integers, and we shall give the same meaning to division applied to fractions, an operation which hitherto has not been defined. Let Y-- — , = » ; then Y=a;x -.= -^i therefore -7- = xc. and d ' da b — = cc. Thus we obtain the rule for dividing one fraction by another ; invert the divisor j and proceed as in multiplication, 145. Hitherto we have supposed, in the present Chapter, that the letters represented whole numbers ; and have thus only recalled rules and proofs which are familiar to the student in Arithmetic. But in virtue of our extended definitions it may be proved that all the rules and formulse given are true when the letters denote any numbers whole or fractional. Take, for example, the foi-mula T =f~ t and suppose we wish to shew that this is true when m . p 1 r a = — , = - . and c= - . n q s n q n p np also ac = — . and hc= — \ ns qs ac _ mr pr mr qs 7nrqs Tnq he ns ' qs ns pr nspr np ' Thus the fonnida is shewn to be true. G4 FRACTIONS. Moreover these formulae and rules hold when the letters de- note negative quantities by virtue of the remarks already made in Chapter v. 146. By means of the foregoing rules and formulae we can simplify algebraical fractions, in which the numerator and de- nominator are themselves fractional expressions. For example, a h a{a-\-h) + lf b a + b b (a + b) a^ + ab + b^ a(a-b) _a(a^- P) a b " a'-b{a~b) " b{a + b) "" a'-alTb' ~ b {a' + U) ' a — b a a{a — b) 147. The beginner requires to be warned that in reducing fractional expressions he should keep the simplest forms which are admissible, in order to avoid unnecessary labour. For exam- ple, suppose we haA^e to reduce the following expression to a single fraction, a b c + 7^ r-„ ^. ,-. + (a — b) {a — c){x — a) [b — a){b — c) {x — b) (c — a) [c - b) [x - c) ' "We might take the product of all the denominators for a com- mon denominator and transform the three fractions accordingly ; but a little consideration will shew that there is a much simpler common denominator which we may put in the following sym- metrical form, (a — b)(b — c) (c — a){x — a) (x — b)[x — c). We may write the proposed expression thus, a b c (a - 6) (c — a) (x -a) [a- b) {b — c) (x — b) {c — a)(b — c) {x — c)' then by reducing to the common denominator we find aib — c)[x-b){x-c) -\-b{c — a) (x-a) (x-c) + c(a-b){x — a) (x-b) {a -b){b- c) (c -a){x- a) (x - b) (x-c) EXAMPLES. VIII. 65 On working out the numerator we find that it reduces to X {a (c^ -¥)-¥h{a'- c') + c{b'- a% and we shall also find that - {a {c' -b') + b(a'- c') + c{b'- a')} = {a-b){b- c) {c - a\ Thus the proposed expression becomes X (x — a) (x — b) {x — c) * As another example it may be shewn that a' b' c' + 71 ^-7-. T^ r. + (a -b)(a- c) {x -a) (6 - a) (6 — c){x- b) (c -a){c- b) {x - c) (x — a){x— b) [x — c)' EXAMPLES OF FRACTIONS. Simplify the following fractions : x^ + 2x-3 ^, x'^-^x-i 1. 2' x' + Qx-T 3. a;'-6a;^+lla:-6 4 x'-Zx^^l • 5. x*+l0x^+2>ox'+50x-¥U a a;' + 9^'+ 26a: + 24 7. (Sx^-Dx'+i 8. 2x'-x'-x + 2' 9. Zx'+Ux+O x^ + bx'' + G 10. 11. x' + 2a;' + 9 12 ic"-4a;' + 4a;'-9' 13. X* -x^ -x+ 1 14. x'-2x^-x'-2x+ 1' lo. bx + 2 IG 26 -\-(ff-^)x- 2bx' ' T. A. x^ -ix-6' a' + 3a'b+ 3ab'+b^ '^If+Yab + b' Zx^-Ux'^2?>x-(S 2a;'-lla;'+17a;-6' 2x^ + 9a;- + 7.^-3 3a;^ + Dx"^ - 1 5a; + 4 * a;3_(3^8_37a;4-210 a;'+4a;'-47.c--210* x^ + 2a;' + 2a; a* + 4a; a'-a'b-ab* + b' a" - a'b - a%' + ab' * {x + i/Y — of - ?/ {x + yy-x'-i/' 5 66 EXA3IPLES. VIII. Perfomi the additions and subtractions indicated in the fol- lowing examples from 17 to 37 : a h a+b a-o 18. .-^-^ ' 19. 20. 21. 22. 2a -2b 'lb- 2c6 2 3 1x- X 2x-l 4:x'^ - 1 ' \?n nj ^ ^ \ m n J x-l x + 2 {x+'2f 5 1 24 2(x+l) lO(aj-l) 5 (2a; + 3)* 6-« a -2b 3x{a-b) x-b £c + 6 tc-o ^, 3+2x' 2-3::c 16a; -a;" 24. — s +^ — r-- 2-x 2 + x X - -i 25. 3 7 4 -20a; 1 - 2a; ~ 1 + 2a; ~ '4a;' - 1 1 h a 26. r + 27. ^,. 1 28 (^'' + ^T « ^ 3, a6 (c6 - bf b a a 3a 2ax 29. + — -2 a- X a + x a - x" „^ 3a-45 2a-b-c ISa-Ac a-Ab 3^- —^ 3 ■^'T2~" 21 • (6-c)(c-a) (c-a)(a-6) {a-b){b-c)' EXAMPLES. VIII. 67 -^ a^-hc h^-ca c^-ah 32. 7 TV-: : + 71 r^T ; + (a + 6) (« + c) (6 + c) (6 + a) [c + a) (c + b) -^ a^-bc b^ + ca c^ + ab OO. , r^, r + -z r-r, c + {a -b){a- c) {b + c)(b- a) {c -a){c-\- b)' oi be ca ab (c -a) {a- b) {a -b){b- c) (b - c){c- a)' 35. 1 I L , 1 a{a-b) {a — c) b{b-c) ib — a) c{c — a) (c-b)' - - a — b b — c c — a (a — b) {b — c) (c — a) a + b b + c c +a (a + b) {b + c) (c+ a)' _ 2 2 2 (a-by+(b-cy + (c-ay 37. r + 1 + + a — b b — c c — a (a — b) (b — c){c— a) 38. Multiply ^^^ by ^ b + a x{a — b)' 39. Multiply ^^^ by ^'~^\ . — iA T»T ix- 1 . ^1 2^^ a^-x^ bc + bx . c-x 40. Multiply toe^etiier -. , -^ , , —5 and — — . 46?/ c — X' a' + ax a-x 41. Prove that (Uf)\(u'^)\(u')'=i.(U^^(^-.^-)(u'-). \c b) \a cj \b aj \c bj \g aj \b aj \ rjf^ \ y^ nt 42. Multiply together , ^ and 1 + ^^ . 1 + y x + x l-x An Tir IX- 1 x{a-x) . a(a + x) 43. Multiply ^, \_ ', by ^ ^ a^ + 2ax + £c* *^ a^ - 2ax + x' AA a- vf o^^-b* a-b 44. bimplify — — ^—: — =-» x -r- — .- . ^ -^ a^-2ab-{- b' a^ + ab ^g cj: 1,-A. /^ + 2/ x-y ^y' \^ + y ). Simplify ( 22/ n Xx-y x + y X —y / 2y i/. ci. -../. a^-^^ a + b /a" -ab + b\' 46. Simplify -5 — r^ . . -^ r — ^2 . ^ ■^ a^ + b^ a-b \« + ab -\- 67 5—2 68 EXAMPLES. VIII. 47. Multiply - - - + 1 by -, + - + 1. 48. Multiply x'-x + l by - + - + 1. x^ -k-x{a + h) + ah x^ — a^ ^ '' X' -x{a + b) + ab x^ -h^ ' 2 2 Cvju ~~ JO . JO 50. Divide -. r^ by — , „ . 51. Divide -^3 TTrby-,^ — ^. 6 (a + 6) *^ a~ ~ 52. Divide -^'^^ by ^ X + y y -v X 53. Divide ~ + ^^^ T, T, by , -^^ . a; + 2/ ic-j/ X' — y x —y 54. S;n.plify(J.l)^(j_Ul). 55. Simplify (-^^ + ^ U f « _ * ) . rs- TP fx + 2y x\ fx+'2y x \ 56. Simplify ( + - ) -4- ( ^ ) . 57. Divide x^ — , by a; + - . X X 58. Divide a;^ + - 4- 2 by oj + - . 59. Divide a;' + 1 + -^ by - - 1 + a?. a; a; 60. Divide d'-h'-c' + 2hc by ^"^ "^ . •^ a + 6 + c -p.. ., a^ + 3a^a; + 3«a; V a;^ , (a + xf 61. Divide 5 3 by £c^-2/^ ' x^ + xy^y 62. Divide a'-b'- c' - 2bc by EXAJVIPLES. VIII. 09 a + b + c a + b -0 63. Divide x--'3ax-2a^+ -^— - by 3x-6a- X + 3a x-^3a n ^ -p. . . , cc 6a X 3a 0-i. Divide -— -4 + -^ by . 2a' x^ ^ la X a+b a—b + 65) Simplify 114— ^-Li. ^ ^ -^ a + b a-b c — d c + d a+x a—x /66.) Simplify i^illlf a+x a—x a—x a+x a-1 b-l c-l + — 1 — + nT CI- i-i* 3abG a b - d7. Simplify bc + ca~ ab 1 1 1 a b c 68. Simplify ^— + -^^j "^ (^^ - ,~.^) • 69. Simplify ^-^^- -,^3) ^(^H^ ^.3^). 70. Simplify ( , 2 -— — =^o)-^ •" \ic — 2/ £c" + 2/ V \^ — y x + y 71. Simplify ^^^ + a + h a — b\ fa' + b^ a^ -b -b a + b) ' \a:'- b' a' + by ' 2 . 1 m + 71 — 771 72. Simplify —4^—= X %^, ^ -^ 1 _ 1 m^ + n^ 71 in 70 EXAMPLES. VIII. X + a X — a ^ X X x-ax+a i 3. Simiihfy — - x—a x+a x+a x—a + x—a x+a 1 1 + 74. Smiphfy^ -|l+__^^— j. 75. Simplify a b + G l__ 1 X + - 76. Simplify X + 1 1 + o O —X a 6+ IX. EQUATIONS OF THE FIRST DEOREE. 148. Any collection of algebraical symbols is called an eoCr- pression. When two expressions are connected by the sign of equality the whole is called an equation. The expressions thus connected are called sides of the equation, or members of the equa- tion. The expression to the left of the sign of equality is called the^rs^ side, and the expression to the right the second side. 149. An identical eqiiation is one in which the two sides are equal whatever numbers the letters stand for ; for example, {x+h){x-h)=x'-b' is an identical equation. An identical equation is called briefly an identity. Up to the present point the student has been almost entirely occupied with identities. Thus the results given in Arts. 55 and 68 are identically true ; and so also are those which will be ob- tained by solving the examples to Chapters iii and iv. EQUATIONS OF THE FIRST DEGREE. 71 150. An equation of condition is one which is not true for every vakie of the letters, but only for a certain number of values ; for example, cannot be tiTie unless x=Q>. An equation of condition is called briefly an equation. 151. A letter to which a particular value or values must be given in order that the statement contained in an equation may be true is called an unhnoicn quantity. Such particular value of the unknown quantity is said to satisfy the equation, and is called a root of the equation. To solve an equation is to find the parti- cular value or values. 152. An equation involving one unknown quantity is said to be of as many dimensions as is denoted by the index of the liighest power of the unknown quantity. Thus, if x denote the unknown quantity, the equation is said +0 be of one dimension when X occurs only in the first power ; such an equation is also called a simj^Ie equation, or an equation of the first degree. If x^ occurs, and no power of x higher than x' occurs, the equation is said to be of two dimensions ; such an equation is also called a quad- ratic equation, or an equation of the second degree. If x^ occurs, and no power of x higher than x^ occurs, the equation is said to be of three dimensions ; such an equation is also called a cubic equor tion, or an equation of the third degree. And so on. It must be observed that these definitions suppose both mem- bers of the equation to be integral expressions so far as relates to X, and not to contain x under the radical sign. 153. We shall now indicate some operations which may be perfoiTQed on an equation without destroying the equality which it expresses. It "will be seen afterwards that these operations are useful when we have to solve equations. 72 EQUATIONS OF THE FIRST DEGREE. 154. If evei'y term on each side of an equation he muUijyIied or divided by the same quantity the results are equal. This follows from Ai-ts. 100, 101. 155. The principal use of the preceding Article is to clear an equation of fractions ; this is effected by multiplying every term by the product of all the denominators of the fractions, or, if we please, by the least common multiple of those denominators. Suppose, for example, JMultiply every term by 2 x 3 x 4 ; thus, 3x4xrc + 2x4xa; + 2x3xa:;=13x2x3x4j that is, 12a; + Sx + 6x = 312. Divide every term by 2 ; thus, 6x + 4:X + 3x= 156. Instead of multiplying every term by 2 x 3 x 4 Ave may multi- ply by 12, which is the L. c. m. of 2, 3 and 4. Thus we obtain at once 6x+ Ax + 3x= 156. 156. Any quantity may he transposed fvm one side of an equation to the other side by changing its sign. Thus suppose x — a = h-y. Add a to each side (Art. 98) ; then x — a + a = h - y + a, that is, x — h + a — y. Now subtract h from each side ; thus, x-h = h-{-a—y-h = a — y. Here we see that — a has been removed from one side of the equation, and appears as + a on the other side ; and + h has been removed from one side and appears as — 6 on the other side. EQUATIONS OF THE FIRST DEGREE. 73 157. Jf the sign of every term in an equation he changed the equality still holds. This follows from the preceding Article by transposing every term. Thus suppose x — a = b — 'i/. By transposition, y—h=a — x, that is, a — x — y-h-j this result is what we shall obtain if we change the sign of every term in the original equation. 158. We can now give a rule for the solution of any simple equation with one unknown quantity. Let the equatioyi first he cleared of fractions ; tlien transpose all the terms ivhich involve the unknown quantity to one side of the equation, and the known quantities to the other ; divide hoth sides hy the coefficient or the sum of the coefficients of the unknown quantity, and the value required is ohtained. Tlie truth of the rule will be obvious from the principles of the preceding Articles, and we shall now apply it to some examples ; in these examples the unknown quantity will be de- noted by X, and when other letters occur, they are supposed to represent known quantities. 159. Solve 3a;- 4 =24 -a;. By transposition, 3a; + a; = 24 + 4; thus, 4a; =28; 28 by division, x— — =7. "We may verify the result by putting 7 for x in the original equation. The first side becomes 3x7-4, that is, 21 - 4, that is, 17 j the second side becomes 24 - 7, that is, 17. 74 EQUATIONS OF THE FIRST DEGREE. , ^^ r. , 5x Ax ^. 5 X 160. Solve Y~y~ 8'*"32' Multiply by 96, which is the l. c. m. of the denominatoi'S ; thus, 5x48xa;-4x32xa:-13x96 = 5xl24-3a;; that is, 240x - 128a: - 1218 = 60 + 3a; ; by transposition, 240a; - 128a; - 3a; = 1248 + 60 ; thus, 109a; =1308; by division, x = -r-y^ = IJ. We may verify the residt by putting 12 for x in the original equation ; it will be found that each side of the equation then becomes 1. 161. Sometimes it is convenient to clear of fractions par- tially, and then to effect some reductions before getting rid of the remaining fractional coefficients. For example, solve x + 7 2a;-16 2a;+5_ 3a; + 7 "Ti 3~''~~i~~^^'^~ir' Here we may conveniently multiply by 12; thus, I?J^±1) _ 4 (2a; - 1 6) 4- 3 (2a; + 5) = 1 6 X 4 + 3a; + 7 ; ^^(nr A-7) that is, Yi - 8a; + 64 + Oa; + 15 = 64 + 3a; + 7. By transposition and reduction. Multiply by 1 1 ; thus, 12a; +84 + 88 = 55a;; by transposition, 1 72 = 43a; ; 172 by division, x = -j^ = ^* We may verify this result as before. EQUATIONS OF THE FIRST DEGREE. 75 The student should notice one point in this example very 2x— 16 1 carefully. The fraction ■ ^ — is equivalent to - (2a;- 16) . This fraction is preceded by the sign — ; and when we multiply by 12 and remove the brackets we obtain —8x + 64. Thus when we clear of fractions we must regulate the signs of the terms which stood in any numerator in the same way as if they had been be- tween brackets. 162. Solve 2x+l 5x-S' Multiply by (2a; + 1) {ox - 8) ; thus, 5{5x-S) = 2{2x+l); that is, 25x - 40 = 4a; + 2 ; by transposition, 21a; = 42 ; ,. 1- . . 42 „ by division, a? = -— = 2. We may verify this result as before. icfj "a 1 2a; -3 4a; - 5 163. Solve . j = - „, oa; — 4 6a;— 7 Multiply by (3a;- 4) (6a; - 7) ; thus, (2a; - 3) (6a; - 7) = (4a;- 5) (3a;- 4) ; that is, 12a;'- 32a; + 21 = 12a;' - 31a; + 20. Take away 12x^ from both sides ; thus, 21 -32a; = 20 -31a;; by transposition, 21 — 20 = 3 2a; -31a;; thus, a; = 1. We may verify this result as before. 164. Solve l-^ = ~-l- Multiply by 6 ; thus, 3a;- 48 = 20a; -14; 76 EQUATIONS OF THE FIRST DEGREE. by transposition, 17a; = — 34 ; by division, x = — — = — 2. We may verify this result ; each side of the equation will be found to become — 9. 165. Solve ax + h = cx + d. By transposition, ax — cx = d — b; that is, (a- c)x = d-b ; , ,. . . d — b by division, x . a — c Verijicaiion ; put this value for x in the original equation ; then the first side becomes — + b, that is, — ^^ -I — , a—c a— c a—c that is, . And the second side becomes — ^^ + d. that a—c a—c . c(d-b) d(a — c) ,. . da-cb IS, -^ ^ + — ^ ^ , that IS, . a—c a—c a—c 166. An equation of the first degree cannot have more than one root. For any equation of the first degree will take the form ax = b if the unknown quantity is brought to one side of the equation, and the known quantities to the other, and to make this true X must be equal to - , and to nothing else. a The result is sometimes obtained thus. Suppose, if possible, that this equation has two different roots a and /? ; then by supposition, aa = b, ajS = b ; therefore, by subtraction, a(a-l3) = 0; but this is impossible, since by supposition a — j8 is not zero, and a is not zero. Thus an equation of the first degree cannot have more than one root. EXAMPLES. IX. 77 EXAMPLES OF EQUATIONS OF THE FIRST DEGREE. 2x+l _7x+5 X X X ^ aj + 1 3aj— 4 1 6cc + 7 ox — 11 x—\ llaj — 1 • 4 10"" 12 • ^ X X X \ - ic + 1 a; + 2 -. a + S ^- 2^3-4=2- ^- -T-^^-=^^--^- 7. aj + — o — ^ o ' S- 19^ + 2(7a;-2) = 4ic+ ^. - a;-3 a; — 4 x — fi cc+l ^' "~4~'^~T~""2~'*'~8~- 10. ^^_?^.3.-14. x-Z 2x-5 41 3x-8 5x + 6 ' ~4 6~~60'^~5 15 • 12. ^-^.50.-10. 13. l(8-a.) + a.-li = ^-f. ,, a+3 a;-2 3a;-5 1 ^^' 2 3 12 4 3a;-l 13-aj 7x ll(a; + 3) 10, 16. 5 2 3 6 5x-3 9-x 5x 19, ,, -7 ^ = T^-6-("-^)- ,^ 5x-l 9x-5 dx-7 17. -^_+-^ = -^ 78 EXAMPLES. IX. 18. !^_^ + 10-^ = 0. 7 5 19. --^=^-. 20. 2^-^=__-. 21. '-^-U-'-^V-^- 22. I±^-fi-^^).r. 4 \ y / 4 V 9 2 4 3* „, 7a; -8 15ic + 8 ^ 31 -a: ^^- -rr^-i3— ^" — 2-- 25. ^^-^^ = 4.-14|. 4 8 * ^^ 2a; -1 3a; -2 5a; - 4 7a; + 6 2o. G 12 2a; — 9 x a; - 3 2^"*'l8 27. -,,^ + — -— — =84-a;. a;— 1 4a; -| 7a;— 6 ^ a; — 2 3a; — 9 28. 1 - = 2 H + . 3 5 8 2 10 3a; -7 25 - 4a; 5a;- 14 ^^' ~5~"*"~~9~-""^^- 2a; + 5 40 - a; _ 10a; -427 ^'-^ ■T3~'"""8~- 19 • «, flj-l aj-2 a;+ 3 a;+ 4 _ „p. a;-l a;-2_a; — 5 a; — 6 a;-2 a;-3 x—^ x-7' 36. (a;- 5) (a;- 2) -(a;- 5) (2a;- 5) + (a; + 7) (a;- 2) = 0. 37. 3 - a; - 2 (a; - 1) (a; + 2) = (a; - 3) (5 - 2a;). EXAMPLES. IX. 79 38. x-3-(3-x){x+l) = {x-3){l+x) + 3-x. 40. ^oj + I) ('^ - 9) - (^ + ^) (^-3) + 1=0. 41. (x-^)(x-^^^-{x-5)(x + 3)-^^l = 0. ,« 9a; + 5 8x-7 3Qx+15 lOJ U 6aj + 2 56 14 ,„ 6a; + 7 2£c-2 2cc+l ,, Gx+l 2x-4: 2x-l A. -} . r:^ A. A. __ ^^ ______ 15 7x-Q 5 • "15 7a;- 16 5 * 45. r + tc + 2 x + 3 x' + ox + 6 ' 4G. (:r+lf ={6-(l-a;)}ic-2. 47. 1 1 ^ ^ x-2 x — 4: x — Q x — S .0 2 1 6 48. T^ =■ + 2x- b X- 3 3aj - 1 * ,^ 25 -ia; 16ic + 4i 23 49. f-+-^ ^ = T- + ^- x+l 3cc + 2 x+ I 51. (a + x) {h +x) = {c + x) {d + x). w /-» XX Cb <^ n .m X \. 52. -+-, =- . 53. ax + o = -+j. a -a + a a ^ , X- a x-b x-G X- (a + h + c) 54. — T- + + = —T -' oca aoo 55. {a + x) {h + x) - a(b -\- c) = ^ ■¥ x^. a + h a b ^^ ax^ + bx + c ax + h 5(j. = + 7. 57. — o = — x-c x-a x-b ;;jc- + ^x + r px-]-q ,,, 3abG a%^ (2a + b)b'x ^ bx 58. T + '. ^ + ^^ — -. — ^ = 3ca; + — . a + 6 {a+bf a{a + b) a 80 " EXAMPLES. IX. m(x + a) n(x+b) /x-a\' x-2a-h 59. J— + ■ — '- =m + n. 60. ( . ) = ^, . x + b x + a \x + b/ x + a + 2b GI. {x-af+{x-by+{x-cy = 3{x-a){x-b)(x-c). 62. '15x + 1-575 - •875x= -06250;. -. « 'ISa;- '05 63. \-2x ^ --40; + 8-9. •0 64. 4-8a: - ''^^"""'^'^ = VQ>x + 8-9. X. PROBLEMS WHICH LEAD TO SIMPLE EQUA- TIONS WITH ONE UNKNOWN QUANTITY. 167. We shall now apply the methods already given to the solution of some problems, and thus exhibit to the student speci- mens of the use of Algebra. In a problem certain quantities are given, and certain others, which have some assigned relations to them, are to be found. The relations are usually expressed in ordinary language in the enunciation of the problem, and the method of solving the problem may be thus described in general terms : denote the unknown quantities by letters, and express in algebraical language the relations which hold between the un- known quantities and the given quantities; we shall thus obtain equations from, which the values of the unknown quantities may be derived. We shall now give some examples. In the present Chapter we confine ourselves to problems which may be solved by using only one unknown quantity. 168. The sum of two numbers is 89 and their difference is 31 : find the numbers. Let x denote the less number, then the greater number is 31 + a;; thus since their sum is 89, we have 31+£c + a; = 89, that is, 3l4-2a;=89; PROBLEMS WHICH LEAD TO SLMPLE EQUATIONS. 81 hj transposition, 2a;=89-31 = 58j 58 bj division, x = -^ = 29. Thus the less number is 29, and the gi-eater number is 29 + 31, that is, GO. 169. A bankrupt owes B twice as much as he owes A^ and C as much as he owes A and B together : out of £300 which is to be divided among them, what should each receive ? Let X denote the number of pounds which A should receive • then 2ic is the number of pounds B should receive j and x + 2a;, that is Zx, is the number of pounds C should receive. The whole sum they receive is £300 ; thus, a; + 2a; + 3£C = 300 ; that is, 6a; = 300 ; and aj = -^=50; therefore A should receive £50, ^ £100, and C £150. 170. Divide a line 21 inches long into two parts, such that one may be three-fourths of tho other. 3a; Let X denote the number of inches in one part, then -j- denotes the number of inches in the other part ; thus, clear of fractions ; thus, 4a; + 3a; = 84 ; that is, 7a; = 84 ; 84 therefore, x— — =12. Thus one part is 12 inches long and the other part 9 inches. 171. If ^ can perfoiTQ a piece of work in 8 days, and B in 10 days, in what time will they perform it together ? T. A. 6 6x ^ a; + — = 21 : 4 82 PROBLEMS WHICH LEAD TO SIMPLE EQUATIONS Let X denote the number of days required. In one day A can 1 . X perfoim ^ th of the work, therefore in x days he can perform - ths of the work. In one day B can perfoi*m . th of the work, there- X fore in x days he can perform ^Tv ths of the work. Hence since A and £ together perform the whole work in x days, we have 8 "^ To " -^ ^ clear of fractions by multiplying by 40 j thus, 5x + Ax = 40, that is, 9a; = 40 ; 40 therefore, ^-'^-^i- 172. A workman was employed for 60 days, on condition that for every day he worked he should receive 15 pence, and for every day he was absent he should forfeit 5 pence ; at the end of the time he had 20 shillings to receive : required the number of days he worked. Let X denote the number of days he worked, then he was absent QO - x days; then lox denotes his pay in pence, and 5 (60 - cc) denotes the sum he forfeited. Thus, 15x-5{60-x) = 24.0; that is, 15x- 300 + 5x= 240 ; therefore, 20a; = 240 + 300 = 540 ; therefore, -f = 27- Thus he vrorked 27 days and was absent 60-27 days, that is, 33 days. 173. How much rye at four shillings and sixpence a bushel must be mixed with fifty bushels of wheat at six shillings a bushel, that the mixture may be worth five shillings a bushel 1 WITH ONE UNKNOWN QUANTITY. 83 Let X denote the number of bushels required ; then 9x is the value of the rye in sixpences, and 600 is the value of the wheat. The value of the mixture is 10 (50 + x). Thus, 10 (50 + re) = 9a; +600; that is, lOo; + 500 = dx+ 600 ; and a; =100. 174. A smuggler had a quantity of brandy which he expected would produce £9. 18s. ; after he had sold 10 gallons a revenue officer seized one-thii'd of the remainder, in consequence of w^hich the smuggler makes only £8. 2s. : required the number of gallons he had and the price per gallon. 198 Let X denote the number of gallons ; then is the value X of a gallon in shillings. The quantity seized is ^^ — gallons, and the value of this is — x shillinsrs ; thus, o X £^ML8 = 198-162 = 36. O X Multiply by 3x ; thus, 198 (a; -10) = 3a; X 36 = 108a; j therefore, 198a;-108a; = 1980; that is, 90a; =1980, and x = --~— = 22. 90 Thus 22 is the number of gallons, and the price of each . 198 gallon IS — shillings, that is, 9 shillings. 175. Tlie student may now exercise himself in the solution of the following problems. We may remark that in these cases the only difficulty consists in translating ordinary verbal state- ments into Algebraical language, and the student should not be discouraged if at first he is sometimes a little perplexed, since nothing but practice can give him readiness and cei-tainty iu this process. • 6—2 8-4 EXAMPLES. X. EXAMPLES OF PROBLEMS. 1. The property of two persons amounts to £3870, and one of them is twice as rich as the other ; find the property of each. 2. Divide £420 among two persons so that for every shilling one receives the other may receive half-a-crown. 3. How much money is there in a purse when the fourth part and the fifth part together amount to £2. 5s. ? 4. After paying the seventh part of a bill and the fifth part, £92 is still due ; what was the amount of the bill 1 5. Divide 46 into two parts, such that if one part be divided by 7 and the other by 3, the sum of the quotients shall be 10. 6. A company of 266 persons consists of men, women and children j there are four times as many men as children, and twice as many women as children. How many of each are there 1 7. A person expends one-third of his income in board and lodging, one-eighth in clothing, and one-tenth in charity, and saves £318. "What is his income ? 8. Three towns. A, B, C, raise a sum of £594 j for every pound which B contributes, A contributes twelve shillings, and C seven- teen shillings and sixpence. What does each contribute 1 9. Divide £1520 among A, B, and (7, so that B shall have £100 more than A, and C £270 more than B. 10. A certain sum is to be di^dded among A, B, and G, A is to have £30 less than the half, B is to have £10 less than the third part, and C is to have £8 more than the fourth part. What does each receive 1 11. The sum of two numbers is 5760, and their difference is equal to one-third of the greater : find the numbers. 12. Two casks contain equal quaitities of beer ; from the first 34 quarts are drawn, and from the second 80 j the quantity remaining in one cask is now twice that in the other. How much did each cask originally contain ] EXAMPLES. X. 85 13. A person bought a print at a certain price, and paid the same price for a frame ; if the frame had cost £1 less and the print 155. more, the price of the frame would have been only- half that of the print. Find the cost of the print. 14. Two shepherds owning a flock of sheep agree to divide its value ; A takes 72 sheep, and £ takes 92 sheep and pays A £35. Required the value of a sheep. 15. A house and garden cost £850, and five times the pnce of the house was equal to twelve times the price of the garden : find the price of each. 16. One-tenth of a rod is coloured red, one-twentieth orange, one- thirtieth yellow, one-fortieth green, one-fiftieth blue, one- sixtieth indigo, and the remainder, which is 302 inches long, \dolet. Find the length of the rod. 17. Two-thirds of a certain number of persons received eighteenpence each, and one-third received half-a-crown each. The whole sum spent was £2. Ids, How many persons were there ] 18. Find that number the third part of which added to its seventh part makes 20. 19. The difference of the squares of two consecutive numbers is 15. Find the numbers. 20. Of a certain dynasty one-third of the kings were of the same name, one-fourth of another, one-eighth of another, one- twelfth of a fourth, and there were five besides. How many kings were there of each name ? 21. A crew which can pull at the rate of nine miles an hour, fiinds that it takes twice as long to come up a river as to go down j at what number of miles an hour does the river flow ] 22. A and B play at a game, agreeing that the loser shall always pay to the winner one shilling more than half the money the loser has ; they commence with equal quantities of money, but after B has lost the first game and won the second, he has twice as much as A : how much had each at the commencement 1 86 EX.\3IPLES. X. 23. A person who possesses £12000 employs a poiiion of the money in building a house. One-thii'd of the money which re* mains he invests at 4 per cent., and the other two-thirds at 5 per cent., and from these investments he obtains an income of £392. What was the cost of the house ? 24. A farmer has oxen worth £12. 105. each, and sheep worth £2. 6s. each; the number of oxen and sheep being 35, and their value £191. IO5. Find the number he had of each. 25. A and B find a purse with shillings in it. A takes out two shillings and one-sixth of what remains ; then B takes out three shillings and one-sixth of what I'emains ; and then they find that they have taken out equal shares. How many shillings were in the purse, and how many did each take 1 26. A hare is eighty of her own leaps before a greyhound ; she takes three leaps for every two that he takes, but he covers as much ground in one leap as she does in two. How many leaps will the hare have taken before she is caught 1 27. The length of a field is twice its breadth ; another field which is 50 yards longer and 10 yards broader, contains 6800 square yards more than the former ; find the size of each. 28. A vessel can be emptied by three taps ; by the first alone it could be emptied in 80 minutes, by the second alone in 200 minutes, and by the third alone in 5 hours. In what time will the vessel be emptied if all the taps are opened ? 29. If an income tax of 7d. in the jDound on all incomes below £100 a year, and of Is. in the pound on all incomes above £100 a year realise £18750 on £500000, how much is raised on incomes below £100 a year ] 30. A person buys some tea at 3 shillings a pound, and some at 5 shillings a pound ; he wishes to mix them so that by selling the mixture at 3s. Sd. a pound he may gain 10 per cent, on each pound sold : find how many pounds of the inferior tea he must mix with each pound of the superior. EXA^IPLES. X. 87 31. A fruiterer sold for 195. 6cl. a certain number of oranges o and apples, of which the latter exceeded the former by 180. He sells the apples at the rate of 5 for 3d., and 15 oranges brino- him in l^cl. more than 35 apples. How many are there of each sort ? 32. A cask A contains 12 gallons of wine and 18 gallons of water ; and another cask J5 contains 9 gallons of wine and 3 gal- lons of water ; how many gallons must be drawn from each cask so as to produce by their mixture 7 gallons of wine and 7 gallons of water ? 33. A can dig a trench in one-half the time that £ can ; B can dig it in two-thirds of the time that C can ; all together they can dig it in 6 days ; find the time it would take each of them alone. 34. A person after paying sevenpence in the pound for In- come Tax has £408. 45. SUl left. What had he at first ? 35. At what time between one o'clock and two o'clock is the long hand of a clock exactly one minute i6 advance of the short hand? 36. A person has just a hours at his disposal ; how far may he ride in a coach which travels b miles an hour, so as to return home in time, walking back at the rate of c miles an hour 1 37. A certain article of consumption is subject to a duty of G shillings per cwt. ; in consequence of a reduction in the duty the consumption increases one-half, but the revenue falls one-thii'd. Find the duty per cwt. after the reduction. 38. A ship sails with a supply of biscuit for 60 days, at a daily allowance of a pound a head; after being at sea 20 days she encounters a storm in which 5 men are washed overboard, and damage sustained that will cause a delay of 24 days, and it is found that each man's daily allowance must be reduced to five- sevenths of a pound. Find the original number of the crew. 88 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE XI. SIMULTANEOUS EQUATIONS OF THE FIRST DEGKEE WITH TWO UNKNOWN QUANTITIES. 176. Suppose we have an equation containing two unknown quantities x and ?/, for example 5x-2i/ = 4. For every value \\'liich we please to ascribe to one of tlie unknown quantities we can determine the corresponding value of the other, and thus find as many paii's of values as we please which satisfy the given /> equation. Thus, for example, if ^ == 1 we find oj = ^r ; if y — 2 we find X = -^- ; and so on. 5 Also, suppose that there is another equation of the same kind, as for example, 4:X+3i/—17. We can also find as many parrs of values as we please which satisfy this equation. But suppose we ask for values of x and y which satisfy both equations ; we shall fi.nd then that there is only one value of x and one value of y. For multiply the first equation by 3 ; thus, 15x- Gy = 12 ; multiply the second equation by 2 ; thus, 8x+6y = 34:. Therefore, by addition, 15a; - (yy + 8x + 6y = 12 + oi ; that is, 23a; = 46, and, x = 2. Thus if both equations are to be satisfied x must equal 2 ; put this value of x in either of the two given equations j for example, in the second equation ; thus we obtain 8 + 32/ = 17; therefore, 3y = 17 - 8, and, y = 3' WITH TWO UNKNOWN QUANTITIES. 89 177. Two or more equations which are to be satisfied by the same values of the unknown quantities are caUecl simultaneous equations. We are now about to treat of simultaneous equations involving two unknown quantities where each unknown quantity occurs only in the first degi'ee, and the product of the unknown quantities does not occur. 178. There are three methods which are usually given for solving these equations. The object of all these methods is the same, namely, to obtain from the two given equations which contain two unkno^Ti quantities a single equation containing only one of the unknown quantities. By this process we are said to eliminate the unknown quantity which does not appear in the single equation. 179. Fii'sf method. The first method is that which we adopted in the example of Ai*t. 176; it maybe thus described: Tnultiply the equations hy such numbers as loill m,ake the coefficient of one of the unknown quantities the same in the two resulting equations ; then hy addition or subtraction ive can form an equa- tion containing only the other unknown quantity. Example. 4aj + 83/ = 22 j 5x-7y^6. If we wish to eliminate y we multiply the frst equation by 7, which is the coefiicient of y in the second, and the secojid equation by 3, which is the coefficient of y in the first equation. Thus we obtain 28x + 21y = lo4:; 15x-21y = 18. Then by addition, 28x + 16x = 15i + 18; that is, 43x= 172, and, X = — ^ = 4. 90 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE Then put this value of a; in either of the given equations, in the first for example j thus, lG + 3y = 22; therefore, 3y = 6, And, 2/ = 2. If we wish to solve this example by eliminating x we multiply the firet of the given equations by 5, and the second by 4 ; thus, 20aj + 15?/ = 110; 20x-2Si/ = 24:. Then by suhiraction, 20x + 15y - (20a; - 283/) = 1 10 - 24 ; thus, 433/ = 86, and, y = 2. 180. Second meiliod. Express one of the unknown quantities in terms of the other from either equation, and substitute this value in the other equation. Thus, taking the same example, we have from the first equation 4a; = 22 -32/; r -^ 1 A 22 - 3y divide by 4, x = ; substitute this value of x in the second equation and we obtain 5(22-3,) 4 -^ ' multiply by 4, 5 (22 - 3^/) - 28y = 24; that is, 110-15y-282/ = 24; by transposition, 43?/ = 86, and, 2/ = 2. Then substitute this value of y in either of the given equations and we shall obtain a; = 4. Or thus ; from the first equation we have 32/ = 22-4a;; r -1 K •? 22-4a;. divide by 3, y= o — ; WITH TWO UNKNOWN QUANTITIES. 91 substitute this value of y in tlie second equation and we obtain o multiply by 3, 1 5x - 7 (22 - 4aj) = 1 8 j that is, 15a:-154 + 2Sa; = 18; that is, 43ic= 172, and, a? = 4. Then substitute this value of x in either of the given equa- tions and we shall obtain y = 2. 181. Third method. Express the same unknown quantity in terms of the other from each equation and equate the expressions thus ohtained. 22 - 3v , , , , ■ . 6 + 7y x = - — -. — ^ , and from the second equation x = — ^ — • 4: Thus, taking the same example, from the firet equation the second equati .1 22-33/6 + 72/ clear of fractions, 5(22 — 3y) = 4(6 + 7y) ; that is, 110 - 15y = 24 + 28?/; by transposition, 43y = 86, and, 2/ = 2. Hence, as before, we deduce x = i. 22 -Ax Or thus; from the first equation we obtain y= — — — -, See — 6 - and from the second equation y = — = — ; thus, 22-4a; _5ic- 6 3 ~~T~' Ilence as before we shall obtain a = 4 and then deduce ?/ = 2, 92 EXAMPLES. XI. EXAMPLES OF SIMULTANEOUS SIMPLE EQUATIONS WITH TWO UNKNOWN QUANTITIES. 1. a; + y=15, x-y — 7. 2. 3a;-2y = l, 3y-4a; = l. 3. 3x-5i/= 13, 2x+77/ = 81. 4. 2a; + 3y-43, 10cc-2/=7. 5. 5x-7y=33, llo; + 12y= 100. 6. 3y-7x = 4:, 27j + 5x = 22. 7. 21y + 20:c=165, TTy- 30a; = 295. 8. 5a; +72/ = 43, llaj + 9^=69. 9. 8a;-21y=33, 6x + 35y = 177. 10. llx-10?/ = ]4, 5a:+7y = 41. 11. 16x + 17y = 500, 17a;- 32/ = 110. 14. ^+^=1 ^ + ^-1 2 3 ' 3 4" 15. ^-^^=8, ^_±^ + ^--2^.11. 16. 11^^=^^, 8^-5y = l. ^^ 2x . y _3yl 1/^^.1^^ a. 18. 4a; + 8y=2-4, 10-2x- 6y = 3-48. 19. a;=4y, ^(2x + 7y)- 1 =|(2k- 6y + 1). 23. EXAMPLES. XI. 93 21 ^^-% ^3^j^±y a;-2y a; y 2 5 ' 4 "2 3* 22 ---^-^ = -^^-^ 2.: ^-^ 2/..^^ 10 15 9 12 18' 3~l2"r5^T0" ix-^y-7 _Zx 2y 5 5 ~IO~r5 ~6' y — 1 X Sy y — X X 11 ^3~ "^ 2 ~ 20 " TT "^ 6 "^ 10 * 2x 5y 3x y 24. J il_-^ 1 = 2 ^-y ^1 7 23 ' X +y b' i T 25 ^^-^y +i + ^iy::2g ^ 4^-3^ + 5 45 -a? 3 8 7 ^~"5~' ,K_ 4£c- 2 _ 55ic + 71y+ 1 ~3 ~ l8 • 26. 2-4a.-.-.3-2^- Jg^ILgg-.g,^ , 2-6+;005y .5 ^~ . , •04y+-l •07a; -•1 •3 •6 27. 13aj+lly = 4a, 12a;-6y = a. 28. — +- = 1, X y 01 m - - + — = 1. X y 29. ^+^ = 1 a b ' X y 2 3a "^ 66 " 3 • 30. ax + hy = c, mx -ny = d. 31. X y + c a + c = 2, ax — by {a — b)c~ 32. « ^ 2/ a + b a- b = ^- ^^-- 94! SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE XII. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE WITH MORE THAN TWO UNKNOWN QUANTITIES. 182. If there be three simple equations and three unknown quantities, deduce from two of the equations an equation con- taining only two of the unknown quantities by the rules of the preceding Chapter ; then deduce from the third equation and either of the former two, another equation containing the same two unknown quantities; and from the two equations thus ob- tained the unknoAvn quantities which they involve may be found. The third quantity may be found by substituting the above values in any of the proposed equations. Example, suppose, 2x + Sy + 4:Z = 16 (I), 3x + 2y-5z = S (2), 6x-67/ + 3z = 6 (3). For convenience of reference the equations are numbered (1), (2), and (3), and this numbering is continued as we proceed with the solution. Multiply (1) by 3, and (2) by 2; thus, 6x + 92/ + I2z = A8, 6x + 4:i/-lOz^lQ; by subtraction, 5y+22«=32 (4). Multiply (1) by 5, and (3) by (2); thus, 10a;+152/ + 20;s = 80, I0x-l2y+ez =12; by subtraction, 27y + Uz=6S (o). Multiply (4) by 27, and (5) by 5 ; thus, WITH MORE THAN TWO UNKNOWN QUANTITIES. 95 135y + 594«=864, 135y+ 70;2; = 340j by subtraction, 52iz = 524, therefore, z=l. Substitute the value of z in (4) ; thus, 5y + 22 = 32; therefore, y = 2. Substitute the values of y and z in (1) ; thus, 2a; + 6+4 = 16; therefore, x = 3. Sometimes it is convenient to use the following rule : from two of the equations express the values of two of the unknown quantities in terms of the third, and substitute these values in the thii'd equation ; hence the third unknown quantitv can be found, and then the other two. Example, sujipose 3x+4:7/-lQz = (1), 5x-8i/ + 10z = (2), 2x+6y+ 7s-52 (3). Multiply (1) by 2, and add to (2) ; thus llflj — 22;5; = 0; therefore x = 2z. Multiply (1) by 5, and (2) by 3, and subtract ; thus 5z 44?/ — 11 0:2; = 0; therefore y= — . Substitute in (3) ; thus 4:z+15z + 7z = 52'j that is 26-^ = 52 ; 5z therefore z = 2 ; and x = 2z = 4, y = -^ = 5. The same methods may be applied when the number of simple equations and of unknown quantities exceeds three. 96 EXAMPLES, xir. EXAMPLES OF SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE WITH MORE THAN TWO UNKNOWN QUANTITIES. ® 3x+2i/-iz = 15, 6x-3i/ + 2z = 2S, 3y + 4:Z-x=2i. @. X +1/ -z= I, 8x + 3y - 6z-l, Sz—4:X — y = l. $i 2x-7y + 4:z = 0, 3x-3i/ + z = 0, dx + 5y + Zz = 28. 4. 4:X-37/ + 2z^9, 2x + 5?/ - 3z = 4:, 5x + Gy-2z = l8. 5. 2x-4y+dz = 28, 7x+3]/-5z = 3, 9x -{■ 10^/ - Uz = 4:. 6. x-2y + 3z = 6, 2x + 37/ - 4:Z = 20, 3x-27/ + 5z=2Q. 7. 4:x-3i/ + 2z = 4:0, 5x + 9i/-7z^4:7, 9x + 8y-3z = 97. 8. 3x + 27/ + z=23, 5x + 2y + iz = 4:6, lOx + 5i/ + 4:Z = 7 5. 9. 5x-6y + 4:z= 15, 7x + 4:y-3z=l9, 2x + y-\-6z^4:Q. 11,11^113 X y X z y z Z 10. -,213 3 2 _ 1 1 4 11. - + -=-, = 2, -+- = -5. X y z z y X z o 3_ 4 1_38 J^ 1^ ?_^ A_J_ ^-Hl w x~5^^z~'5'3^'^2y'^z~6'5x 2y'^ z~ 10 3y — l_Qz X 9 14. 6x iz T"^T"^ ■ 6 + ^=2/+«' Sic + 1 z 1 _ 2;^ y ~7 l4'^6"2T'^"3* lOx + iy - 5z 4:X + 6y- 3z lOx + iy — 5z = ix + 6y — 3z - 8, lOx + 4y-5z 4:X+ Qy-3z x + y -\-z 10 "^ ~~~3 " 4 EXMIPLES. XII. 9' 15. 7a;-3y = l, ll;^-7w=l, 4«-7y-l, 19a;-3M = l. 16. 3^-2y = 2, 5:c-72r = ll, 2a;+3y = 39, 4y + 3^=41. - 17. 2a;-3y + 2^ = 13, 4y+ 22; =14, 4w-2ic=30, 5i/ + 3u = 32. 18. 7u- \3z= 87, lOy- 3a;- 11, 3u + lix = 61, 2jc-1U=50. ■ 19. 7x-2z + 32i=17, 20. 3a; — 4y + 3^ + 3v - - 6ii = = 11, 4i/-2z+ i? = ll, 3x - 5y + 2;^ - -4:U = = 11, 5i/-3x-2u = 8, \0y-3z^3u- -2v = ; 9 4y - 3w + 2v = 9, 5z 4- 4zt + 2v - -2x = ^3, 3;s+8i^ = 33. 6ii- 3?; + 4a;- -2y = :6. 5- X y ^ X z -+^ = 1, + - a b a c 1, ^ c 22, ay + hx = c, ex + a ;:; = 5, I )z + cy^ a. 3 a h ^ he - +- = 1, - +- = X y y z 1, c - + z X 24. x-\-y + z = Qi, (6 + c) a; 4- (c + «) y + (a + 6) 2; = 0, hex + cny + abz =^1. 25. rta; + 6y + C2; = J, a^a; + b^y + c^z = A\ a^x + b^y + c''z = A\ 26. xyz = a{yz—zx- xy) -b {zx - xy - yz) = c {.nj -yz- zx). 27. x-\-y-\-z = a+b + c, bx + cy + az= ex ^- ay ^- bz = a' + b' + c'. 28. X- ay + a^z ^ a^, x-by + b^z = 6^ ,2, a; - c?/ + c 2; = c' . T. A. 7 93 PROBLEMS WHICH LEAD TO SIMPLE EQUATIONS XIII. PROBLEMS WHICH LEAD TO SIMPLE EQUATIONS WITH MORE THAN ONE UNKNOWN QUANTITY. 183. We shall now give some examples of problems which lead to simple equations with more than one unknown quantity. A and B engage in play ; in the first game A wins as much as he had and four shillings more, and finds he has twice as much as -S ; in the second game B wins half as much as he had at first and one shilling more, and then it appears he has three times as much as A : what sum had each at first 1 Let X he the number of shillings which A had, and y the number of shillings which B had ; then after the first game A has 2a; + 4 shillings and B has y — x — 4: shillings. Thus by the question, 2a; + 4 = 2 (y - a: - 4) = 2?/ - 2x - 8 ; therefore, 27/ — 4x— 12; therefore, y-2x== Q. Also after the second game A has 2a; + 4 — ■! — 1 shillings, and B has 2/ — a; — 4 + ^ + 1 shillings. Thus by the question, y_a;_4 + | + 1^3(2a; + 4-^-l)=6a:+12-^-3; . therefore, 2y - 2a: -8 + 2/4-2 = 12a; + 24 -3?/ -6; therefore, 6y — 14a; = 24, and, 3?/ -7a; = 12. And from the former equation, 3?/ — 6u; = 18 ; hence by subtraction, x = Q ; therefore, y = \S, WITH MORE THAN ONE UNKNOWN QUANTITY. 99 184. A sum of money was divided equally among a certain number of persons ; had there been three more, each would have received one shilling less, and had there been two fewer, each would have received one shilling more than he did : required the number of persons, and what each received. Let X denote the number of persons, y the number of shillings which each received. Then xy shillings is the sum divided ; thus by the question, {x + 3){y-l) = xy, and also, {x — 2) (y + 1) = xy. The fii-st equation gives xy + 3y — x — S = xy ; thus, 3y-x = 3. The second equation gives xy — 2y + X — 2 = xy ; thus, x-2y = 2. By addition, 3y-x + x — 2y = 5 ; ' that is, y = 5. Hence, x = 2y+ 2 = 12. 185. What fraction is that which becomes equal to | when its numerator is increased by 6, and equal to ^ when its denom- inator is diminished by 2 1 Let X denote the numerator and y the denominator of the fraction ; then by the question, x+Q 3 Clear the first equation of fractions by multiplying by 4y ; thus, i{x + Q) = Sy; therefore, 3^ - 4a; = 24. < — :: 100 EXAMPLES. XIII. Clear the second equation of fractions by multiplying by ^(y-Vj-y thus, therefore, y-2x = 2, and, 3?/ - Ga: = 6. By subtraction, 2>y-ix-{'iy-(Sx) = U- 6; that is, 2-c=18, and, a; = 9. Hence, 2/ = 2 + 2a; = 20. 9 Thus the required fractiqii is -^ . 4\j EXAMPLES OP PROBLEMS. ~1. A cei*tain fraction becomes 1 when 3 is added to its nu- merator, and J when 2 is added to its denominator. What fraction is it? 2. A and B together possess <£570. If ^'s money were three times what it really is, and ^'s five times what it really is, the sum would be .£2350. What is the money of each ? 3. i If the numerator of a certain fraction is increased by one the value of the fraction becomes ^ ; if the denominator is in- creased by one the value of the fraction becomes \. What is the fraction ? 4. Find two numbers such that if the first be added to four times the second, the sum is 29 j and if the second be added to six times the first the sum is 36. 5. If ^'s money were increased by 36s. he would have three times as much as B ; but if jB's money were diminished by 5s. he would have half as much as A. Find the sum possessed by each. 6. A and B lay a wager of IO5. j if A loses he will have twenty-five shillings less than twice as much as B will then have ; EXA3IPLES. XIII. 101 but if B loses lie will have five-seventeenths of M]\8i\' A .Will then have : find how much money each of them has. - , . . ^ . ., ^~7r Find two numbers, such that twice che'fli'st plus the' second is equal to 17, and twice the second plus the first is equal to 19. 8. Find two numbers, such that one-half the first and three- fourths of the second together may be equal to the excess of three times the fii'st over the second, and this excess equal to 11. 9. For five guineas can be obtained either 32 pounds of tea and 15 pounds of coffee, or 36 pounds of tea and 9 pounds of cofiee : find the price of a pound of each* 10. Determine three numbers such that their sum is 9 ; the sum of the first, twice the second, and three times the third, 22 ; and the sum of the first, four times the second, and nine times the thii'd, 58. 11. A pound of tea and three pounds of sugar cost six shil- lings, but if sugar were to rise 50 per cent, aiid tea 10 per cent, they would cost 7 shillings. Find the price of tea and sugar. 12. A person has £2550 to invest. The three per cent, con- sols are at 81, and certain guaranteed railway shares which pay a half-yearly dividend of IO5. on each original share of £25 are at £21. Find how many shares he must buy that he may obtain the same income from the railway shares as from the rest of his money invested in the consols. 13. A person possesses a certain capital which is invested at a certain rate per cent. A second person has £1000 more capital than the fii'st person and invests it at one per cent, more ; thus his income exceeds that of the first person by £80. A third person has £1500 more capital than the first and invests it at two per cent, more ; thus his income exceeds that of the first person by £150. Find the capital of each person and the rate at which it is invested. 14. A sum of money is divided equally among a certain num- ber of pei-sons ; if there had been four more each would have 102 EXAJMPLES. XIII. received a &liillmg less than he did ; if there had been five fewer -each would have received two shillings more than he did : find the number of persons and what each received. 15. Two plugs are opened in the bottom of a cistern con- taining 192 gallons of water; after three hours one of the plugs becomes stopped, and the cistern is emptied by the other in eleven more hours ; had six hours occurred before the stoppage, it would have requii^ed only six hours more to empty the cistern. How many gallons will each plug hole dischai'ge in. an hour, sup- posing the discharge uniform 1 16. A person after paying a poor-rate and also the income- tax of 7d. in the pound, has £486 remaining ; the poor-rate amounts to X22. IO5. more than the income-tax : find the original income and the number of pence per pound in the poor-rate. 17. A certain number of persons were divided into three classes, such that the majority of the first and second together - over the third was 10 less than four times the majority of the ' second and third together over the first ; but if the first had 30 vN more, and the second and thii'd together 29 less, the first would have outnumbered the last two by one. Find the number in each class when the whole number was 34 more than eight times the majority of the third over the second. 18. A farmer would spend all his money by buying 4 oxen and 32 lambs ; instead of doing this he bought the same number of oxen and half as many lambs, and had a surplus of £9 after paying for them and for their conveyance by railway at an average cost of six shillings per head. Each ox cost as many pounds as its carriage by railway was shillings, and the lambs altogether cost three times as many pounds as the carriage of each was shillings. How much money had the farmer to begin with ? 19. A and £ play at bowls, and A bets £ three shillings to two upon every game ; after a certain number of games it appears that A has won three shillings ; but if A had bet five shillings to two and lost one game more out of the same number, he would have lost thirty shillings. How many games did each win 1 EXAMPLES. XIII. 103 20. Five persons, A, B, C, D, E play at cards ; after A lias won half of J5's money, B one -third of C's, C one-foiii'tli of Z>'s, i> one-sixth of ^'s, they have each £1. 10*\ Find how much each had to begin with. 21. If there were no accidents it would take half as long to travel the distance from ^ to jB by railroad as by coach ; but three hours being allowed for accidental stoppages by the foiTner, the coach will travel the distance all but fifteen miles in the same time ; if the distance were two-thirds as great as it is, and the same time allowed for railway stoppages, the coach would take exactly the same time : required the distance. 22. A and B are set to a piece of work which they can finish in thirty days working together, and for which they are to receive £1. 10s. When the work is half finished A intermits working eight days and B four days, in consequence of which the work occupies five and a half days more than it would otherwise have done. How much ought A and B respectively to receive 1 23. A and B run a mile. Fii-st A gives B a start of 44 yards and beats him by 51 seconds; at the second heat A gives B a start of 1 minute 15 seconds, and is beaten by 88 yards. Find the times in which A and B can run a mile separately. 24. A and B start together from the foot of a mountain to fo to the summit. A would reach the summit half an hour before B, but missing his way goes a mile and back again need- lessly, during which he walks at twice his former pace, and reaches the top six minutes before B. C starts twenty minutes after A and B and walking at the rate of two and one-seventh miles per hour, arrives at the summit ten minutes after B. Find the rates of walkins: of A and B, and the distance from the foot to the summit of the mountain. 25. A railway train after travelling for one hour meets with an accident which delays it one hour, after which it proceeds at three-fifths of its former rate, and ai'rives at the terminus three hours behind time ; had the accident occuried 50 miles further on, 104 EXAMPLES. XIII. the train would have arrived 1 hour 20 minutes sooner. Required the length of the line, and the original rate of the train. 26. A, B, and C sit down to play, every one with a certain number of shillings. A loses to B and to C as many shillings as each of them has. Next B loses to A and to (7 as many as each of them now has. Lastly C loses to A and to -5 as many as each of them now has. After all every one of them has sixteen shillings. How much had each originally ] 27. Two persons A and B could finish a work in m days ; they worked together n days when A was called off and B finished it in p days. In what time could each do it ] 28. A railway train running from London to Cambridge meets on the way with an accident, which causes it to diminish its speed to - th of w^hat it was before, and it is in consequence n a hours late. If the accident had happened h miles nearer Cam- bridge, the train would have been c hours late. Pind the rate of the train before the accident occurred. 29. The fore- wheel of a carriage makes six revolutions more than the hind-wheel in going 120 yards; if the circumference of the fore- wheel be increased by one-fourth of its present size, and -the circumference of the hind- wheel by one-fifth of its present size, the six will be changed to four. Hequii^ed the circumference of each wheel. 30. There is a number consisting of two digits ; the number is equal to tlu^ee times the sum of its digits, and if 45 be added to the number the digits interchange their places : find the number. 31. There is a number consisting of two digits ; the number is equal to seven times the sum of its digits, and if 27 be sub- tracted from the number the digits interchange their places : find the number. 32. A person proposes to travel from A to B, either direct by coach, or by rail to C, and thence by another train to B. The trains travel three times as fast as the coach, and should there be EXAMPLES. XIII. 105 no delay, the pei'son stai-ting at the same hour could get to B 20 minutes earlier by coach than by train. But should the train be late at 6', he would have to wait there for a train as long as it would take to travel from C to £, and his journey would in that case take twice as long as by coach. Should the Coach how- ever be delayed an hour on the way, and the train be in time at (7, he would get by rail to B and half way back to C, while he would be going by coach to B. The length of the whole circuit ABC A is 76| miles. Required the rate at which the coach travels. 33. A offers to nin three times round a coui^e while B runs twice round, but A only gets 150 yards of his thii'd round finished when B wins. A then offers to run four times round for B's thrice, and now quickens his pace so that he iims 4 yards in the time he foi-merly ran 3 yards. B also quickens his so that he runs 9 yards in the time he foi-merly ran 8 yards, but in the second round falls off to his original pace in the first race, and in the third round only goes 9 yards for 10 he went in the first race, and accordingly this time A wins by 18A yards. Determine the length of the course. 34. A man starts p hours before a coach, and both travel uni- formly ; the latter passes the former after a cei-tain niunber of hours. From this point the coach increases its speed to six-fifths of its foi-mer rate, while the man increases his to five-fourths of his foi-mer rate, and they continue at these increased rates for q hours longer than it took the coach to overtake the man. They are then 92 miles apai-t ; but had they continued for the same length of time at their original rates they would have been only 80 miles apart. Shew that the original rate of the coach is t^vice that of the man. Also if ^ + 5- =16, shew that the original rate of the coach was 10 miles per hour, and that of the man 5 miles per hour. lOG DISCUSSION OF SOME PROBLEMS XIY. DISCUSSION OF SOME PROBLEMS WHICH LEAD TO SIMPLE EQUATIONS. 186. We propose now to solve some problems which lead to Simple Equations, and to examine certain peculiarities which present themselves in the solutions. We begin with the following problem : What number must be added to a number a in order that the sum may be 6 ? Let x denote this number ; then, a ■k-x = h \ therefore, x = h - a. This formula gives the value of x corresponding to any as- signed values of a and b. Thus, for example, if a =12 and b = 25, Ave have x= 25 -12 = 13. But suppose that a = 30 and b = 24: ; then cc = 24 - 30 = - 6, and we naturally ask what is the meaning of this negative result ? If we recur to the enun- ciation of the problem we see that it now reads thus : What number must be added to 30 in order that the sum may be 24 ? It is obvious then, that if the word added and the word sum are to retain their arithmetical meanings, the proposed problem is impossible. But we see at the same time that the following problem can be solved : What number must be taken from 30 in order that the difference may be 24 ? and 6 is the answer to this question. And the second enunciation differs from the first in these respects ; the words added to are replaced by taken from^ and the word sum by difference. 187. Thus we may say that, in this example, the negative result indicates that the problem in a strictly Arithmetical sense is impossible ; but that a new problem can be formed by appro- priate changes in the original enunciation to which the absolute value of the negative result will be the correct answer. 188. This indicates the convenience of using the word add in Algebra in a more extensive sense than it has in Arithmetic. Let X denote a quantity which is to be added algebraically to a ; WHICH LEAD TO SIMPLE EQUATIONS. ' 107 then the Algebraical sum is a +x, whether x itself be positive or negative. Thus the equation a + x = b will be possible algebraically whether a be greater or less than b. We proceed to another problem. 189. ^'s age is a yeai^, and ^'s age is h years ; when will A be twice as old as B ? Supposed the requii-ed epoch to be x yeara ti'om the present time j then by the question, a + x=^2{b +x) ; hence, x = a-2b. Thus, for example, if a = 40 and 6 = 15, then a; =10. But suppose a = 35 and b = 20, then x = — 5; here, as in the pre- ceding problem, we are led to inquire into the meaning of the negative result. Now with the assigned values of a and b the equation which we have to solve becomes S5 + x = 4:0 + 2x, and it is obvious that if a strictly arithmetical meaning is to be given to the symbols x and +, this equation is impossible, for 40 is greater than 35, and 2x is greater than x, so that the two members cannot be equal. But let us change the enunciation to the fol- lowing: ^'s age is 35 years, and ^'s a^e is 20 years, when was A twice as old a& £ i Let the required epoch be x years from the present time, then by the question, 35 - X = 2(20 -x) = iO -2x; thus, x = 5. Here again we may say the negative result indicates that the problem in a strictly Arithmetical sense is impossible, but that a new problem can be formed by appropriate changes in the original enunciation, to which the absolute value of the negative result will be the correct answer. We may observe that the equation corresponding to the new enunciation may be obtained from the original equation by chang- ing X into — X. 190. Suppose that the problem had been originally enun- ciated thus; As age is a years, and J5's age is b years; find the 108 DISCUSSION OF SOME PROBLEMS epoch at wliieli ^'s age is twice that of B. These words do not intimate whether the required epoch is before or after the present date. If we suppose it after we obtain, as in Ai*t. 189, for the requii-ed number of yeai-s x-a—'ih. If we suppose the required epoch to be x years before the present date we obtain a: = 26 - a. If 26 i!5 less than a, the first supposition is correct, and leads to an arithmetical value for x', the second supposition is incorrect, and leads to a negative value for x. If 26 is greater than a, the second supposition is correct, and leads to an arithmetical value for x'y the first supposition is incorrect and leads to a negative value for x. Here we may say then that a negative result indi- cates that we made the wrong choice out of two possible supposi- tions which the problem allowed. But it is important to notice, that when we discover that we have made the wrong choice, it is not necessary to go through the whole investigation again, for we can make use of the result obtained on the wrong supposition. We have only to take the absolute value of the negative result and place the epoch hefore the present date if we had supposed it after, and after the present date if we had supposed it before. 191. One other case maybe noticed. Suppose the enuncia- tion to be like that in the latter part of Art. 189; -^'s age is a years, and ^'s age is 6 yearsj when was A twice as old as -51 Let X denote the required number of years ; then « - 03 =-- 2 (6 — a?), hence, aj = 26 — a. Now let us verify this solution. Put this value for x) then a — x becomes a — (26 — a), that is, 2a — 2b; and 2 (6 — a;) becomes 2(6 — 26 + a), that is, 2a -2b. If 6 is less than a, these results are positive, and there is no Arithmetical difficulty. But if & is greater than a, although the two members are algebraically equal, yet since they are both negative quantities, we cannot say that we have arithmetically verified the solution. And when we recur to the problem we see that it is impossible if a is less than 6; because if at a given date ^'s age is less than -S's, then ^'s age never was twice ^'s and never will be. Or without proceeding to WHICH LEAD TO SIMPLE EQUATIONS. 109 verify the result, we may observe that if 6 is gi-eater than «, then X is also greater than a, which is inadmissible. Thus it appeai-s that a problem may be really absurd, and yet the result may not immediately present any difficulty, though when we proceed to examine or verify this result we may discover an intimation of the absurdity 192. The equation a + x='2{b+x) may be considered as the symbolical expression of the following verbal enunciation: Sup- pose a and h to be two quantities, what quantity must be added to each so that the fii'st sum may be t"s\'ice the second 1 Here the words quantity, sum, and added may all be understood in Alge- braical senses, so that x, a, and h may be positive or negative. This Algebraical statement includes among its admissible senses the Arithmetical question about the ages of A and B. It appears then that when we translate a problem into an equation, the same equation may he the symbolical expression of a more comprehen- sive problem than that from whicli it was obtained. We will now examine another problem' 193. A and B travel in the same direction at the rate of a and h miles respectively per houi-. A arrives at a certain place P at a certain time, and at the end of n hours from tJiat time B arrives at a cei-tain place Q. Find when A and B meet. p Q R Let c denote the distance PQ; suppose A and B to travel in the direction from P towards Q, and to meet at R at the end of x hours from the time when A was at P ; then since A travels at the rate of a miles per hour, the distance PR is ax miles. Also B goes over the distance QR in x — n hours, so that QR is b{x — n) miles. And PR is equal to the sum of PQ and QR ; thus, ax = c + b{x — n) = c + hx- hn; therefore, x = -,- . a — "VVe shall now examine this result on different suppositions as to the values of the given quantities. 110 DISCUSSION OF SOME PROBLEMS I. Suppose a greater than h, and c greater than hn] then the value of X is positive, and the travellers will meet, as we have supposed, after A arrives at P. For when A is at P, the space which B has to travel before he reaches Q is hn miles, and since hn is less than c, it follows that when -4 is at P he is hehind B ; and A travels more rapidly than B, since a is greater than h. Hence A must at the end of some time ovei-take B. rrr, ,. . ■r.T^ a(c — hi) r_,. The distance PR = ax = —^ ~ . Ihus, a — o a{c-hn) a{c — hn) — c(a — h)_ch-^ahrh_h(c-an) a-b a—b a—b a—b Now if c be greater than an, this ex^Dression is a positive quantity, so that R falls, as we have supposed, beyond Q; we see that this must be the case, for since c is greater than an, it will take A more than n hours to go from P to Q, so that he cannot ovei-take B until after passing Q. If, however, c be less than an, the ex- pression for QR is a negative quantity, and this leads us to suj)- pose that some modification is required in our view of the problem. In fact A now takes less than n hours to go from P to Q, so that he will overtake B before arriving at Q. Hence the figure should now stand thus: And now, since PR = PQ — RQ, the equation for determining X would natui'allj be written av = c — b (71 — x) = c — hn + bx. This, however, we see is really the same equation as before. Again, if c be equal to an the value of RQ is zero. Thus R now coincides with Q ; and c - bn an — hn X = , = ^— = 71. a—o a—b Hence A and B meet at Q at the end of n hours after A was at P. II. Next suppose that a is greater than h, and c less than hn. The value of x is now negative, and we may conjecture WHICH LEAD TO SIMPLE EQUATIONS. Ill from what we have hitherto observed respecting negative quanti- ft 0)1/ ties that A and B instead of meeting 7- hours after A was a— at P, will now really have met j- hours before A was at P. ' ^ a — b And in fact, since c is less than hn it follows that B was behind A when A was at P, so that A must have passed B before arriving at P. Hence the correct solution of the problem would now be as follows : R P Q Suppose that A and B meet x hours hefore A arrives at P ; let P be the point where they meet. Then PP =■ ax, and PQ = b(x + n). Also PP = PQ-PQ; thus, ax = b (x + n) — c ; therefore, x~ . . a — III. Next suppose that a is less than b, and c greater than bn. In this case also the expression originally obtained for x is negative, and we shall accordingly find tliat A and B met before A was at P. For B now travels more rapidly than A, and is before A when ^ is at P ; so that B must have passed A before A was at P. The result now is, as in the second case, that A and B met -^ hours before A was at P. b~a '' lY. Last suppose that a is less than b, and c less than bn. Here the expression originally obtained for ic is a 'positive quantity, for it may be written thus, -y . Now B traA'els more rapidly than A and is behind A when A is at P ; thus B must at some time overtake A . If we suppose A and B to meet after A is at Q, the figure will stand thus : Here we should naturally write the equation thus, ax = c + b{x — 7i) = c + bx— bn. 112 DISCUSSION OF SOME PROBLEMS If we suppose A and B to meet he/ore A is at (?, the figure will stand thus : P R Q Here we should naturally write the equation thus, ax = c — h(n — x)=c — hn + hx. In the two cases we have, however, r^eally the same equation, and we obtain x = , . — a 194. The preceding problem may be variously modified ; for instance, instead of supposing that A and JB travel in the same dii'ection, we may suppose that A travels as before, but that B travels in the opposite direction. In this case, if we suppose, as before, that A and B meet x hours after A arrived at P, we shall find that x = ;- . Thus the time of meeting will necessarily a + b be after A leaves F, and the travellers meet at some point to the right of F. The student should notice that the value of aj in the present case coincides with the result obtained by writing — b for b in the original value of a; in Art. 193. "O 195. Or instead of supposing that the arrival ot B a,t Q occurs 71 hours after the arrival of A at F, we may suppose it to occur n hours before ; and we suppose A and B to travel in the same direction. In this case if x have the same meaning as before, we shall find that x — 7- . This is a positive quantity if a is greater than b, and the travellers then really meet after the arrival of A at F. If, however, a is less than b, the value of x is a negative quantity ; this suggests that the travellers now meet -^ hours before the arrival of A at F, and on examination this b-a '' ' will be found correct. The student should notice that the value of X in the present case coincides with the result obtained by writing — n for n in the original value of x in Art. 193. WHICH LEAD TO SIMPLE EQUATIONS. 113 196. Again, let us suppose that A and £ travel in opposite directions, and that the arrival of -4 at P occui'S n hours before that of j5 at Q ; and suppose the positions of F and Q in the former figures to be interchanged, so that now A reaches Q before he reaches P, and B reaches P before he reaches Q. If x have the same meaning as before, we shall now find that x =i r- . If then h7i is greater than c, the value of a; is a positive quantity, and the travellers meet, as we have supposed, after the arrival of A at P. If however hn is less than c, the value of cc is a negative quantity, and it will be found that the travellers meet 7- hours be/ore the arrival of A at P, The student should notice that the value of x in the present case coincides with the result obtained by wi-iting — c for c in the value of x in Art. 19-1; it also coincides with the result obtained by wiiting — b for b, and — c for c in the original value of x in Art. 193. 197. From a consideration of the problems discussed in the present Chapter, and of similar problems, the student will acquii-e confidence and accuracy in dealing with negative quantities. We will lay down some general principles wliich have been illustrated in the preceding Articles, and the truth of which the student will find confirmed as he advances in the subject. (1) A negative result may arise from the fact that the enunciation of a problem involves a condition which cannot be satisfied ; in this case we may attribute to the unknown quantity a quality directly opposite to that which had been attributed to it, and may thus fonn a possible problem analogous to that which involved the impossibility. (2) A negative result may arise from the fact that a wrong supposition respecting the quality of some quantity was made when the problem was translated from words into Algebraical symbols ; in tliis case we may correct our supposition by attri- bviting the oj)posite quality to such quantity, and thus obtain a positive result. (3) Wlien we wish to alter the suppositions we have made T. A. 8 Il4i DISCUSSION OF SO>LE PROBLEMS respecting tlie quality of the known or unknown quantities of a problem, and to attribute an opposite quality to them, it is not necessary to form a new equation ; it is sufficient to change in the old equation the sign of the symbol representing each quantity which is to have its quality changed. 198. "VVe do not assert that the above general principles have been demonstrated ; they have been suggested by observation of particular examples, and are left to the student to be verified in the same manner. Thus when a negative result occurs in the solution of a problem the student should endeavour to interpret that result, and these general principles will serve to guide him. "When a problem leads to a negative result, and he wishes to form an analogous problem that shall lead to the corresponding positive result, he may proceed thus : change x into — a; in the equation that has been obtained, and then, if possible, modify the verbal statement of the problem, so as to make it coincident with the new equation. "VVe say, if j^ossible, because in some cases no such verbal modification seems attainable, and the problem may then be regarded as altogether impossible. 199. We will now leave the consideration of negative quan- tities, and examine two other singularities that may occur in results. In Art. 193 we found this result, x = r- Suppose that a— a = h, then the denominator in the value of x is zero ; thus, denot- ing the numerator by JV, we have x= — , and we may ask what is the meanins: of this result? Since A and B now travel with equal speed, they must always preserve the same distance ; so that they never meet. But instead of supposing that a is exactly equal to h, let us suppose that a is very nearly equal to b ; then T may be a very large quantity, since if a - 5 is very small Cli — compared with N, it will be contained a large number of times in K ; and the smaller a - h is, the lari^er will z be. This is ' ' "^ a-b WHICH LEAD TO SIMPLE EQUATIONS. 115 N . . . . . • . abbreviated into the phrase " — is infinite," and it is written thus, — = 00 . But the student must remember that the phrase is onli/ an abbreviation, and no absolute meaning can be attached to it. 200. The student should examine every problem, the result N of which appears under the form ^ , and endeavour to interpret that result. He m&j expect to find in such a case that the pro- blem is impossible, but that by suitable modifications a new problem can be formed which has a veiy great number for its result, and that this result becomes greater the more closely the new problem approaches to the old problem. 201. Again, let us suppose that in Art. 193 we have a-b, and also c = bn', then the value of x takes the form ^ . On examining the problem we see that, in consequence of the suj^- positions just made, A and B are together at P, and are travelling with equal speed, so that they are always together. The question, when are A and B together, is in this case said to be indeterminate, since it does not admit of a single answer, or of a finite number of answers. 202. The student should also examine every problem in which the result appeai-s under the form ^, and endeavour to intei-pret that result. In some cases he will find, as in the ex- ample considered above, that the problem is not restricted to a finite number of solutions, but admits of as many as he pleases. We do not assert here, or in Art. 200, that the interpretation of iV^ the singularities yr and tt will always coincide with those given in the simple cases we have considered ; the student must there- fore consider separately each distinct class of examples that may occur. 116 EXAMPLES. XIV. MISCELLANEOUS EXAMPLES. CHAPTER XIV. 1. Simplify tlie expression 3a-[6+{2«-(6-c)l]+| + |^i. 2. Reduce to its lowest terms the expression ~~3x' + Ux' + 22x + 2l • CC —^ Qi OC "— u Ob 3. Find the value of — -, when x — 7 . a a — o 111 4. Simplify ^—-^^-^——^ + jpZcy^^^ + (^_^)(^_j) • K Shew th.t <^'''{^-^){^-c) + h-{a-d){c-d) _h-d 5. fehew that ,,«(,,_^,)^^_^^^^.(^_,)(,_^) - ^3c when m = l, or 2. « -P , X -^ • 1 X r. a^ + 6^ + c^ - 3a5c D. Jtieduce to its simplest form z rrs — 71 r^ — ; r?. ^ (a - 6)- + {b- cf + (c - a)^ 7. 1^ xy + yz + zx = \, shew that a? 2/ z _ AiXyz n^ "*" i^ '^ T^' ~ {\-x'){i-y'){i-z'y 8. Solve the equation {x-2af + {x-2hf = 2{x-a^h)\ 9. Solve the simultaneous equations x + y + z=a + h + c, hx + cy + az = cx + ay+hz = ah + hc + ca. 10. Find the least common multiple of x^+6x^+nx + 6, x''+7x'+Ux + 8, x^ + 8a;" + 19a; + 12, and a;' + 9a;» + 26a; + 24. ANOMALOUS FORMS. 117 XY. ANOMALOUS FORMS WHICH OCCUR IN THE SOLUTION OF SIMPLE EQUATIONS. 203. We have in the preceding Chapter referred to the forms iV , . — and ^ which may occur in the solution of an equation of the first degree. We shall now examine the meaning of these forms when they occur in the solution of simultaneous equations of the first degree. We will first recall the results already obtained. 204. Every equation of the first degree with one unknown quantity may be reduced to the form ax = h. Now from this we 7 7 obtain aj = - . If a = the value of x takes the form -^ : in this a U case no finite value of x can satisfy the equation, for whatever finite value be assigned to x, since ax = 0, we have = 6, which is impossible. If a = and 6 = 0, the value of x takes the fonn p: ; in this case every finite value of x may be said to satisfy the equation, since whatever finite value be given to x we have = 0. If 6 = and a is not = 0, then of course a; = ; this case calls for no remark. 205. Suppose now we have two equations with two unknown quantities j let them be ax + hy — c and a'x + h'y = c. We will first make a remark on the notation we have here adopted. We use certain letters to denote the known quantities in the first equation, and then we use corresponding letters vnth accents to denote corresponding quantities in the second equation; here a and a' have no necessary connexion as to value, although they have this common point, namely, that each is a coefficient of Xj one in the fii'st equation and the other in the second equa- tion. Experience will establish the advantage of this notation. Instead of accents subscript numbers are sometimes used ; thus (Tj and a, might be used instead of a and a respectively. 118 ANO^klALOUS FORMS WHICH OCCUR IN THE By solving the given equations we obtain Vc — he' ac — ac h'a — ha' ' ah — ah' ' I. Suppose that h'a — ha'—O; then the vahies of x and y take A B the forms — and - ; we should therefore reciu' to the given equa- tions to discover the meaning of these results. From the relation h'a — ha' =0 we obtain - = -=h suppose ; thus a'= ha and h'— kh. By substituting these values of a and h' we find that the second of the given equations may be written thus : Jcax + hhy = c', whence, ax + hy --j-. Now if ^ be different from c, the last equation is inconsistent with the first of the given equations, because ax + hy cannot be equal to two different quantities. We may therefore conclude A B that the appearance of the results under the forms — and -r indicates that the given equations are inconsistent, and therefore cannot he solved. II. Next suppose that h'a — ha' = 0, so that — = t , and also that — = — , and therefore of course - y . In this case the nu- c a merators in the values of x and y become zero as well as the denominators, so that the values of x and y take the form - . Now by what we have shewn above, the second of the given equations may be written ax + by = j^ . C But now J =c, so that the second given equation is only a SOLUTION OF SIMPLE EQUATIONS. 119 repetition of the first ; we have thus really only one equation involving two unknown quantities. We cannot then determhie X and y, because we can find as many values as we please which will satisfy aie equation involving two unknown quantities. In this case we say that the given equations are not indeiyendent, and that the values of x and y are indeterminate. 206. We have hitherto supposed that none of the quantities a, 6, c, a\ h', c can be zero ; and thus if the value of one of the A unkno^vn quantities takes the form - or ^ the value of the other takes the same form. But if some of the above quantities are zero, the values of the two unknown quantities do not necessarily take the same form. For example, suppose a and a' to be zero ; A then the value of x takes the form - , and the value of y takes the form ^ . Now in this case the given equations reduce to by = c, and Vy = c ; these lead to y = |, and 2/ = ^,. Thus we have two cases. Firet, if -r is not equal to -=-, the c c two equations are inconsistent. Secondly, if , is equal to -, the two equations are equivalent to one only. In the second case, c c' since the relation r = r/ makes the numerator of x also vanish, u the values of both x and y take the form - ; in this case x is in- determinate but y is not, for it is really equal to j . 207. Before we consider the peculiarities which may occur in the solution of three simultaneous simple equations involving three unknown quantities, we will indicate another method of solvmg such equations. 120 ANOMALOUS FORMS WHICH OCCUR IN THE Let the equations be ax -\-hy +cz= d, a'x + h'y + c'z - d'j a"x + h"y + c"z = d". Let I and wi denote two quantities, the vahies of which are at present undetermined ; multiply the second of the given equations by Z, and the third by m ; then, by addition, we have ax + hy + CZ + 1 {a'x + h'y + c'z) + m {a"x + h"y + c"z) = d + Id' + md", that is, a; (a + la'+ma") + y{b + Ib'+ ml") + z(c-\-lcf+ mc") = d+ld'+ md''. Now let such values be given to I and m as will make the coefficients of y and z in the last equation to be zero ; that is, let h + W ^mh"=% c + lc'+mG"=0. Thus the equation reduces to £c (a 4- la + ma") =d+ ld'+ md" ; d + ld'+md" therefore, x= r-7 7,, a + la + ma "We must now find the values of I and m, and substitute them in this expression for x, and then the value of x will be known. "We have h + lh' + mh"=0, c + lc+mc'=(); from these we shall obtain y'c-hc" hc-h'c l — t / // T77~/ i ^^ = b'c"-b"c" b'c"-b"c" substitute these values in the expression for x, and after simplifi- cation we obtain _ d {b' c"- V'c') 4- d! {b"G - be") + d" {be' - b'c) ^ ~ a\b'c" - b"J) + a! {b"c - bo") + a" {be' - b'c) ' By a similar method the values of y and z may also be obtained. 208. The above method of solution is called the method of indetermiyiate muUijMers^ because we make use of multipliei-s which we do not determine beforehand, but to which a convenient value is assigned in the coui'se of the investigation. The multi- pliers are not finally indeterminate ; they are merely at first un- determined, and if it were possible to alter established language, SOLUTION OV SIMPLE EQUATIONS. 121 the word undetermined might here with propriety be substituted for indeterminate. 209. "We now proceed to our observations on the values of X, 2/, and z which are obtained from the equations arc -f by + cz = d, ax + h'y + c'z = d', a"x + h"y + d'z — d". The value of x has been given in Art. 207 ; if the student investigates the value of y he will find that the denominator of it is the same as that ichich occurs in the value of x, or can be made to be the same by changing the sign of every term in the nume- rator and denominator. The same remark holds with respect to the denominator in the value of z. 210. We may however obtain the values of y and z from the expression found for the value of x. For the original equations might have been written thus : by + ax + cz= d, b'y + ax + c'z = d', b"y + a"x + c'z — d" y we may say then that the equations in this form differ from those in the original form only in the following particulars ; x and y are interchanged, a and b are interchanged, a' and b' are interchanged, and a!' and b" are interchanged. We may therefore deduce the value of y from that of x by the follo^ving rule : for a, a' , and a!' write 6, b\ and b" respectively, and conversely. Thus, from. _ d{b'c" - b"c') + d' {b"c - be") + d" {be' - b'c) ^~ a {b'c" - b"c') + a' {b"c-bc")+a"{bc'-b'c) we may deduce that _ d {a'c" - a!'c') + d' {a"c - ac") + d" {ac' - a'c) ^ " 6 {a'c" - a"c') + b' {a"c - ac") + b" {ac' - a'c) ' It will be found on comparison that the denominator of the value of y is the same as that of the value of x with the sign of every term changed. Similarly by interchanging a, a', and a" with c, c', and c" respectively, we may deduce the value of z from that of a; ; or by interchanging b, b', and b" with c, c', and c" respectively, we may deduce the value of z from that of y. 122 ANOMALOUS FORMS WHICH OCCUR IN THE 211. There is another system of interchanges by which the values of y and z may be deduced from that of x. The given equations are ax-\-h\j ^ cz = d, a'x + h'y + c'z = d', a"x + h'-y + c"z = d" ; they may also be written thus, hy^cz-vax^ d, h'y + c'z + a'x = d\ h"y + c"z + a"x = d". We may say then that the second form differs from the first only in the following particulars j x is changed into y, y into z^ z into Xy a into 6, h into c, c into a, a' into 6', and so on. We may therefore deduce the value of y from that of x by this rule : change a into 6, h into c, c into a, and make similar changes in the letters with one accent, and in those with two accents. The value of z may be deduced from that of y by again nsing the same, rule. 212. These methods of deducing the values of ?/ and z from that of X by interchanging the letters may perhaps appear difiicult to the student at first, but they deserve careful consideration, especially that Avhich is given in Art. 211. We shall now proceed to examine the peculiarities which may occ\ir in the values of the unknown quantities deduced from the equations ax + hy -\- cz = d, a'x + h'y + c'z = d', a"x + h"y + c"z = d" . 213. The most important case is that in which d^ d', and d" are all zero. The given equations then become ax + hy + cz= 0, a'x + h'y + c'z = 0, a"x + h"y + c"z = 0. It is obvious that x = 0, y — 0, z=0 satisfy these equations ; and from the values found in Art. 210 it foUow^s that these are the only values which will satisfy the equations unless the deno- minator there given vanishes, that is, unless a {h'c" - h"c') + a' {h"c - he") + a" {he' - h'c) = 0. If this relation holds among the coefficients, the values found SOLUTION OF SIMPLE EQUATIONS. 123 for X, 7/, and z take the form - , and we must recur to the given equations for further information. We observe that when this relation holds the equations are not independent ; from any two of them the third can be deduced. For multiply the first of the given equations by h"c — b'c\ the second by be' — b"c, and the third by b'c — be', and then add the results. It will be found that by vii'tue of the given relation we arrive at the identity = 0; thus, in fact, if the first equation be multiplied by b''c - b'c", and the second equation by be" - b"c, and the two added, the result is equivalent to the third equation, for it may be obtained by multiplying that equation by be' — b'c. Suppose then that this relation holds ; we may confine our- selves to the first two of the given equations, for values of x, y, and z which satisfy these will necessarily satisfy the third equa- tion. Divide these equations by x ; thus XX XX y ca' — c'a z ab' ~ a'b ^^"^""^ X " be' -b'c ' X ^ bT^b'^ ' "We may therefore ascribe any value ive please to x, and deduce corresponding values of y and z. Or we may put our result more symmetrically thus ; let p denote any quantity whatever, then the given equations will be satisfied by x = p {be' - b'c), y = p {ca' — c'a), z^p {ab' — a'b). "We might in the same way have used the second and third of the given equations, and have omitted the first; we should thus have deduced solutions of the form x=q {b'e" - b"e'), y = q {c'a" - c"a'), z = q {a'b" - a"b'), where q is any quantity. These values however are substantially equivalent to the former ; for it will be found that by virtue of the supposed relation among the coefiicients, p{bc' — b'e) p {ca' - c'a) 2^ {^^' ~ ^'^) q(b'c"~b"c') " q{(/cr^"a:) " q{a'b"-a"b') ' 124 ANOMALOUS FORMS WHICH OCCUR IN THE 214. We shall now consider the peculiarities wliich may occur when d, d', and d" are not all zero. We shall first shew that if the value of any one of the un- N . . known quantities takes the form - , the given equations are inconsistent. Suppose, for instance, that the value of x takes this form, that is, suppose that a {h'c" - h"c) + a' {b"c - he") + a'' {be - h'c) is zero. Of course if the given equations were consistent, any equation legitimately deduced from them would also be true. Now multiply the first of the given equations by h'^c' — h'c", the second by he" — h"c, and the thii'd by h'c — he' and add. It will be found that the coefficients of y ^^^ ^ '^^ ^^ resulting equation vanish ; and the coefficient of x is zero by supposition. Thus the first member of the resulting equation vanishes, but the second member does not ; hence the resulting equation is impossible, and therefore those from which it was obtained cannot have been con- sistent. 215. We cannot however aflarm certainly, that if the value of one, of the unknown quantities takes the form - , the equations are consistent, but not independent. For it is possible that the value of one of the unknown quantities should take this form, while N the value of another takes the form — ; and, as we have shewn N . in the preceding Article, the occurrence of the form — is an indi- cation that the given equations are inconsistent. For example, suppose the equations to be ax + hi/+ cz = dj a'x + hy + ez = d\ ax + hj/ + cz = d". Here it will be found that the values of y and z take the form N — , and that of x takes the form - . SOLUTION OF SIMPLE EQUATIONS. 125 Moreover, if the values of all the unknown quantities take the form - , we cannot affirm certainly that the given equations are consistent, but not independent. For example, suppose the equations to be ax-\-by + CZ — d, ax + hi/ + cz = d\ ax + hi/ + cz= d"; here it will be found that the values of all the unknown quan- tities take the form - , but the equations themselves are obviously inconsistent, unless d^ d', and d" are all equal. 216. "We may shew that if the numerators in the values of 07, yy and «, all vanish, the denominator will also vanish, assuming that d, d', and d" are not all zero. For supposing these numerators to vanish we have d {h'c" - h"c') + d' (h"c - be'') + d" {be' - b'c) = 0, d {ca" - c'a!) + d' {c"a - ca") + d" {ca! - c'a) = 0, d {a!b"- a"b') + d\a!'b-ab") + d'! {ab' - a'b) = 0. Let us denote these relations for shortness thus, Ad + Bd'+Cd"=0, A'd + B'd'+.C'd"=0, A"d + B"d' + C"d" = 0. By Aii:. 213, since d, d' and d" are not all zero the following relation must also hold, A {B'C" - B"G') + A' {B"0 - BC") + A" {BC - B'C) = 0. It will be found that B'C" - B"C' = a{a {b'c" - b"c) + a' {b"c - be") + a" {be' - b'c)] ; and B"C - BC" and BC - B'C may be similarly expressed, so that finally the relation becomes {a {b'c" - b"c') + a {b"c - be") + a" {be' - b'c)}' = 0. Tliis establishes the required result. 217. If we adopt the method of indeterminate niultipliers given in Art. 207, it may happen that the two equations for find- ing I and m are inconsistent ; we will examine this case. Suppose then b"c — b'c" = 0, so that these two equations are inconsistent (Art. 205). In this case the value of x may be obtained from the 126 ANOMALOUS FORMS WHICH OCCUR IN THE second and third of tlie given equations, witliout using the first. For multiply the second of the given equations by d\ and the third by c', and subtract ; thus the coeflEicients of y and z vanish, and we have an equation for determining x. For example, sup- pose the equations to be 4a: + 2?/+ 3^ = 19, a; + 2/+4« = 9, a; + 2?/ + 8«= 15. Here the value of x may be found from the second and third equations ; we shall obtain a; = 3 ; substitute this value of x in the three given equations ; from the first we have 2y + 3z = 7, and from the second or third y + iz^Q; hence y = 2 and z = 1. Again, the values of I and m may take the form - , so that the two equations for finding them are not independent j we will examine this case. Here we have b'^c' — h'c' = 0, hd' — h"c = 0, and h'c — 6c'' = ; these suppositions are equivalent to the two relations h' c' h" c' T = ~ and -^r =— ' Suppose then that h' = pb, and therefore c b G ' c=pc, and that b" = qb, and therefore c" = qc. Thus the given equations are ax + bt/ + cz = d, a'x + 'pby + jpcz = d\ a"x + qby + qcz = d''^ and they may be written thus, , a' , d' a!' , d" ax + by + cz-d. — x + by + cz = — , — x + bi/ + cz= — . Here x may be found from any two of the equations ; if w^e do not obtain the same value from each pair, the given equations are of course inconsistent ; if we do obtain the same value for x^ then the given equations are not independent ; and in fact we shall in the latter case have only one equation for finding by + cz^ so that the values of y and z are indeterminate. For example, suppose the given equations to be a; + 22/ + 3;^;= 10, 3^ + 4?/ + 6,^ = 23, x-¥^yA- 9-^ = 24. From any two of these equations we can find a; = 3 ; then substituting this value of x in any one of the three equations we obtain 2y + 3;2; = 7, and thus y and z are indeterminate. If, how- ever, the right-hand member of one of the given equations be SOLUTION OF SIMPLE EQUATIONS. 127 altered, we shall not obtain the same value of x from each pair of the equations, and thus the given equations will be inconsistent. 218. In the preceding Articles we have supposed the given equations to be solved, and from the peculiar forms of the solu- tions have drawn inferences as to the nature of the given equa- tions. We will now take one example of investigating a relation between the equations without solving them. Suppose, as before, that the equations are ax + by + cz = d, a'x + h'y + c'z = d\ a'x + Vy + c"z = d"-y and let us find the relations wliich must exist among the known quantities, in order that the third equation may be deducible from the other two bj multiplication by suitable quantities and addition. Suppose then that by multiplying the first equation by X, and the second by /x, and adding, we obtain a result which is coincident with the third equation. Thus, (Xa + ika!)x + (X6 + ^V)y + (Xc + ilc')z = Xd + fxd' is equivalent to a"x + h"y + c"z = d"; that is, we suppose that \a + ii.a' _ a" \h + fxh' _ h" Xc + /mc' c" \d + ixd'~W'* Xd+fjid^'W" Xd+fjid'^W'' From the last three equations we deduce X _ a"d'-a'd" X _ h"d'-h'd" X _ c"d'-c\r ~^~ ad"-a"d' Ji ~ hd"- h"d ' ji" cd"-c"d ' Hence in order that the third equation may be deducible from the other two in the manner proposed, we must have the follow- ing relations among the known quantities, a"d'- a'd" _ V'd'- h'd" _ d'd'-cd" ad" -ad ~M'-h"d~ cd"-c"d ' It is easy to shew that if these relations hold, the values of x, y, and z take the foiTu - . For by multiplying up we obtain results which shew that the numerators in the values of x, y, and z vanish; and then by Art. 216 the denominator will also vanish. 128 EXA3IPLES. XV. MISCELLANEOUS EXAMPLES. CHAPTER XV. 1. Keduce -^ — t-^. — ——^7 — n rr to its simplest form. 2. Shew that {a + h + c) {a^+ b^+ c^+ ahc) - {ah + hc + ca) {a' + b^ ^- c') = a* + h* + c\ 2 2 2 2 relation between t and x. 4. \i 2s = a + h + c, shew that » 1 1 1 1 ahc + — , + s — a s — b s — c s s(s — a){s — b)(s — c)' 5. Shew that the g. c. 5L of two quantities is the L, c. M. of their common measures. 6. Solve the equation (x -9){x- 7) (x -5)(x- 1) = {x -2){x- 4) (x-6){x-l 0). 7. Solve the simultaneous equations x + y + z = Oy ax + by + cz= 0, hex + cay + abz + (a — b) {b- c) (c — a) = 0. 8. If - + 7 + - = 1 , shew that a c a + + c 1 1 1\'"+' 1 \a cj ^" + 1 JL. ?)2"+' 4- /.2"+' * a 9. A person leaves £12670 to he divided among his five children and three brothers, so that after the legacy duty has been paid, each child's share shall be twice as great as each brother's. The legacy duty on a child's share being one per cent, and on a brother's share three per cent., find what amounts they respectively receive. 10. Solve the equation 12 3 6 aj + 6a x — Za x + 2a x + a INVOLUTION. 129 XVI. INVOLUTIOK 219. If a quantity be continually multiplied by itself, it is said to be involved or raised, and the power to which it is raised is expressed by the number of times the quantity has been em- ployed in the multiplication. The operation is called Involution. Thus, as we have stated (Art. 16), ax a or a^ is called the second power of a; a x a x a or a^ is called the third power of a; and so on. 220. If the quantity to be involved have a negative sign prefixed, the sign of the eveji powers will be positive, and the sign of the odd powers will be negative. For, — a X — a = a', — a x — a x — a = a^ x — a = — a^y — a X — a X — a X — a = — a^ X — a = o^f and so on. 221. A simple quantity is raised to any power by multiply- ing the index of every factor in the quantity by the exponent of that power, and prefixing the proper sign determined by the pre- ceding Article. Thus cr raised to the ?i^^ power is oT"; for if we foiTQ the product of n factors, each- of which is a"*, the result by the ride of multiplication is cC"". Also (ah)" = ah x ah x ab... to n factoi-s, that is, a X a X a .. to n factors xh xh x b... to n factors, that is, a" X 6". Similarly, a'h^c raised to the fifth power is a'%^^c^. Also — a"* raised to the n^'^ power is ± a""', where the positive or negative sign is to be prefixed according as n is an even or odd number. Or as - a" = - 1 x a"*, the n^^ power of - a"* may be written thus (- 1)" x a"" or (- l)^*'"". 222. If the quantity which is to be involved be a fraction, both its numerator and denominator must be raised to the pro- posed power. (Art. 142.) T. A. 9 130 INVOLUTION. 223. If the quantity wliich is to be involved be com^yound, the involution may either be represented by the proper index, or may actually be performed. Let a + b be the quantity which is to be raised to any power, a + b a' + 2ab +b' a' + 3a'b + 3ab' + b' a + b a + b a + b a^ + ab a^ + 2a'b + ab' a' + 3a'b + 3a'b'+ab' + ab + b' + a'b + 2ah' + b' + a% + 3a'b' + 3ab' + b' a'+2ah- 4-6' a' + da'b + 2>ab' + b' a* + 4ra'b + 6a'b' + iab' + b' Thus the square or second power of a + b is a^ + 2ah + ¥, the cube or third power of a + b is a^ + 3a^6 + 2>ab'^ + b^, the fourth power of a + 6 is a* + ia^b + Qa^b^ + iab^ + U, and so on. Similarly, the second, tliird, and fourth j^owers oi a — b will be found to be respectively a' — 2ab + b', a^ - 2>a^b + 3a6^ — b^, and a* — ia% + Qa'b'— iab^ + b*} that is, wdierever an odd power of b occurs, the negative sign is prefixed. "We shall hereafter give a theorem, called the Binomial Theo- rem, which will enable us to obtain any power of a binomial ex- pression without the labour of actual multiplica^tion. 224. It is obvious that the n^^ power of a"* is the same as the power of a", for each is a"'"; and thus we may arrive at the m th same result by different processes of involution. We may, for example, find the sixth poY\^er oi a + b by repeated multiplication by a + b ', or we may first find the cube of a + b, and then the square of this result, since the square of {a + by is (a + bf ; or we may first find the square of a+b and then the cube of this result, since the cube of (cc + by is (a + by. 225. It may be shewn by actual multiplication that (a+b + cy = a^ + b- + c' + 2ab + 2bG + 2ac, {a + b + c + dy = a' + b' + c'+d' + 2ab + 2ac +2ad + 2bc + 2bd+ 2cd, The following rale may be observed to hold good in the above and similar examples : the square of any rindtinomial consists of EXAMPLES. XVI. 131 the square of each term, together with twice the product of every pair of terms. Another fonn may also be given to these results, (a + 6 + c)- = a' + 2a {h + c) + h' + 2bc + c", {a+b + c + ciy = a^ + 2a {b +c + d) + h- + 2b (c + d) + c' + 2cd + d\ The following inile may be observed to hold good in the above and similar examples : the square of any multinomial consists of the square of each term, together with twice the p)^'oduct of each term hy the sum of all the terms which follow it. These iiiles may be strictly demonstrated by the process of mathematical induction, which will be explained hereafter. 226. The following are additional examples in which we employ the first of the two rules given in the preceding Ai-ticle. (rt - 6 + cy = a^ + b^ + c- - 2ab - 2bc + 2ac, (1 - 2a^+ 3.r-y = 1 + 4a;' + 9a;'- 4a:- 12a/ + 6a;' = l-4a; + 10a;'-12a;^ + 9a>*, (1 + a; + a;' + a;')' = 1 + a;' + a;* + a' -f 2a; + 2a;' + 2x^ + 2x^ + 2a;* + 2x' = 1 + 2a; + 3a;'+ 4a;' + 3 a;* + 2a;'' + a;'. 227. The results given in Art. 55 for the cube of a + 6, the cube of rt - 6, and the cube of a + 6 + c should be carefidly noticed. The following may also be verified. {a -vb + c + df = a^ + W + c^ + d"^ + 3a' {b + c + d) + 36' {a + c + d) + 3c' {a + b + d) + 3 J' {ct^b + c) + (jbcd + Qacd 4- Qabd + Oabc. EXAMPLES OF INVOLUTION. 1. rind (1 + 2a; + 3a;')'. 2. Find (1 - a; + a;' - a/)'. 3. Find {a + b- c)^ 4. Find ( 1 + 2a; + a;')'. 5. Find (1 + 3a; + 3a;' + x'Y + (1 - 3a; + 3a;' - x^. r Qi .1. . (27a*-18a'6 '- bj ^ (9a' - 6')" (6' - a') G. Shew that ' ^. .», -f ^-^ /m-^a =o. 64a b4a 6 9 - 2 132 EXAMPLES. XVI. 7. Shew that (ax' + ^b^iytj + ci/) {aX^ + 2hXY + cY') = [axX+ cyY+ b {xY + yX)Y + {ac - J/) {xY- yX)\ 8. Shew that {x' + ;?a:y + qy') {X ' + pX Y + qY') = (a;X+;;y.r+ (/y F)' +;; {xX+p7jX+ qyY) {xY- yX)+q{xY- yXf and also = {xX+pxY+ Qy^^y +2){xX+2JxY+ qyY)(yX-xY) + q{xY-yXy. 9. Simplify (1 - IQj;'^ + 5x'} (5 - 30x' + 5x') + {5x - lOx^ + x') (20a; - 2Qj;^) {5x - 10a;' + xj + (1 - lUa;' + 5xy ' 1 0. Shew that {a' + h' + c' + d') (// + q' + r + s') = [ap — hq + cr — dsf + {aq +hp — cs — dry + {ar — hs~ cp + dq)' + {as + hr + cq + dpf. X VII. E YOLUTIOK 228. Evolution, or the extraction of roots, is the method of deter min ing a quantity, which when raised to a proposed power ^vdll j^roduce a given quantity. 229. Since the n^^ power of cC^ is oT", an ?i'^ root of a""" must he 0-"* ; that is, to extract any root of a simple quantity, we divide the index of that quantity by the index of the root requii'ed. 230. If the root to be extracted be expressed by an odd . number, the sign of the root will be the same as the sign of the proposed quantity, as appears by Art. 220. Thus, l!(-a^)=-a. 231. If the root to bo extracted be expressed by an even number, and the quantity proposed be positive, the root may be either positive or negative ; because either a positive or negative tiuantity raised to an even pov/er is positive by Art. 220. Tlius, J{a-) = ± a. 232. If the root proi)osod to be extracted be expressed bv' an even number and tlio sign of the proposed quantity be negative, EVOLUTION. 133 the root cannot be extracted ; because no quantity raised to an even power can produce a negative result. Such roots are called impossible. 233. A root of a fraction may be found by taking that root of both the numerator and denominator. Thus, V \FJ b \/ \ by b 234. We will now investigate the method of extracting the square root of a compound quantity. Since the square root of ct^ + 2a5 + b^ is a-\-b, we may be led to a general rule for the extraction of the square root of an alge- braical expression by observing in v/hat m'lnner a aul b may be derived from a' + 2ab + b'. a^ + lab + b- ^a + b a' la + b) 2ab + b' 2ab + b' Arrange the terms according to the dimensions of one letter a, then the first term is a^, and its square root is a, which is the first term of the requii'ed root. Subtract its square, that is a*, from the whole expression, and bring down the remainder 2ab + b'. Divide 2ab by 2a and the quotient is b, which is the other term of the required root. Multiply the sum of twice the first terai and the second term, that is 2a + b, by the second term, that is 6, and subtract the product, that is 2ab + h^, from the remainder. This finishes the operation in the present case. If there were more terms we should proceed with a-\-b as we did formerly \\dth a ; its square, that is a^ + 2ab + b^, has already been sub- tracted from the proposed expression, so we should divide the remainder by the double of a + 6 for a new term in the root, and then for a new subtrahend we should multiply this tenii by the sum of t\vice the former terms and this term. The process must be continued until the required root is found. 134 EVOLUTION. 235, For example, required the square root of tlie expres- sion Ax* - 12u;^ + 6x^ + Ca; + 1. Ax* - 12a:' + 5x' + 6a; + 1 (^2x' - 3a' - 1 4a;'' 4:o'' - 3a;; -1 2a;' + 5a;* + 6a: + 1 -12a;'+9x« 4a;'-6.r-i; -4a;'+6a; + l - 4a;" + 6a; + 1 Here the square root of 4a;* is 2a;^, which is the first term of the requii'ed root. Subtract its square, that is 4a;'*, from the whole expression, and the remainder is — 12a;' + 5a;^ + 6a; + 1. Divide — 1 2a;' by twice 2a;^, that is by 4a;^, the quotient is — 3a;, which will be the next term of the required root ; then mul- tiply 4a;^ — 3a; by — 3a; and subtract, so that the remainder is — 4a;' 4- 6a; +1. Divide by twice the portion of the root already found, that is by 4a;^ — 6a; ; this leads to — 1 ; the product of 4a;^ — 6a; — 1 and — 1 is — 4a;^ + 6a; + 1, and when this is subtracted there is no remainder, and thus the required root is 2a;^— 3a;— 1. For another example, required the square root of the expres- sion x^ - Qax^ + 15aV - 20ct'a;' + loaV - Qa^x + a^. The operation may be arranged as before, x' _ 6aa;'+ 15aV- 20a'a;'+15aV- 6a'x+a' {x'-3ax'+3a'x - a^ x' 2x' - dax'J - 6aa;' + 15aV - 20a'a;' + 15aV - 6a^a; + a' — 6aa;' + 9a V 2x^ - Gax' + Za'xJ Qa'x* - 20a'a;' + 15aV - Ga'x + a' 6aV-18a'a;'+9aV 2a;' - 6ax' + 6a'a; - a' J - 2a'a;' + 6a*a;' - 6a'a; + a' - 2a'x' + 6a*x' - (Sa'x + a' EVOLUTION. 135 236. It lias been already remarked, that all even roots admit of a double sign. (Art. 231.) Thus in the first example of Art. 235, the expression 2x'' — 3a; — 1 is found to bo a square root of the expression there given, and — 2x' + 3a; + 1 will also be a square root, as may be verified. In fiict, the process commenced by the extraction of the square root of 4a;*, and this might be taken as 2a;' or as — 2a;^ ; if we adopt the latter and continue the opera- tion in the same manner as before, we shall arrive at the result — 2.x'*+ 3a; + 1. Similarly in the second example of Art. 235 wo see that — a^ + 3aa;^ — 3a'a; + a^ will also be a square root. 237. The fourth root of an expression may be found by ex« tracting the square root of the square root. Similarly the eighth root may be found by three successive extractions of the square root, and the sixteenth root by four successive extractions of tho square root, and so on. For example, required the fourth root of the expression 81a;* - 432a;' + 864a;' - 76Sa; + 256. Proceed as in Art. 235, and we shall find that the square root of the proposed expression is 9a;*— 24a; + 16; and the square root of this is 3a; — 4, which is therefore the fourth root of the proposed expression.' 238. The preceding investigation of the square root of an Algebraical expression will enable us to prove the rule for the extraction of the square root of a number, which is given in Arithmetic. The square root of 100 is 10, of 10000 is 100, of 1000000 is 1000, and so on ; hence it will follow that the square root of a number less than 100 must consist of only one figure, of a number between 100 and 10000 of two places of figures, of a number be- tween 10000 and 1000000 of three places of figures, and so on. If then a point be placed over every second figure in any number beginning with the units, the number of points will shew the number of figures in the square root. Thus the square root of 4356 consists of two figures, the square root of 611524 of three figures, and so on. 136 EVOLUTION. 239. Suppose the square root of 435G required. Point the number according to . . the rule : thus it appears that the 4 3 5 6 (^ GO + 6 root consists of two places of figures. '^ Let a + b denote the root, whore a is 120 + 6^)750 the value of the figure in the tens' 7 5 G place, and b the figure in the units' place. Then a must be the greatest multiple of ten which has its square less than 4300 ; this is found to be 60. Subtract a", that is the square of 60, from the given number, and the remain- der is 756. Divide this remainder by 2a, that is by 120, and the quotient is 6, which is the value of b. Then (2a + b) b, that is 126 X 6 or 756, is the quantity to be subtracted; and as there is now no remainder, we conclude that 60 + 6 or 66 is the required square root. It is stated above that a is the greatest multiple of ten which has its square less than 4300. For a evidently cannot be a greater multiple of ten. If possible suppose it to be some multi- ple of ten less than this, say x ; then since x is in the tens' place, and b in the units' place, x + b is less than a ; therefore the square ofx + b is less than a^, and consequently x + b is less than the true root. If the root consist of three places of figures, let a represent the hundreds and b the tens; then having obtained a and b as before, let the hundreds and tens together be considered as a new value of a, and find a new value of b for the units. The c^-^Dhers may be omitted for the sake of brevity, and the following rale may be obtained from the process. Point every second figure beginning with the units' place, and thus divide the whole 4 d o b (^b b number into several periods. Find the great- '^ " est number whose square is contained in the 12 6^756 first period; this is the first figure in the 7 5 6 root ; subtract its square from the first period, ' EVOLUTION. 137 and to the remainder bring down the next period. Divide this quantity, omitting the last figure, by twice the part of the root already found, and annex the result to the root and also to the divisor, then multiply the divisor as it now stands by the part of the root last obtained for the subtrahend. If there be moi-e periods to be brought down the operation must be repeated. 240. Extract the square root of G11524 ; also of 1024G401. 6il524(^782 l624G40i(,3201 4 9 9 1 4 8J 1 2 1 5 6 2^124 118 4 12 4 ir)G2;3124 6401;G4 1 3 12 4 G 4 1 In the second example the student should observe the occur- rence of the cypher in the root. 241. The rule for extracting the square root of a decimal follows from the preceding iiile. We must observe, however, that if any decimal be squared there will be an even number of decimal places in the result, and therefore there cannot be an exact square root of any decimal which in its simplest state has an odd number of decimal places. The square root of 21 wG is one-tenth of the square root of 100 X 21 "7 G, that is of 217G. So also the square root of -OSGl is one-hundredth of that of 10000 x -OSGl, that is of 3G1. Thus w-e may deduce this rule for extracting the square root of a decimal : put a point over every second figure beginning at the units' place, and continuing both to the right and left of it ; then proceed as in the extraction of the square root of integers, and mark oflT as many decimal places in the result as the number of periods in the decimal part of the proposed number. 138 EVOLUTION. 242. The student will probaLly soon acquire tlie conviction that many integers have strictly speaking no square root. Take for example the integer 7. It is obvious that 7 can have no integer for its square root ; for the square of 2 is less than 7, and the square of 3 is greater than 7. Nor can 7 have any fraction as its square root. For take any fraction which is strictly a fraction and not an integer in a fractional form, and multiply this fraction by itself ; then the product will be a fraction : this statement can be verified to any extent by trial, and may be demonstra,ted by the principles of Chapter Lii. Thus 7 has no square root, either integi'al or fractional. In like manner no integer can have a square root unless that integer be one of the set of numbers 1, 4, 9, 16, ... which are the squares of the natural nimibers 1, 2, 3, 4, ..., and are called square numhcrs. 243. In the extraction of the square root of an integer, if there is still a remainder after we have arrived at the figure in the units' place of the root, it indicates that the proposed number has not an exact square root. We may if we please proceed wdtli the approximation to any desired extent Ly supposing a decimal point at the end of the proposed number, and annexing any even number of cyphers and continuing the operation. We thus obtain a decimal part to be added to the integral part already found. It may be observed that in such a case by continuing the process w^e shall not arrive at figures in the root which circulate or recur. For a recurring decimal can be reduced to a fraction by a rule eriven in books on Arithmetic, and which will be demon- strated in Chapter xxxi ; and therefore, if the square root were a recun-ing decimal it could be expressed as a fraction, and so there would be an exact square root, which is contrary to the supposition. Similarly, if a decimal number has no exact square root, we may annex cyphers and proceed with the approximation to any desired extent. EVOLUTIOX. 13.) 244. The following is tlie extraction of tlie square root of twelve to seven decimal places. 1 2-0 6 . . . (^3-4 G 4 1 1 6 9 6 4^3 2 5 G G8GJ4400 4 1 1 G G92 4;28400 2 7 G 9 6 69281^70400 6 9 2 8 1 G92820i;ill9000 0^ 6 9 2 8 2 1 6928202 G;42617 9 9 4 15 6 9 2 15 6 10 4 8 7 7 4 4 Thus we see in what sense we can be said to approximate to the square root of 12 : the square of 3-4641016 is less than 12, and the square of3'4641017 is gi-eater than 12 ; the former square differs from 12 bv the fraction which has 10487744 for numerator and 10'* for denominator. 245, It can be demonstrated by the principles of Cliapter Lir. that no fraction can have a square root unless the numerator and denominator are both square numbers when the fraction is in its lowest terms. But w-e may approximate to any desii'ed extei.t to the square root of a fraction. 140 EVOLUTION. r ^ Suppose for example we require the square root of = . y3 /3 - = '^ - ; then approximate to the square root of 3 and to the square root of 7, and divide the former result by the latter. But the following methods are preferable. 3 Convert j- into a decimal to any required degi'ee of approxi- mation ; and approximate to the square root of this decimal. Or proceed t,„.: /^ = /^.^^I = .Ji^= ^^l; then approximate to the square root of 21 and divide the result i>y 7. 246. When n + 1 figures of a square root have been obtained hy the ordinary method, n Tiiore r)iay he obtained by division only, supposing 2n + 1 to be the whole number. Let N represent the number whose square root is required, a the part of the root already obtained, x the pai^t which remains to 1 )e found ; then J^= a + x, so that iV = a" + 2ax + x^, therefore, N — a^ = 2ax + x^, A — a x and — = ^ + ?r- • '2a la Thus N — a^ divided by 2 a will give the rest of the square X' Ou root requii-ed, or a;, increased by ■^; and we shall shew that — is a proper fraction, so that by neglecting the remainder arising from the division we obtain the part required. For x by sup- position contains n digits, so that x^ cannot contain more than X' 271 digits ; but a contains 2?i -t- 1 digits, and thus — is a proper Aa fraction. EVOLUTION. 14 1 The above demonstration implies tliat N is an integer with an exact square root : but we may easily extend the result to other cases. For example, suppose we require the square root of 12 to 4 places of decimals. We have in fact to seek the square root of 1200000000, and to divide the result by 10000. Now the process in Art. 244 shews that 1200000000 - 1119 = (34G41)'. Here JV may stand for 1200000000-1119; and then a may stand for 34600 and b for 41. Thus the demonstration assures us that we can obtain 41 by dividing 2840000 by 69200, that is bv di\ddinG: 28400 bv 692 : and this coincides with the mle 2:iven in books on Arithmetic. In like manner if we require the square root of 12 to 6 places of decimals, the last three figures, namely 101, can be obtained by divicUn^ 704000 bv G928. 217. We will now investigate the method of extracting the cube root of a compound quantity. The cube root of a^ + oa'^b + 3aO^ + h^ is a + b, and to obtain this we mav proceed thus : Arran-j^e _ „ ,,. „ ,„ ,., , " ^ ,. , .. a^ + da'b-rdab^ + b' La + b tiio terms accordinij to the dimen- , ci .sions of one letter a, then the first . tei-m h a^ and its cube root is a, ^a'^ oa'b 4- 3ab^ + b^ which is the first tenn of the re- 3a-b + 3ab' + b^ quired root. Subtract its cube, that is a^, from the whole expression, and bring down the remainder oa^b -f oab^ + b^. Divide the first tenn of the remainder by Sa", and the quotient is b, which is the other term of the required root ; then subtract Sa'^b + oab^ + b^ from the remainder, and the Avhole cube of a + b has been subtracted. This finishes the oi)era- tion in the present case. If there were more terms we should proceed with a + 6 as we formerly did with a ; its cube, that is a^ -4 Sirb + 3ab' + b^, has already been subtracted from the j)ro- itoscd expression, so we should divide the remainder by 3 (a + bf ibr a new term in the root ; and so on. H2 EVOLUTION. 218. It v/ill be convenient in extracting the cube root of more complex algebraical expressions, and of numbers, to iirran^-e the process of the preceding Article in three columns, as follows : 3a + h 3rt' a' + ocirh + 3a6' + h^ {a + h (3a + h)h a^ ia'-¥'dah~+h' 3a'b + dab' + b' 3a'b + 3ab' + b' Find the first term of the root, that is a ; put a^ under the given expression in the third column and subtract it. Put Za in the first column, and 3ct^ in the second column ; divide 3a*6 by 3a.*, and thus obtain the quotient b ; add b to the quantity in the first column ; multiply the expression now in the first column by b, and place the product in the second column and add it to the quantity already there; thus w^e obtain Sa^ + Sab + b"; multiply this by b and we obtain 3a'b + 3a¥ + 6^, which is to be placed in the third column and subtracted. We have thus com- pleted the process of subtracting (a + by from the original ex- pression. If there were more terms the process would have to be continued. 249. In continuing the operation we must add such a quan- tity to the first column as to obtain there three times the part of the root ahead?/ found. This is conveniently effected thus : we have already in the first column 2>a + b ; <^, > place 2b under the b and add ; so we obtain 3a + 36, which is three times a + b, that is, three times the ^^ + ^^ part of the root already found, Moreover, we must add such a quantity to the second column as to obtain there three times the square of the imrt of the root already found. This is conveniently efiected thus : we have ^ ^ ^ already in the second column (3a + b) b, and ^"' "^ ^"^ + ^^ below that 3a^ -f- 3a6 + b' ; place b^ below and add tJie expressions in the three lines ; so we Za^ + Qah + 36* obtain 3a^ + 6a6 + 36^, which is three times EVOLUTION. l-tS (a + 5)', that is, tlirec times the sqiiaiv of the part of the root already found. 250. Example; extract the cuue root of 6a;' — 3cx| 12a;* — Gojj — 3cjc (6.x;' — 3t;a:;) 12a;*-18c;r-^+9cVj> Go;'^ — dcx + c" + 9cV. 12x*-36ca;'+27cV + c'"(6a;'- ■ Ocx* + c") 1 2^" - 3Gca;' + 33c V- ^c^x f c* 8a;' - 36ca;* + 66cV - G3cV + 33cV - cx + c' ^x' - 36ca;' + Q>Qc"x' - G3cV + 33cV - Oc'^u; + c" -36ca;^+54cV-27cV 12cV - 36cV + 33cV - ^c'x + c" 1 2cV - 3Gc^^;' + 33cV - ^c'x + c' The cuhe root of 8a;® is 2x^ ^yhich will be the first term of the root; put 8a;'' under the given expression in the third column and subtract it. Put three times 2x' in the first column, and three times the square of 2a;^ in the second column ; that is, j^ut Q)x^ in the first column, and 12a;^ in the second column. Divide — 36ca;* by 12a;^, and thus obtain the quotient — 3ca;, which will be the second term of the root; place this term in the first column, and multiply the expression now in the first column, that is, 6a;' — 3ca; by — 3fa;; place the product under the quantity in the second column and add it to that quantity; thus we obtain 12.«*- 18ca;^+ 9c'V; multiply this by - ocx, and jJace the product in the third column and subtract. Thus we have a remainder in the third column, and the part of the root already found is 144< EVOLUTION. We must now adjust the first and second cohunns in the manner exphiined in Art. 249. "VVe put twice — 3cx, that is, — 6cx, under the quantity in the first column, and add the two lines ; so we obtain 6a;' — 9cx, which is three times the part of the root already found. We put the square of — 3cx, that is, 9c^^•', under the quantity in the second column, and add the last three lines in this column; so we obtain 12x^— 36cx^ + 27c'x'', w^hich is three times the square of the part of the root already found. l!s^ow divide the remainder in the third column by the ex- pression just obtained, and we arrive at c" for the last term of the root; proceed as before and the operation closes. 251. The preceding investigation of the cube root of an Algebraical expression will enable us to deduce a rule for the extraction of the cube root of any number. The cube root of 1000 is 10, of 1000000 is 100, and so on; hence it will follow that the cube root of a number less than lOOO must consist of only one figure, of a number between 1000 and 1000000 of two places of figures, and so on. If then a point be placed over every third figure in any number beginning with the units, the number of points will shew the number of figures in the cube root. 252. Suppose the cube root of •105221: requii-ed, 405224(^7 + 4 3 4 3 2 1 + 4 14 7 8 5 6 I 5 D 6 6 2 2 2 4 6 2 2 2 4 Point the number according to the rule ; thus it appears that tlie root consists of two places of figures. Let a + b denote the root, where a is the value of the fig-ure in the tens' place, and b the figure in the units' place. Then a must be the greatest multi- ple of ten which has its cube less than 405000; that is, a must be EVOLUTION. 145 70. Place the cube of 70, that is 313000, iii the third column under the given number and subtract. Place three times 70, that is 210, in the first column, and thi-ee times the square of 70, that is 14700, in the second column. Divide the remainder in the third column by the number in the second column, that is, divide G2224: by 14700; we thus obtain 4, which Ls the value of b. Add 4 to the first column ; multiply the sum thus formed by 4, that is, multiply 214 by 4; we thus obtain 856; place this in the second column and add it to the number already there. Thus we obtain 15556; multiply this by 4, place the product in. the third column and subtract. The remainder is zero, and therefore 74 is the re- quired root. The cyphers may be omitted for brevity, and the process will stand thus : 2 1 4 1 4 7 8 5 6 15 5 5 6 4052 24(74 6 2 2 2 4 6 2 2 2 4 253. Example; extract the cube root of 12812904. 6 3 6 6 9 4 1 2 1 8 9^ 13 8 9 9J 15 8 7 2 7 7 6 16 14 7 6 12 812 904(234 4 8 12 4 16 7 6 4 5 9 04 6 4 5 9 4 After obtaining the first two figures of the root 23, we adjust the first and second columns in the manner exj^lained in Art. 249. We place twice 3 under the first column and add the two lines giving 69, and we place the square of 3 under the second column and add the last three lines giving 1587. Then the operation is continued as before. The cube root is 234. T. A. 10 14G EVOLUTION. 254 Example; extract the cube root of U4182818617453. 144182818617453 (52437 125 19182 15G08 1572 3) 8112 3574818 g/ 625 6^ 3269824 157297 817456> 304994617 leJ 247259907 823728 57734710453 47169^ 57734710453 82419969 9 82467147 1101079 8247815779 The cube root is 52437. 255. If the root have any number of decimal places the cube will have thi'ice as many; and therefore the number of decimal places in a decimal nimiber, which is a perfect cube, and in its simplest state, will necessarily be a multiple of three, and the number of decimal places in the root will be a third of that niimber. Hence if the given cube number be a decimal, we place a point over the units' figure, and over every third figure to the right and left of it ; then the number of points in the decimal part of the proposed number will indicate the number of decimal places in the cube root. If a number have no exact cube root we may, as in the ex- traction of the square root, proceed with the approximation to any desired extent. See Art. 243. EVOLUTION. 256. Kequir ed the cube root of 1481*544. 31) 2/ 3 3 n i 4 8 i -5 4 4 (^11-4 1 3 34 3 3 1^ 1. 48 1 3 3 1 3 G 3 13 3 15 5 4 4 15 5 4 4 3 7 6 3 G Tlie cube root is 11-4. 147 257. Whe?i n + 2 figures of a cube root have been obtained by tlie ordinary viethod, n 7)iore may be obtained by division only, sujjposing 2n + 2 to be the whole number. Let N represent the number whose cube root is required, a the part of the root already obtained, x the part which remains to be found ; then 1^1 X = a + X, so that N = a? + Zaix + Zaoi? + o? ; therefore, X — a? — 3a~x + 3ax^ + x^, , X-a^ ic^ x^ and - ^^ ^ ^ + _ + oa a 3a Thus X-a^ divided by 3a' will give the rest of the cube .2 3 root required, or x, increased by h ^^^ ; and we shall shew that the latter expression is m j^roper fraction, so that by neglect- ing the remainder arising from the division, we obtain the part required. For by supposition, x is less than 10", and a is not x^ 10^" 1 less than 10^""^'; thus — is less than „^^ , that is, less than y^ . x^ 10^" 1 And ^-^ is less than - — ,.^^ , that is, less than - — ycv^^ ' -^^^^^ x' a? . 11 — + ;r— o is less than r— + - — _ „„.„ , and is thus less than unitv. a oa 10 o X iU Remarks similar to those in the latter part of Art. 246 apply here. 10—2 148 EXAMPLES. XVII. EXAMPLES OF EVOLUTION. Extract the square roots of the expressions contained in the following examples from 1 to 15 inclusive : 1. a;*-2a;' + 3a;'-2x' + l. 2. a;' - 4a;' + 8x + 4. 3. ix'+Ux^' + dx'-Qx + l. 4. 4a;' - 40;" + 5a;'- 2a; + 1. 5. 4a;* - 12aa;' + 2oaV - 24a''a; + 1 6a\ 6. 25a;' - 30aa;' + 49a V - 24a'a; + IQa*. 7. x' - Qax' + 1 oa V - 20a^x^ + 1 5a V -Qa^x+ a\ 8. (a -hY-2 (a» + h') {a - 6)' + 2 {a* + h'). 9. 4 {{a' - h') cd + ah (c' - d')Y + {{a' - ¥) {c' - d') - iahcd]'. 10. a' + b* + c' + d*- 2a' (6» + d') - 2b' {c' - d') + 2c' (a' - d'). 11. (..1)'-, (._!). 12. a;'-a;' + ^ + 4a;-2 + 4 4 X .^ a* a a x 13. — +— +— „-aa;-2 +—2 4 a; ar or 14. a' + 2 (26 - c) a' + (46= - 46c + 3c=) a' + 2c= (26 -c)a + c\ 15. (a - 26)' x" -2a {a- 2b) x^ + (a' + ^ab - 6a - 86' + 126) x^ -(4a6-6a)a;+46'-126 + 9. 16. Find the square root of the sum of the squares of -2, "4, •6, -86. Extract the cube root of the expressions and numbers in the following examples from 17 to 23 inclusive : 17. a;' - 9a;' + 33a;' - 63a;' + 66a;' - 36a; + 8. 18. 8a;' + 48ca;' + 60c''a;' - 80cV - 90cV + lOSc'a; - 27c^ 19. 8a;" - 36ca;^ + 102c'a;' - 17 IcV + 204cV - 144c*a; + 64c'. 20. 167-284151. 21. 731189187729. 22. 10970-645048. 23. 1371742108367626890260631. 24. Extract i\\e fourth root of (a;' + -^ j - 4 (« + - j + 12. EXAMPLES. XVII. 149 25. If a number contain n digits, its square root contains i{2;i + l-(-l)''} digits. 26. Shew that the following expression is an exact square : {x^ - yzf + (y** - Z3^ + (^ - ^y - 3 (a;- - yz) i^f - zx) {z^ - xy). XYIII. THEOKY OF INDICES. 258. We have defined a"*, where m is a positive integer, as the product of m factors each equal to a, and we have shewn that m 1 oT -K a"* = a""^", and that — = a-"'"" or —^^:z^ according as m is greater or less than n. Hitherto then an exponent has always been a positive integer ; it is however found convenient to use exponents Avhich are not positive integers, and we shall now explain the meaning of such exponents. 259. As fractional indices and negative indices have not yet been defined, we are at liberty to give what definitions we please to them ; and it is found convenient to give such definitions to them as will make the important relation oT x a" = a""^" always true, whatever m and n may he. For example j required the meaning of a'-'. By supposition we are to have a^ x a^ — a' = a. Thus a^ must be such a number that if it be multiplied by itself the result is a ; and the square root of a is by definition such a nimiber ; therefore a^ must be equivalent to the square root of a, that is, a^ = J a. Again ; required the meaning of a^. By supposition we are to have or y. or y. or = ar*^""^ = a^ = a. Hence, as before, or must be equivalent to the cube root of a, that is a* = ^a. loO THEORY OF INDICES. 3 Again ; required the meaning of or. 3 3 3 3- By supj)osition, a' x a' x a* x a' = a ; therefore c^ = >,J/a\ These examples would enable the student to understand what ia meant by any fractional exponent ; but we will give the defini- tion in general symbols in the next two Ai'ticles. 260. Required the meaning of iv^ where n is any positive lohole number. By supposition, 1 -1 - ^ .f. ^ ^. ^ 4- ... to )» t«nii3 , a" X a" X a" x ... to n factors = a" '* ** =a ^a ; therefore a" must be equivalent to the n^^ root of a, that is, a» = H^a. 261. Required the meaning o/ a''* where m and n are any j^osi- tive whole numhers. By su2:>position, m on in ni in m , , — . „ . +- H (-. . . to n terius a" xa" X a" x ... to n factors = a" » " =^ a ; therefore a" must be equivalent to the n}^ root of a'", that is, a^ = ^a". Hence a" means the n}^ root of the 'Wi*^ power of a ; that is, in a fractional index the numerator denotes a power and the denominator a root. 262. We have thus assigned a meaning to any positive index, whether whole or fractional ; it remains to assign a meaning to negative indices. For example, required the meaning of «~^ By sui)position, a^ x a~^ — a^~' = n} = a, therefore a~' = — = — . a a We will now give the definition in general symbols. THEORY OF INDICES. 151 2G3. Required the meaning of a~" ; where n is any positive number whole or fractional. By supposition, whatever m may be, we are to have Now we may suppose m positive and greater than w, and then, by what has gone before, we have and therefore a"""" — — . Therefore a"* x a"" = a a"* therefore " " 1 a = — . a' In order to express this in words we will define the word reciprocal. One quantity is said to be the reciprocal of another when the product of the two is equal to unity ; thus, for example, X is the reciprocal of - . Hence «"" is the reciprocal of a" ; or we may put this result symbolically in any of the foUo^ving ways, 264. It will follow from the meaning which has been given to a negative index that a'^-^-a" = a"*"" when in is less than n, as well as when m is gi-eater than n. For suppose m less than n ; we have a"* 1 Suppose ra = n', then a"^-^a" is obviously = 1 ; and a"'~" = a''. The last symbol has not hitherto received a meaning, so that we are at liberty to give it the meaning which naturally presents itself; hence we may say that a"— 1. 152 THEORY OF INDICES. 2G5. Thus, for example, according to these definitions, J = ^a\ a^ = Ja\ a^ = J a' = a% -a 1 -A 1 1 -i 1 _ 1 a =-^, a -="7=,, «- = — = —. Thus it will appear that it is not ahsolutehj necessary to intro- duce fractional and negative exponents into Algebra, since they merely supply us with a new notation for quantities which we had already the means of representing. It is, as we have said, a con- venient notation, Avhich the student will learn to appreciate as he proceeds. The notation which we have explained will now be used in establisliiiig some propositions relating to roots "and powers. 2(S^. To shew that a" x V = (ahY. Let a" X b" =x; tlierefore x" = (a" X 6"Y = fcA X flf^ , (by Art. U), =axb. Tlius x" = ah, therefore x= labY} which was to be proved. In the same manner we can prove that 1^ 1 267. As an example of the preceding proposition we have J a X ^b = /^(tti). Nov»', as we have seen in Art. 236, a square I'oot admits of a double sign ; hence strictly S})eaking our result should be stated thus : the product of one of the square roots of a into one of the square roots of b is equal to one of the square roots of ab. A similar remark a2')plies to other propositions of the present Chapter. In the higher parts of m.athematics the matter liere noticed is discussed in more detail : see Theory of Eqimtions, Cliapter xi. THEORY OF INDICES. 153 208. Hence a" x h" x c" = \cLh\ x c" = iahcy. And by proceeding in this way we can prove that 111 1 / ^ 1 a" X 6" X c" X X Ic" = {ahc....h V . Suppose now that there are m of these quantities a^h, c, ... k, and that each of them is equal to a ; then we obtain (^r -{■■}■ But fcA'' is, by Arts. 260, 2G1, a^; thus a" ] =a". Hence comparing this with Art. 261 we see tliat the ?i"' root of the m^^ power of a is equivalent to the m^^ power of the n^^ root of a. . , 269. To sheAV that UrV = ar\ Let ic =(«"')"; therefore a" = a'" j therefore a;'""^aj there- — / i-\i J fore X = a™". Thus ( a"* )" = a'"", which vras to be proved. •ni mp 270. To shew that a" = a"P. m Let cc = a" ; therefore x" = a*" ; therefore a;"^ = a*"^ ; therefore mp m mp X = aP^. Thus a" = a"^, which was to be proved. 271. The student may infer from what we have said in Art. 265, that the propositions just established may also be established without using fractional exponents. Take for example that in Art. 2QQ ] here we have to shew that ::ja X ;'6 = ;/(a5). 154< THEORY OF INDICES. Proceed as before ; let a; = i^a x ^b ; therefore a" = Ua X ;/6)" = CJay X (^5)", (by Art. 41), = a x 6. Tims x" = ah J therefore x = ^(ab), which was to be proved. 272. We have been led to the definitions of Arts. 2G0...265 as consequences of considering the relations a"* x a" = a"*"^" and (a'")" = a*"" to be universally true, whatever m and n may be ; we shall now proceed to shew conversely that if we adopt these defi.- nitions the relations a^x a" =^a"'^" and (a*")" = «"*" are universally true, whatever ui and 7i may be. £ !1 f + - 273. To shew that a^ x a' = a^ '. a'x a' --a" x a''j by Art. 270, = (a^'V X (a'' J", by definition, = (a"" X aA^, by Art. 266, 27-4. In the same way we can shew that 275. Thus the relation a"' x a" = ct'"^" is shewn to be true when ni and ii are positive fractions, so that it is true when m and n are any positive quantities. It remains to shew that it is also true when either of them is a negative quantity, and when both are negative quantities. (1) Suppose one to be a negative quantity, say ?^ ; let n = — V, 1 ^"» Tlien a" X a" = a"* X «-"::= a" X - = — = a""-", (by Art. 274), a a ^ THEOEY OF INDICES. 155 (2) Suppose both to be negative quantitiea ; let m ~ — fx, and n — — v, Tlien 276. Similarly a"' x a" x a^ = a"'+" x a^ = a"'+"+^ ; and so on. Thus if we suppose there to be r quantities m, n, j), •••, and that each of the others is equal to m, we obtain whatever la may be. 277. To shew that ( a'^ V = a'i\ Let x=ia'iy; therefore x' = {cv^j = ««, by Ai-t. 270 ; there- pr fore a;'^' = a^''; therefore a; = a*', which was'to be proved. 278. To shew that (a"')" = a""* universally. By the preceding Article this is true when 711 and n are any positive quantities ; it remains to shew that it is tnie when either of them is a negative quantity, and when both are negative quantities. ( 1 ) Suppose n to be a negative quantity, and let it = — v. Then (a"'T = (nry - - — - = — . = a"'"" = a"'". \ / {ay a (2) Suppose m to be a negative quantity, and let it = — fx. Then (a''r = («-^)" = (^y = ^^ = rt"^" = rt'"". (3) Suppose both m and n to be negative quantities ; let m = — fx and n = — v. Then {a-r = (a"'^)-" = ^, = ^^ = a^^ = a"-. 15G EXAJVIPLES. XVIII, EXAMPLES OF INDICES. 1. Simplify {x^xx^)^i ■/ 2. Find the in-oduct of a^ a '\ a \ and a \ 3. Find the product of f^\\ f^j and ^J^J • 4. Simplify the product of J, cr^, l!a\ a"i\ IJa^', and (a~^)l f/cA' 5. Simplify 6. Multiply a- + h^ + a" -6 by ah~'- - a- + &-. S 11 8 11 7. Multiply x^ - xy- + x'-y - y^ by a; + x-y-^ y. 8. Multiply a- - cc" + a"^' - fr + a^ - a + a^ - 1 by «- + 1. 9. Multiply a^ - a^ + 1 - a~^ + a~3 by rr + 1 + ct"*. 10. Multiply - 3a-5 + Scr^'^"^ by - 2^"' - Sa-^S. 8 11 Z. 11 11. Divide XT - xy"^ + ic-y - y by x~ — y\ 12. Divide x^ + x-Vt^ + a^ by x^ + a;^cr + a", 8n 8n n n 1 3. Divide a^ - a " hj a^ - a ■ . U. Divide '2xSj-^ - 6xY' + 7^'V~' - ^^' + ^.-cy by x^y~^ — x'^y'" + xy~\ 15. Divide a- - ah + ab^ - 2Jb' + b^ by J - ab^- + ah - Z/l 1 0. Simplify a^ — ax' + a^:-c — x^ a^ — a^x'^ + Zg^x - Zax- + a^x^ — x^ EXAMPLES. XVIII. 157 , 7/« x' 2l/ - x^ 1 7. Extract the sqiiare root ot — + -j- + r— . ^ X ^y {xyf 1 8. Extract the square root of ^a-Ua'h^ + 96' + IGa'^c^ - 246^ c^ + 16cl 19. Extract the square root of 256a;^ - 512a; + 640a;^ - 5Ux^ + 304 - 128a;-^ + 40a;- ■' - 8a;-' + x'^, 20. If a^ = 6% shew that (t)*^^*' '> ^"^^ i^ a = 26, shew that h = 2. XIX. SUEDS. 279. When a root of an Algebraical quantity which is re- quired, cannot be exactly obtained, it is 'called an irrational ov surd quantity. Thus IJa' or a^ is called a surd. But IJa or a*, though apparently in a surd form, can be expressed by a"^, and so is not called a surd. The rules for operations with surds follow from the proposi- tions established in the preceding Chapter, as will now be seen. 280. A rational quantity may he expressed in the form of a given surd, hy raising it to the poicer whose root the surd expresses^ and affixing the radical sign. n Thus a - Ja^ = ^a^, &c. ; and a + x = (a + a;)^. In the same manner the fomi of any surd may be altered ; thus {a + x)'^ ^(a + xy^{a+xy The quantities are here raised to certain powers, and the roots of those powers are again taken, so that the values of the quantities are not changed. 158 SURDS. 281. The coefficient of a surd may he introduced under the radical sign, by first reducing it to the form of the surd and then multiplying according to Art. 271. For example, a Jx = Ja' X Jx = J{a^x) ; ay^ = i^Y)' > X J{2a -x) = J{2ax' - x') ; ax (a- xf = {a' (a - a:)'}^ ; 4.J2=:J{16x2) = J32. 282. Conversely, any quantity may be made the coefficient of a surd, if every part under the sign be divided by the quantity raised to the power whose root the sign exjjresses. Thus sJic'^ — d^) = <^'" X v/(<^ — ^) ] J{cb^ — a-x) = a J (a — x) ; (a' - x'f = / X (l - ^^n . JQO ^ ^(4 X 15) = 2 V15 ; 1 _ iV- Vl - -V'- - (--lY- (^'-^')' ,6"^ x^J b \ x'J x \b'^ J xb ' 283. When surds have the sa7ne irrational p)art, tlieir sum or difference is found by affixing to that irrational part the sum or difference of their coefficients. Tlius a Jx ± b ^/x = (a ± b) Jx ; V300±5V3 = 10^/3±5^/3 = 15^3 or 5^/3; J{?a%) + J{3x'b) = a J{3b) + x J{3b) = {a + x) J (3b). 284. If two surds have the same index, their product is found by taking the product of the quantities under the signs and retain- ing tlie, common index. Thus (I" X b" = {abf, (Art. 266) ; J2xJ3 = JQ', {a + b)^x{a-b)"' = {a'-b')K 285. If the surds have coefficients, the product of these coeffi- cients must be prefixed. Tlius ajxxbjy=abj(xy); 3^8x5^^2 = 15^/16 = 60. SURDS. 159 286. If the indices of two surds have a common denominator, let the quantities he raised to the powers expressed hy their respective numerators, and their product may he found as hefore. Tims 2^ X 3- = 8^ X 3* = (24)^ ; {a + x)'- x{a-xY ={{a + ^) {^^ ~ ^T] - 287. If the indices have not a common denominator, they may he transformed to others of the same value unth a common deno- minator, and their product found as in Art 28G. Thus (a- - x')^ x{a- x)^ = (a' - x'f x {a - x)' = {{a' - x') {a - x)'^^ ; 2^ X 3* = 2^ X 3^ = 8^ X 9^ = (72)*. 288. If two surds have the same rational quantity under the radical signs, their product is found hy making the sum of the indices the index of that quantity. Thus fl" X a'" = a" "*, (Art. 2 / 3) ; V2x^2 = 2^x2^ = 2^^^2l 289. If the indices of two surds have a common denominator, the quotient of one surd divided hy the otiier is obtained hy raising them respectively to tlie powers expressed hy the numerators of their indices, and extracting that root of the quotient ichich is expressed hy the common denominator, 1 m Thus, «l.g)MAH.26G);^=(f);; 4'*!i.r|,)' 290. If the indices have not a common denominator, reduce them to others of the same value vnth a common denominator, and proceed as hefore. Tlius (a' - xi-^{a? - x')^ = (a? - a=')U(a' - x'f = ^(^^^^- IGO SURDS. 291. If the surds have the same rational quantity under the radical signs, their quotient is obtained by making the difference of the indices the index of that quantity, \ 1 11 Thus, ft^^a" = a^ ", (Art. 274) ; /0_L. 3/.2 _ 9^^9^ - 9^~3 _ 9^ 292. It is sometimes useful to put a fraction wliicli has a slm})le surd in its denominator into another form, by multiplying both numerator and denominator by a factor which will render the denominator rational. Thus, for example, 2 ^ 2^3 JJZ J-6 J'ZxJZ 3 • If we wish to calculate numerically the approximate value of 2 . . — -r it will be found less laborious to use the equivalent form 2 v/3 Q. -1 1 a a Jb 293. It is also easy to rationalise the denominator of a frac- tion when that denominator consists of two quadratic surds. For ^ ^ ^^ (Jb ^ Jc) ^ a (Jb =f Jc) Jb^Jc Wb^Jc)[^Jb=Fjc) b-G • So also ^^ = ^ (^ ^ n/^') ^ ajb^Jc) SimiKrlv ^±^ = (^ ^ J'^) (3 + V^) = ^^^K/5 ^ 7 + 3^5 294. By two operations we may rationalise the denominator of a fraction when that denominator consists of thi'ee quadratic surds. For suppose the denominator to be J a + Jb + Jc ; first multiply both numerator and denominator by Ja + Jb - Jc, thus the denominator becomes a + b ~ c + 2 J{ab) ; then multiply both numerator and denominator by a + b- c—2 J(ab), and we obtain a rational denominator, namely (rt + 6 — c)^ — iab, that is, a' + b' + c'- 2ab ~ 2bc - 2ca. SURDS. 161 295. A factor may he found which will rationalise any binomial. 11 1^ 1 (1) Suppose tlie binomial a^ + h'' . Put x = aP, y = l'^; let n be the least commou multiple of 2^ and q ; then cc" and y" are both rational. Now (a; + 2/)(a;"-^-a;"-^2/ + a:"-y-...±2/""') = ^-"'^2/", where the upper or lower sign must be taken according as n is odd or even. Thus x"-'-x''-'y + x''-y- ^y"-' is a factor which will rationalise x + y. , 1 1 (2) Suppose the binomial a^ — 6'' . Take x, y, and n as be- fore. Kow {x - y) (x""^ + x'^'-y + 03""'?/^ + + 3/""^) = a;'' - 2/".- Thus jc""^ + x^'-'y + ic"-^^^ + +2/" is a factor which will rationalise x- y. , ,n-l 1 ,1 Take, for example, a- + 6^ ; here n =- 6. Thus we have as a rationalising factor 'o x^ — x*y + x^y" — x'y^ + xy* - y^, 6 4^1 S2 23 14 5 that is, a^ - a^ 6^ + a^ 6^ -d^-Jj^ + a^ 6^ - Z;-, that is, a^ — a'h'^ + a- b'-^ — ab + a- b" — b". The rational product is x^ — y^, that is, a- — 5-, that is, a^ — b'. 296. The square root of a rational quantity cannot be partly rational and partly a quadratic surd. If possible let ^Jn = a -\- Jm ; then by squaring these equal quantities we have n — a' + 2a ^Jm + m ; thus 2a Jm = n — a^ — m, 2 M fll ,_^ A-»l and ^Ini = ^ ^ , r. rational quantity, wliich is contrary to the supposition. See Art. 24S, T. A. 11 162 SURDS. 297. If two quadratic surds cannot he reduced to others which have the same irrational part, their product is irrational. Let fjx and Jy be the two quadi-atic surds, and if possible let sj{xy) = rx, where r is a whole number or a fraction. Then xy = r^x^y and y = r'x, therefore Jy = r Jx, that is, Jy and Jx may be so reduced as to have the same irrational part, which is contrary to the supposition. 298. One quadratic surd cannot he made up of two others which have not the same irrational part. If possible let Jx = Jm + Jn ; then, by squaring, we have x = m + n + 2 J {mil), and J{mn) = ^{x — 7n — n), a rational quan- tity, which is absurd. See Art. 242. 299. In any equation x + Jj = a + ^^b which involves rational quantities and quadratic surds, the rational parts on each side are equal, and also the irrational parts. For if ic be not equal to a, suppose x = a + 7n ; then a + m + Jy = a + Jh, so that m + ^y = Jh ; thus Jb is partly rational and partly a quadratic surd, which is impossible by Art. 296. Therefore x = a, and consequently Jy = Jh. 300. If Jia +Jh)^x + Jy, then J {a -Jh) = x- Jy. For since J{ct + Jh) = x + Jy, we have by squaring a + Jh = x'' + 2x Jy + y ; therefore a = x^ + y, and Jh — 2x Jy, (Art. 299). Hence a - Jh = x^ — 2x Jy + y. and J{a -Jh) = x- Jy. Similarly we may shew that if J{a + Jh) = Jx + Jy, then J {a - Jh) = Jx- Jy. SURDS. - 163 301. The square root of a binomial, one of whose terms is a quadratic surd and the other rational, may sometimes he expressed hy a binomial, one or each of whose terms is a quadratic surd. Let a+ Jb he the given binomial, and suppose J{a + Jb)=Jx + Jy. By Ai-t. 300, J{a - Jb) = Jx- Jy. By multiplication, V(^* — b) = x — y. By squaring both sides of the first equation, a + Jb = x+ 2J{xy) + y ; therefore a — x-\-y. Hence, by addition and subtraction, a + J ice -b) = 2x, a- J {a' -b) = 2y; therefore x = i{a + J{a^ - b)}, y = J {^ ~ V(^* - ^)}- Thus X and y are known, and therefore J{a + Jb), which is Also J(a — ^b) is known, for it is Jx — Jy. 302. For example, find the square root of 3 + 2 J'2. Here a = 3, Jb = 2J2, a*-b = d-S=l; therefore a; = i(3 + l)=2, 2/=|(3-l) = l. Thus J{3+2^2) = ^'2 + Jl=J2 + l. 303. Again ; find the square root of 7 — 2^10. Instead of using the result of Art. 301 we may go through the whole operation as follows : Suppose J {7 -2^10)= ^x- sjy; then, by squaring, 7-2 710 = £c-2 J(xy) 4- y ; hence x + y = 7 (1), 11—2 164- SURDS, and 2J{xy) = 2JlO; therefore {x + y)' -ixy = i^- {IJlOy, that is, («:-2/y = 49-40 = 9, and x-y = Z (2); therefore, from (1) and (2), cc = 5, and y^% Thus V(7-2V10)^V5-n/2. 304. It appears from Art. 301 that hence, unless a^ -h be a perfect square, the values of Jx and ^Z^/ will be complex sui'ds, and the expression ^/a: + Jy will not be so simple as J [a + ^6) itself. 305. A binomial surd of the fomi sj{(i^c) + Jh may be written thus, Jc(a-{- / - ) • If then a^ be a perfect square, the square root of « + / - may be expressed in the fomi ^x + Jy ; and tlierefore the square root of J{a'c) + Jh will be t^lc {Jx + Jy). 306. For example, find the square root of ^/32 + JoO. Here ^/32 + ^30 = ^2 (4 + ^15) ; thus V(V32 + ^30) = :j-2x V(4 + V15) j and it may be shewn that Hence V(V33 + ^30) = ^2 (^5 + y|) = ^^^5 + JS). SURDS. 165 307. Sometimes we may extract the square root of a quantity of the form a + Jh + ^c + Jd by assuming J{a-¥jh+Jc^Jd)=^x + Jy+Jz', then a + Jb+Jc + Jd=x + y + z + 2 J{xy) + 2 J{yz) + 2J{zx) ; "sve may then put 2 V(^2/) = J^ 2 V(H = x/c, 2 J{zx) = Jd, and if the values q/ x, y, and z, found from these, also satisfy X + y ■¥ z=^a, we shall have the required square root. 308. For example, find the square root of 8 + 2,^/2 + 275 + 2^/10. Assume ^(8 + 2^2 + 2jD + 2J\0)=^Jx + Jy + Jz] then 8 + 2V2 + 2V5 + 2V10 = a; + y + ;^ + 2 J{xy) + 2 J{yz) + 2 J{zx). Put 2V(^2/) = 2 72, 2J{yz) = 2J5, 2J{zx) = 2JlO; hence, by multiplication, ,J(xy) x ^(yz) =^10, and J(^oc)= J 10, therefore, by di^T.sion, y =1 ; hence x — 2, and z — 5. These values satisfy the equation x + y + z — S. Thus the required square root is J2 + ^1 + ^-'5, that is, 1 + 72 + J5. 309. If 4/(« + Jb) = x + Jy, then ^(a - Jb) = x~ Jy. For suppose ^/{a + Jb)=x + Jy; then, by cubing, a + Jb =- x"" + 3x^ Jy + ?,xy + y Jy ; therefore a^x^ + Zxy, Jb = Zx^ Jy + y Jy, (Art. 290) ; hence a- Jb ^x"" - 3aj' ^y + Secy - y Jy, and 4/(<^ - >/^) = ^' - v^y- 166 SUKDS. 310. The cube root of a binomial a^Jb may be sometimes found. Assume U{a + ^b) = a; + Jy^ then ^{a - Jb) ^x- ^y. By multiplication, IJif^ -b)—x* — y. Suppose now tliat a'' — 6 is a perfect cube, and denote it by c', thus c = £c' - 2/ ; and, as in Art. 309, a = o;^ + '6xy. Substitute the value of y ; thus a = a;"'' + Zx {x^ — c) ; therefore 4a3^ — Zcx = a. From this equation x must be found by trial, and then y is known from the equation y — x^ — c. Thus it appears that the method is inapplicable unless c^ -b be a perfect cube ; and then it is imperfect since it leads to an equation which we have not at present any method of solving except by trial. The proposition, however, is of no practical importance. 311. For example, find the cube root of 10 + ;yi08. Assume ^/(lO + ^108) = a; + ^y, then ^/(lO - ^1 08) ^ a; - Jy. By multiplication, ^(100 - 108) ^oj'-^/, that is, ^2 = x''-y, Also 10 = x^ + Sxy =x'' + 3x {x" + 2) ; therefore 4a;' + 6a; = 1 0. We see that this equation is satisfied byaj=l; hence y = 3f and the required cube root is 1 + ^JS. Again ; find the cube root of 18 ^3 + 14 J5. 18^3+1^5 = 3^3(6 + ^ /|). Tlie cube root of 3^3 is ^3 ; and the cube root of 6 + ^ Z? 196 5 8 can be found. For here a^-b = 36- -^ x - = -^— : so that 9 3 27 2 c = - o • Hence we have the equation 4a;' + 2a; = 6, which we see is EXAMPLES. XIX. 167 satisfied hj x=l. Thus the required cube root is ^3 ( 1 + / o ) > that is J3 + Jo. 312. We will now solve an equation involving surds which will serve as a model for similar examples : the equation resembles those already solved in the circumstance that we obtain only a single value of the unknown quantity. Solve ^(a3 + 2) + V(»-U) = 8. By transposition, J{x + 2) = 8 - J{x - 14) ; square both sides, ic + 2 = 64-16 J{x -14) + a;-14; transpose, 16 J{x — 14) = 48 ; divide by 16, ^(^-14) = 3; square both sides, ic - 14 = 9 ; therefore a; = 23. EXAMPLES OF SURDS.- 1 2. 1. Find a factor which will rationalise a^ - h^. 2. Find a factor which will rationalise ^^2 - ^/3. 3. Find a factor which will rationalise ^3 + Jo. 1 4. Given ^3 = 1*7320508, find the value of -^ . 5. Shew that (5.^5) (1+^3) gVl^- 7. Extract the square root of 9--24 /- + 34-24 A+9^. 2/ V 2/ \/ X X 8. Extract the square root of {a + bf - -i (a - b) J{ab). 168 EXAMPLES. XIX. Extract the square root of the expressions in the following examples from 9 to 18 inclusive : 9. 4 + 2^3. 10. 7-4^3. n. 7 + 2^/10. 12. 18 + 875. 13. 75-12721. 14. 16 + 5 77. 15. ab + c'+ J{{a' - c') {h' - c% 16. ^27 + 715. 17. -9 4-6 73. 18. l+(l-c'')-i. 1 9. Find the value of 1 +x 1 -X , JS -, 1- -T— — rr^ X when X — \.- . 1 + 7(1+0;) 1 + 7(1-0.-) 2 20. Find the value of 1 + £C 1 - X , 73 + T— r7^ -v when X — l+7(l + a:) l-7(l-a;) 2 21. Extract the square root of 6 + 272 + 273 + 2,^/6. 22. Extract the square root of 5 + JIO — J 6 — J15. 23. Extract the square root of I5-273-2715 + 672-27G + 275-273O. 24. Extract the cube root of 7 + 5 J2. 25. Extract the cube root of 16 + 8 ^0. 26. Extract the cube root of 9 73 - 11 72. 27. Extract the cube root of 21 76 - 23 75. 28. Shew that 7(75 + 2) - 7(75 - 2) = 1. . 29. Solve the equation 7(^ + ^^) ~ J^ — ^• 30. Solve the equation J{Zx + 4) + ^J{2)X - 5) = 9. 31. Solve the equation aj(b — x) =h J {a — x). 32. Solve the equation 7(^ + ^) + 7(^ + ^) = 7^- QUADRATIC EQUATIONS. 169 XX. QUADRATIC EQUATIONS. 313. When an equation contains only the square of the unknown quantity the value of this square can be found by tlie rules for solving a simple equation ; then by extracting the square root the values of the unknown quantity are found. For example, suppose 8x'- 72 + 10x' = 7-2ix' + S0: by transposition, 4:2x^ =168; by division, x^ — i ; therefore x =■ J-i = ± 2. The double sign is used because the square root of a quantity may be either positive or negative. (Art. 231.) It might at first appear that from a;^ = 4 we ought to infer, not that ic = ± 2, but that =j= a: = =fc 2. It will however be found that the second foiTn is really coincident with the first. For ± a; = ± 2 gives either +x = + 2, or + x= — 2, or ~x = + 2, or — aj = — 2 j that is, on the whole, either x = 2, or a: = — 2. Hence it follows, that when we extract the square root of the two mem- bers of an equation it is sufiicient to put the double sign before the square root of one of the members. 3 14-. Quadratic equations which contain only the square of the unknown quantity are called ^>>2«-(3 quadi'atics. Quadratic equations Avhich contain the first power of the unknown quantity as well as the square are called adfected quadratics. We proceed now to the solution of the latter. 315. We shall first shew that every quadratic equation may be reduced to the form ar^ + 2JX = q, where p and q are positive or negative. For we can reduce any quadratic equation to this form by the following steps : bring the tenns which contain the unknown quantity to the left-hand side of the equation, and the known quantities to the right-hand side ; if the coeflScient of x' be nega- tive, change the sign of every term of the equation ; then divide 170 QUADRATIC EQUATIONS. every term by the coefficient of x^. Thus we may represent any (quadratic equation by x^ + ^?:c = q. To solve this equation we add - p^ to both sides ; thus 3 S a" + wx + V = V + Q. The left-hand member is now a complete square; extract the square root of each member j thus f ^^f="x/(?^^)^ transpose the tenn ^ , and we obtain P ^ Iff ^ = -^=^^(7- + ^? 2 V V^ 316. For example, suppose -3a;' + 360^-105=0; transpose, - 3jc'' + 36^ = 105 ; change the signs, Zx^ -36a: = -105; divide by 3, a;' - 1 2a; = - 35 ; /ISX" add to both sides [~^)^ that is, 36 ; thus a;'-12a; + 36 = 36-35= 1; extract the square root of both members ; thus a;- 6- ±1. Therefore a; = 6 ± 1 ; that is, a; = 7, or 5. If either of these values be substituted for x in the expression — 3a;^ + 36a; — 105, the result is zero. 317. Hence the follo'vVT.ng rule may be given for the solution of a quadratic equation : By transposition and reduction arrange tJie equation so that tlie terms involving the unknown quantity are alone on one sidcy QUADRATIC EQUATIONS. 171 and the coefficient of x' is + 1 ; add to both sides of the equation the square of half ilie coefficient of x, and extract the square root of both sides. 318. As another e ixample we will take ax^ + 6x + c = ; transpose, ax' + hx = — c ', divide bj «, o bx c a a „ hx a;- + — a b' y- c h'-iac 4a" 4a^ a 4a* extract the square root, . + ^ = ^ JC" iac) ^ 2a 2a ~b^J{h'-iac) transpose, x — ^^^^ . 2a The particular case in which c = should be noted. Then, taking the upper sign we have cc = j and taking the lower sign we have re = . In fact in this case the equation reduces to ax^ + 6a; = 0, or X (ax + b) = : and it is j^lain that this is satisfied, either when a; =: ; or when ax + b = 0, thab is wlien x=- — . a 319. When an example is proposed for solution instead of going thi"ough the process indicated in Art. 317, we may make use oi the fo7'mida in Art. 318. Thus, take the example in Art. 316, namely, — 3x'+ 36iC — 105 = 0, and by comparing it A\dth the formula in Art. 318 we see that we may suppose a- — 3, 6-36, c = — 105. Hence if we put these values for a, b, and c in the result of Art. 318, we shall obtain the value of x. Here 6''-4ac= (36)' -12x105 = 36; tliereforc x = ^ — = 7, or 5. 36=^. 6 ^ -6 172 QUADRATIC EQUATIONS. 320. For another example take the equation x^ — (jx = — 2 ; add ("^y, x'-Gx + = d-'2=7 ; extract the square root, aj - 3 = ± JT, transpose, a; = 3 ± ^/7. Here ^7 cannot be found exactly ; but we can find an ap- proximate value of it to any assigned degi'ee of accuracy, and thus obtain the value of x to any assigned degree of accuracy. 321. In the examples hitherto considered we have found two different roots of a quadratic equation ; in some cases however we shall find really only one root. Take for example the equation a' — 12:« + 36 = ; by extracting the square root we have x — 6 = 0, and therefore ic = 6. It is however convenient in this case to say that the quadratic equation has two equal roots. 322. If the quadratic equation be represented by ax' + 6a; + c = 0, we know from Art. 318 that the two roots are respectively ~h + J{h'-iac) -h-J{h'-iac) • — — ■ ana ^ . 2a la Now these will be difierent unless h^ — -iac — 0, and then each of them is — TT- . This relation 6^ — 4(»c = is then the condition that Za must hold in order that the two roots of the quadratic equation may be equal. 323. Consider next the example re^— IOjc + 32 = 0. By transposition, x^ — lOx = — ^2 ] by addition, £c" - lOo; + 25 = 25 - 32 = - 7. If we proceed to extract the square root we have x-5 = ^ J-1. EXAMPLES. XX. 173 But the negative quantity — 7 lias no square root either exact or approximate (Art. 232) ; thus no real value of x can be found to satisfy the proposed equation. In such a case the quadratic equation has no real roots ; this is sometimes expressed by saying that the roots are imaginary or impossible. We shall return to this i>oint in Chapter xxv. 324. If the quadratic equation be represented by ax^ + hx + c = 0, we see from Art. 318 that the roots are real if h^ — iac is positive^ that is, if h^ is algebraically greater than 4ac, and that the roots are impossible if h' — -iac is negative, that is, if If is algebraically less than 4«c. EXAMPLES OF QUADRATICS 1. x'-ix + ^^O. 3. 6a;'-13a;+6 = 0. 5. 2a;' - T.'c + 3 = 0. 7. cc'+10;c + 24-0. 9. 14a: — x" = 33. 11. x'-3=hx-Z). 13. 110ic«-21ic + l =0. 15. {x-'\.)(x-2)^(j. 1 7. (3a: - 5) (2x - 5) = (x+ 2,) (x -I). 18. (2a: +1) (a: + 2) = 3a;' -4. 19. (a;+l)(2a; + 3) = 4a:^-22. 20. {X -l){x-2) + {x - 2) (x - 4) = 6 (2a; - 5). 21. (2a; - 3)' = 8a;. 22. (5a; - 3)'- 7= 44a: + 5. 23. (a;-7)(a;-4) + (2a:-3)(a;-5)=103. 2. a;' - 5a: + 4 = 0. 4. 3a:^-7a:=20. G. 3a;' -53a; + 34 = 0. 8. 7a;'- 3a' = 160. 10. 2a;'-2a;-| = 0. 12. 4:{x'-l) = iX-l. 14. 780a;'- 73a: +1 = 0. 16. (3a;- 2)(a;-l)=14, 174 24. 25. 2G. 27. 28. 30. 32. 34. 35. 36. 38. 40. 42. 44. 46. 48. EXi\3IPLES. XX. 5 , 7 73^_ 7* +5^'+14()-^- X - |)(^-5) + (»'-5)(''-3=(^"4)(^--5)- 2'^^"3'^^* _ ,,7 21 + 65a: 8a; + 1 1 + - = ^ X 7 X i 21 a; + 5 1 + 23 7 3 29. 31. 6 X 5(x-l) X 6 4 2(a;-l) a;^-l 4* 33. 21 5 -ic 3 X 23 7 ~ '7 + a; 2(a;^-l) 4(a; + l) 8 a; + ^0 _ 3(10 + a;) 15 ' 3(10"^" 95 2aj 3aj-50 12a; + 70 15 "^ 3 (10+ a;) ' a;^ - 5a; „ 1 — =x-3 + -. X + O X 190 X 3 a:-l 2 "^ a; a;-l a; + 2 X- 2 x — 2 X + 2 x — 6 5 6* a;-12 _5 "6 a; — 12 a;— 6 4 5 + 12 a;+ 1 x+ 2 x+ 3 * 2a;- 3 3a;- 5 5 / 3^^ "^2a;-3~2* a; + 3 a; - 3 _ 2a; - 3 a; + 2 aj-2~a;-l 49. 37. 39. 41. 43. 45. 47. X - x + 2 4 X x-l a; + 4 X 7 3* 10 4 ■ a; + 4"'T* + 2a; a; — 4 X x+l 1 + a;+ 1 13 6 x — 2 x + 2 5 * a; + 2 x a; + 4 * 3a;-2 2x-5 8 "2¥^5 ~3^^ ~3 2 a; + 2_2(a;+3) a; + 2 x — 2 x — 3 EXAMPLES. XX. I75 50. 10 (2x + 3) (x-3) + (7x + 3)^= 20 (x + 3) (x - 1). 51. {7-4.J3)x' + {2-J3)x = 2. 52. x- - 2ax + a''-b-=0. 53. x'-2ax + b' = 0. 54. {3a' + h') (x' -x + l) = (3b' + a') (x' + x+l). 5o. + + = 0. X — a X — X — c 56 1 , 11 1 {x-b){x-c) {a + c){a + b) {a + c){x-c) '^ {a + b){x^)- K^ 1 111 57. J =-+-+-. a + + X a X 58. (ax — b) (bx — a) = c^. ^^ a h 2c 59. + 60. abx X — a X — b X - c' 3a^x Qa^ + ab — 2b^ b^x + -^ X + a x + b X + c 61. + ~ + = 3. X — a x — o X — c „^ a + c(a + x) a + X a oz. : + — a + c (a- x^ X a — 2cx ' XXI. EQUATIONS WHICH MAY BE SOLVED LIKE QUADRATICS. 325. There are many equations which, though not really quadratics, may be solved by processes similar to those given in the preceding Chapter. For example, suppose X* - 9x' + 20 = 0. Transpose, x* - Ox' = - 20 ; 176 EQUATIONS WHICH MAY BE /OX 2 /Ox 8 1 by addition, x' - 9a;' + U j = U) ~^^ '^ i' , 9 1 extract the square root, x - - = ± - ; 9 1 therefore a;' = - ± ^ = ^? ^^' ^ ^ therefore x = ^ Jo, or ±2. 326. Similarly we may sohe any equation of the form ax'" + 6aj" + c = 0. Ti-anspose, . ax^" + hx"" = -c ; 2„ bx" c divide by a, x + — = ; , hx"" /by /bV c b'-iac by addition, x^" + — + Ur- = o~ — = — a~~~- — ^ extract the square root, a;" + — — ^ h^_^J{b'-iac) ^ ■la -b^Jib'-iac) therefore cc" = ^^7^ . za Hence by extracting the n^^ I'oot the value of x is known. 327. Suppose, for example, a; + 4 ^x = 21 ; therefore x+ i Jx+ i = 25 ; therefore Jx + 2=^b; therefore Jx = -2^5 = 2), or - 7 ; therefore £C = 9, or 49. 328. Again, suppose x~' + x~^=- 6; .^ r. _, -1 1 25 therefore x+x'^ + -=—r\ 4, 4, therefore a; ^ + - ^ ^ ; SOLVED LIKE QUADRATICS. x-^ = - 1 - =fc 2 5 2'' = 2, or -3; X-' = 4, or y, X -- 1 1 or 1 9" 177 therefore therefore and 320. Suppose we require the solutions of the equation x + J(5x+ 10) = 8. B J transposition, ^{^x + 1 0) = 8 - a; ; square both sides ; thus 5a; + 10 = 61- 16x + x^ ; therefore x" — 21x = — 54: ; /21\2 /21\' 225 therefore x' - 21a; + (-9-) = (—) - 51 = -^- ; .1 ^ 21 15 thereioro x — ir^^^'-rj -.-*^ 21 15 therefore jy = — ± — = 1 8, or 3. Substitute these vahies of x in the left-hand side of the given equation; it will be found that 3 satisfies the equation but that 18 does not; we shall find however that 18 does satisfy the equation x-J{5x+10) = 8. In iact the equation 5x+ 10 = 64: — IGx + x" which we obtained from the given equation by transposing and squaring might have arisen also from x — ^{5x + 1 0) = 8. Hence we are not sure that the values of x which are finally obtained will satisfy the proposed equation ; they niai/ satisfy the other fonn. 330. Again, consider the example x-2J{af + x + 5)-U=:0. By transposition, a; - 14 = 2 J(x' + x + 5) ; T. A. 12 178 EQUATIONS WHICH IklAY BE by squaring, x" - 28^3 + 196 = 4ic'' + 4a; + 20 ; therefore 2>x^ + 32a3 := 1 76. -44 From the last equation we shall obtain x = i, or . It will, o however, be found on trial that neitlier of these values satisfies the proposed equation ; each of them however satisfies the equation a; + 2 ^(a;" + a; 4- 5) - 14 = 0. From this and the preceding example we see that when an equation has been reduced to a rational form by squaring, it will be necessary to examine whether the roots which are finally obtained satisfy the equation in the form originally given. This remark apj^lies for instance to equations like those solved in Arts. 312, 327, and 328. 331. Suppose that all the terms of an equation are brought to one side and the expression thus obtained can be represented as the product of simple or quadratic factors, then the equation can be solved by methods already given. For example, suppose [x — c) {x~ — Zax -\- 2a") = 0. Tlie left-hand member is zero either when £c — c = 0, or when aj*— 3ax + 2ct^ = ; and in no other case. But if a? — c = 0, we have x = c] and if x^ — ?)ax + 2a^ — 0, we shall find that x = a, or 2a. Hence the proposed equation is satisfied by x = c, or a, or 2^ j and by no other values. 332. Facility in separating expressions into factors wdll be acquired by experience ; some assistance however will be furnished by a principle which v.^e will here exemplify. Consider the example x{x — cf = a{a ~ c)". Here it is obvious that x^a satisfies the equation ; and we shall find that if we bring all the terms to one side x — a will be a factor of the whole expression. For the equation may be written x' - c^ - 2c (x" - a') ■¥c'{x-a) = 0; that is, {x - a) [x"" + ax + a^-2c {x + a) + c^} = 0. SOLVED LIKE QUADRATICS. 179 Hence the other roots besides a will be found by solving the quadratic a' + ax + a^ — 2c (ic + a) + c" = 0. In this manner when one root is obvious on inspection, we may succeed in aiTanging the equation in the manner indicated in Art. 331. 333. We "svill now add some miscellaneous examples of equa- tions reducible to quadratics. (1) Suppose ic' - 7a; + V(x' - 7aj + 18) = 24. Add 18 to both sides; thus x"- - 7x + 18 + J{x- - 7^ + 18) - 42 ; complete the square ; thus a- - 7a; + 18 + f(x' - 7a; + 18) + -], = 42^ - ^'f : * 4 1 13 therefore J(x- — 7a; + 1 8) + ;^ = ± — ; therefore s X-^" — 7a; + 18) = 6, or — 7 ; therefore a;'' - 7a; -f 18 = 36, or 49. Hence we have now two ordinary quadi'atic equations to solve. We shall obtain from the first x= d, or — 2, and from the second a;= ^ (7 ± ^173). It will be found on trial that the lii'st two only are solutions of the proposed equation ; the others apply to the equation x' -7x- J{x' -7x+\ 8) = 24. (2) Suppose x* + x'-ix' + x+\ -0. Divide by x' ; thus a;^ -r a; - 4 + -f- ., - ; X X' 12—2 180 EQUATIONS WHICH MAY BE or x^ + —o + x + --4: = 0: X x therefore (^ "^ ~) + (^ + -) " ^ = ^ ^ therefore f^'''") +(^+-) = ^> and (^ + 1.) ^(•^^3+i = '5i=T' therefore therefore First suppose therefore therefore Next suj)pose therefore therefore ^ -^-Zx-^r -r --7 — \=-7 \ 4 4 4 , . 3 Jo , -3±^/o theiefore £c + ^ = ± -^ , and a; = (3) Siqipose 4 a;^ + 3a: 4 1 = So;'' + - ^^ ^x" Transpose x' — ox^ + 3a; + 1 = —^ ; therefoi'e (x''--^j — j- + 3a; + 1 = -q- ; thcrefo.0 (.'-|y-2(.= 4^)-J+l 1 a: + - a; 1 ^2 5 1 a; + - X = 2, or -3 1 a; + - : X = 2j x'- 2.X' + 1 - j X- ]. 1 a; + - = a; -3; x' + 3x. = -1; 4x' SOLVED LIKE QUADRATICS. 181 4x' 25a^ therefore (a:^ J^J - 2 (x^ -^^) ^ I =^ ^ ^ _ ^^ . Extract the square root, then - 3a; , 5x We have now ordinary quadratics, namely, x^ — 9" "^ ^ ~~a' » 3x , 5x ox 5x and X* — — — 1 = — _ . From the former we sliall obtain li a; --- J (7 ± ^85), and from the latter a; = ^ (1 ± ^/lO). (4) Suppose Ox Jx-Ux+ejx-l = 0. AVe may write tlie equation in the form {x-3 Jxy + 2(x-3 V'C) + 1 = ^'• Hence a; - 3 Jx + 1 = ± x. Take the upper sign ; thus a; — 3 Jx + 1 = a; ; therefore ^x = ^ , and a; = ^ . Take the lower sign ; tJius a; - 3 Jx + 1 =- x; therefore 2a; — 3 Jx + 1 = 0. From this we obtain Jx =1, or -^ , and therefore a; = 1, ov j, (5) Suppose X + C + J(x* - c') 9 (a; + c) .(1). x + c- J(x' - c") 8c In solving this equation we shall employ a principle which often abbreviates algebraical work. 182 EQUATIONS WHICH MAY BE Suppose that a p b=q' a will a + b p + q 6 - 5 ' a-b p—q b - q ' a+b p+q a-b p-q For the first of these three results is obtained by adding unity to each of the given equal quantities, the second is obtained by subtracting unity from each of the given equal quantities, and the third result is obtained by dividing the first by the second. Each result is sometimes serviceable. For the present example we employ the third. Thus from (1) we deduce 2 ^{x' — c') 9x' + c Square both sides, and simplify the left-hand member ; thus X— c {^x + cf .(2). Again, by employing the third of the above results we deduce from (2) X _ (9:c4-17c)^ + (9a; + c)^ _ (9a; + 17c)^+ (9a; + c)^ c ^(9x + 17c)'-(9x + c)''~ lGc(18a;+I8c) * By i-educing, w^e obtain Q>?jx^ — \^xc — Hoc' = 0, , . , . 5c 29c and from this, x= ^ , or x = — ^ . (6) Suppose "r - ^ ) + v/(3a^' -x) = -^ ^(1 - ix). Transpose ; thus ^ ,/(l - 4^) - y (^ - .r) = JiZax - X). SOLVED LIKE QUADRATICS. 183 By squaring, ^ (1 - ^x) - 3a J{1 - ix) V(^ |^ -xj = Sax - ^ = -J(l-44 Di\Ide by ;^/(l - ix) ; thus Oa^ + Sa ,,, , , „ //Sa V(l - -to.) = 3« /(^^ - .) By squaring, (1 + 3a)' (1 - 4a;) = 1G(-j — x) ; therefore 4a: {(1 + 3a)' - 4} = (1 + 3a)' - 12a = (1 - 3a)' j therefore ix (3a + 3) (3a - 1) = (3a - 1)' ; thei-efore x — s-^-r -rr . 12 (a + 1) Also corresponding to the fiictor ^(1 — 4x), v/hich "vras removed, we have the root x= - . 4 This example is introduced in order to di'aw the attention of the student to the circumstance that when both sides of an equa- tion are to be squared, an advantageous arrangement of the terms on opposite sides of the equation shoukl be made before squaring. If in this example as it originally stands we square both sides, no terms will disappear ; but by transposing before squaring we ob- tain a result in which — x occui-s on both sides, and may therefore be cancelled. (7) Suppose We have identically ^2^9_(.y2_9) = 18 = 34-16. Hence, dividing the members of this identity by the cor- responding members of the proposed equation, we obtain V(^^ + y)-V(^'-9)=V(34)-4. 184? EQUATIONS WHICH MAY BE SOLVED LIKE QUADRATIC'S. Tlierefore, by addition, J{x'' + 9) = ^/(3-i) ; therefore x' = 25, and a? = ± 5. Tliis equation is introduced for the sake of ilhistrating the artifice employed in the solution. This artifice may often be em- ployed with advantage ; for instance, example (6) may be solved in this way. (8) J(2cc.i)-2J(2-.)^J0^^. We may write this equation thus. The factor ^{2x + 4) - 2 ^(2 — x) can now be removed from both sides ; thus we obtain J{9x' + 16) =. 2 y{2x + 4) + 2 ^(2 - x)]. By squaring, 9x- + 16 = 4 {12 - 2a; + 4 ^(8 - 2x')] ; therefore x' + Sx=^ i{S - 2x') + 16 J(S -2x'); therefore a;- + 8a; + 1 6 = 4 (8 - 2a;') + 1 6 ^(8 - 2a;') + 16. Extract the square root ; thus ±(a; + 4) = 27(8-2a;') + 4. The solution can now be completed ; we shall obtain 4^2 and also a pair of imaginary values. Also, by equating to zero the factor ^(2a; + 4) - 2 ^(2 - x), 2 which was removed, we shall obtain x = -^. It will be seen that very artificial methods are adopted in some of these examples ; the student can acquii'e dexterity in using such transformations only by practice. More examples will be found in Chapter liv. EXAMPLES. XXI. 185 EXAMPLES OF EQUATIONS REDUCIBLE TO QUADRATICS. 1. ?>x + 2Jx-\=^0. 2. x'' + 31a;' = 32. 3. 3a;' + 42a;t = 3321. 4. 1 1 a"- 13a;"" = 14. 5. a;«_35a;^+ 216-0. G. 1 2 a;" - a;" +2 = 0. 7. x + 2 sj{ax) + c = 0. 8. 3a;'-7a;'-4307G. 9. a;'-14x- + 40 = 0. 10. 2a;* 2r" 11. V(2^)-'J^^ = -52. 12. 3a;"ya;"+ 3^.= 16. 13. aj + 5-V(a^ + 5) = G. 14. 2^a;+4-=5. Va; 15. a;U5a;-^-22-0. IG. 3a;^-4a;^ = 7. 17. 2a; + V(4«5 + 8) = ^. IS. ,2 (a;- + a;~'») = S. 19. ^(2aj + 7) + V(3a:-18) = V(7a;+l). 20. .^^:i:^)h-V(-.3)= ^ 7(a; - 3) ^^ '~ J{x - 3) ' 21. J {a + a;) + J{(i -x) = Jh. 22. 7(a;+9) = 2 Va;-3. 23. a; + V(5^ + 10) = 8. 24. 2'+' + 4' = 80. 25. ^^-11^ + ^ = 39. a;-2 a;+ 1 26, V(^ + ^) ^ v/(^-a^) ^ 28. {a + h) J{cr + h' + x') - (a - h) J {a" + lr- x') = a- + h\ 29. x + Jx+ J{x + 2) + Jix' + 2x) = a. 186 EXAMPLES. XXI. 30. 2x+J{-2 + 2x) = c{l-x). -- a —X a + x . J a + J(ct -x) Ja + J{a + x) ^ /{x+2a)~J( x~2a ) ^ x^ /{x-2a)+J{x + 2a) 2a' 33. J(x+S)-J{x+3) = ^fx. 34. J(x + 3)+J{x+8) = 5jx. x^ — a} x'' + a' 34 30. -: :, + ~ o = T^ . X' + a' X' — a' 1 36. J{a + hx"") -Ja = c ^/(6x'"). 37. J{x^i)-Jx = J{x + l 38. x'+-,-a''-- = 0. 39. qoi- = — o r^. X a' yol X —a 1 11,1 11 .^ cr + rc' a'^-x^ , 43. J{1 -x + x^)- J(l + X+ X-) = m. 45. J{ar - 3ax + a}) + J(x' + 3aaj + a') = J {2a' + 2^/^). 47. ^^(^^)-^^(C/^'4^x)^0. 48. Jx^J{x-J{\-x)] = \. 49. (a: + «)'-(-^-a)' = 242a\ >0. — o — =- = a; + X' —1 \/ x' EXA3IPLES. XXI. 187 51. J{x' + ax + b') + J(x' + Ix + a^) - a + K 25a;^-ir) _ 2>{x'~4.)x ^ ' 10.<;-8 2x - 4 ' 53. J{2x + 9) + J{^x -15)^ ^/(7.x- + 8). V (X \/ ( «6c J 55. J{x' + 2x - 1) + J(x' + x + l):r. J2 + J.]. 56. J{x'' + ax-l) + J(x' +hx-l)= J a + Jb. 57. (.'c-+l)(x'4-2) = 2. 58. {x--¥a){x + b)^ab, 59. (a; - o) (a; - b) (cc - c) 4- a^^c = 0. 1 1 4:X GO. 1 - X 1 -r oc 1 + rc" Gl. 7+— f+ 7+ A = ^' a;4-a + 6 it — «4-6 a: + a-o x — a — b G2. (a - a:) (a; 4- wi) (a •¥ x) {x — m) X + n x — n 63. f^'=l+q. \a — ay ao 64. 2x-\-l+x J{x' + 2) + (cc -f 1) ^/(rc' 4- '2x + 8) - 0. 65. x' + 3 = 2 Jifc' - 2x + 2) + 2x. 66. re' 4- ou; + 4 = 5 J{x' 4- oo; + 2S). 67. J{x'-2x+0)-^=3-x. 68. 3a;' 4- 15a; - 2 ^/(;e' 4- 5a; 4- 1) = 2. 69. (x + d){x- 2) + 3 J{x (x + 3)} = 0. 3 2 • 71. a;(a; + l)4-3 V(2a;^4-6a; + 5) = 25-2a; 3 70. a;' 4- 3 - J(2x' - 3x +2) = ^{x + l). 72. ar - 2 JiZx' - 2ax + 4) + 4 = ?,^ (a: + ^ + 1). 188 EXMIPLES. XXI. 73. x'-x-\-ZJ{2x'-Zx-¥2) = % + 'i. t 4. i = -x — x^, \ + X + X' 75. {x + a) {x + 2«) (a: + 3a) (a; + 4a) ^- c*. 7G. IGx (a; 4- 1) {x + 2) {x + 3) = 9. 78. a - a;* + (1 - a;)\ 80. a;'- 2a;' + a- =132. 81. Jx + ^/(a; + 7) + 2 J{x^ + 7a;) = 35 - 2x. 82. a;'- 8 (a; + 1) ^/a; + 18a; + 1-0. 83. 2 (a;* + axf +Jx + J{a + x) = b- 2x, 84. a)'+2a;'-lla:'4-4a; + 4 = 0. 85. x* ■\- 4a'x = a*. 86. a;* + ax^ + 5a;^ + ca; + — = 0. 87. 1+ /(1-"U /(1+" 77. 70. a' + ax + x^ a' CI? — ax ■\-x' x" - 2x^ + X X' = a. \/ \ xj V \ ^* 88. a^2 + i + 2 fa; + -^ = ^ a; V a;/ 9 V \ xJ v \ ^v ^ (^ + 1) 2 91. a;' + 1=0. 92. nx^ + x + n + 1 = 0. 93. (a;- 2) (a;- 3) (a;- 4) = 1 . 2 . 3. 94. (a;- 1) (a;- 2) {x- 3) - (6 - 1) (6 - 2) (6 - 3) = 0. 95. (a;- 1) (a;- 2) (a;- 3)= 24. 9G. Ga;' - ox' + a; = 0. 97. a;' + a;'' - 4a; - 4 = 0. 98. - + - + _ = !+-+ — . a X X a a' 99. 8.x'' + lGa; = 9. 100. x^-~=.U. 3a; 101. ort;'^ + 8a;* - 8a;- = 3. EXAMPLES. XXI. 189 102. x(x'-2) = m {x"- + Imx + 2). 1 03. {x- - a") {x + a) 5 + (a' - b') {a + b}x + {b" - x") (6 ■¥x)a^ 0. 10:t. x' ^i^x- ■vU-\+ —-^ a; + 1 - 0. 105. {li - Vf x' + 'px' + A; _ 1 + _.-L ^ .,; +1 = 0. XXII. THEOEY OF QUADRATIC EQUATIONS AND QUADRATIC EXPRESSIONS. 334. A quadratic equation cannot have more than two roots. For any quadratic equation will take the form ax"^ + 6x + c ^ if all the teims are brought to one side of the equation ; and then by Art. 318 the value of a: must be either -b + J{b'-\ac) -b~ J{ b^^ iac ) 2a ""'' 2a ' that is the value of x must be one or the other of two quantities. The result is sometimes obtained thus. If possible let tlij-ee different quantities a, fi, y be roots of the quadratic equation ax' + bx + c=0 f then, by supposition, aa' +ba + c = 0, a/^- + b^ + c = 0, ay' + by + c = 0. By subtraction, a {a' -fi') + b{a-/3) = {); di'dde by a — /3 which is, by supposition, not zero j thus a{a+/3)+b = 0. Similarly we have a (a + y) .+ 6 = 0. By subtraction, a (J3 — y) = {); this however is impossible, since by supposition a is not zero, jind /S - y h not zero. Hence there cannot be three diflerent roots to a quadratic equation. 1J)0 THEORY OF QUADRATIC EQUATIONS 335. In a quadratic equation where the coe^cient of the first term is unity and the terms are all on one side, the sum of the roots is equal to the coefficient of the second term with its sign changed^ and the iwoduct of the roots is equal to the last term. For the roots of a:c^ + 6x + c = are - 6 + J{h- - Aac) 1 - & - J{^' - 4ac) _ and - j 2a la hence the sum of the roots is , and the product of the roots is a ^^ — that is, - . And by dividing by a the equation 4cr a JjJC c may be written x' -\ + - = ; and thus the proi:)osition is esta- *' a a blished. 336. Let a and /3 denote the roots of the equation ax^ + 6ic + c = ; he then a + /5 = and aB = -. These relations are useful in findini; ' a a the values of expressions in Avhich a and j3 occur in a symmetrical manner. For example, tv a tv 1 1 _a + /5__^_^c__6 a ./? ayS a ' a c ' The relations demonstrated in Art. 335 are useful in verifvinof the solution of a quadratic equation ; of course if the roots ob- tained do not satisfy these relations we are certain that there is some error in the work. When we know one root of a quadratic equation we can deduce the other root by the aid of either of these relations. Take for example the equation a + c h + G _2(a + h + c) X + a ' X + h X -h a + b AXD QU^M)RATIC EXPFvESSIONS. 191 Here x = c obviously satisfies the equation ; clearing of fractions we obtain {a + h) x^ + {a' + h- -c{a + h)}x-c (a' + h") = 0. Tims the product of the roots is ^-^ — -— - : and as one root a+ 2 12 is c the other must be — a + b 337. We have ax" + hx + c = f o hx c] L a a) be. now put for - and - their values in terms of a and B ; thus a a '^ ' ax^ + bx-\-c = a{x' — {a + /3) x + ajS] = a {x— a) (x — ^). Thus the exjyj'ession ax^ -\-bx + c is identical \\'ith the expres- sion a (x — a)(x — 13) ; that is, the two expressions are equal for all values of x. Hence we can prove the statement of Ai-t. 334 in another manner. For no other value of x besides a and j3 can make (x — a) (x — /3) vanish ; since the product of 'two quantities cannot vanish if neither of the quantities vanishes. The student may naturally ask if the identity ax' + bx -i- c = a (x — a) (x — /3) holds in those cases alluded to in Art. 323, where the roots of ax' + bx + c = are impossible ; we shall return to this point in Chapter xxv. 338. The student must be careful to distinguish between a quadratic equation and a quadratic expression. In the quadratic equation ax' + bx-^ c — we must suj^pose x to have one of tv.-o definite values, but when we speak of the quadratic expression ay? + 6cc + c, without saying that it is to be equal to zero, we may supj)Ose X to have any value we please. 339. We have 3 1 r 2 ^^ ^1 ax +bx + c ^ a{x + ■ — + - r {. a a) (/ bV c K-) (/ bV b'-4ac) (\ 2a J a ia-j [\ zaj ia J 192 THEORY OF QUADRATIC EQUATIONS Now first supiiose that h^ — iac is negative ; then — -— - — is / hV l/-iac . M . . also negative ; hence ( .'C 4- — 1 j-^ — is necessarily positive \ ^CV/ T.Cb for all real values of x. In this case, ax^ +hx + c being equal to the product of a into some positive quantity must have the same sign as a. Thus if h^ — iac be negative, ax" -^-hx ■¥ c has the same sign as a for all real values of x. Next suppose that h^ — iac is zero ; then ax" + hx + c ("^iy- Here, as before, ax"^ + hx + c has the same sign as a ; in this case the expression ax^ + hx + c is a i^erfect square with respect to x^ and its square root is i ^a (x + - \ 2a Last, suppose that h^ - 4«c is positive ; then (, '2a 2a ) ( 2a 2a ) ~a{x— a) (x — /3), ■\\-hei-e a and y8 are both real quantities, namely, -h-^J{K'-Aac) . _ -h-J{Jr-Aac) a = ^ '- and yS = --J^ — . 2a 2a The expression a (x — a) (x — /3) must have the same sign as a except when one of the factors x — a and x — ^ is positive, and the other is negative ; and we shall now shew that this can only be the case when x lies in value between a and /?. Of the two quantities a — ^ and /S — a one must be jDOsitive j suppose the former, so that a is algebraically greater than yS. Now if x is algebraically greater than a, then a; - a is positive, and therefore also a;-^ is positive, a,nd if x is algebraically less than /?, then a — ^ is negative, and therefore also x — a is negative. But if x lies between a and p, then a; - a is negative, and x- p is positive. AND QUADRATIC EXPRESSIONS. 193 For such a value of x the sign of the exj^ression ax^ + 6a: -f c is the contrary to the sign of a. The conclusion of the investigation of the three cases is this : whatever real value x may have ax' -^hx^- c and a never differ in sign, except when the roots of ax^ + 6x + c = are possible and different, and x is taken so as to lie between them. 340. The roots of ax +bx + c^O are ^ ' za and the roots of hx + c=.0 are ^^-^^r '- . 2a It is obvious that the latter roots are the same as the former with their signs changed. Hence if two quadratic equations differ only in the sign of the second term, the roots of one may be obtained by changing the signs of the roots of the other. 341. Suppose we want to divide ax" \-hx + c by x — h. The first term of the quotient is ax, and the next term ah + h, and there is a remainder ah' + hh + c. If this remainder vanish, so that ah' + bh + c ~0, then /i is a root of the equation ax' + bx + c = 0. Thus the expression ax^ + bx + c is divisible hj x — h only when A is a root of the equation ax' + ux + c = 0. 342. Some particular cases of the equation ax^+bx + c=0 may now be investigated. The roots of the equation are -b + Jib'-iac) -b-J(b'-iac) we will first examine the results of supposing a = 0. The numerator of the first root becomes —b + b, that is, ; thus this root takes the form .- . The numerator of the second root becomes — 2b ; thus this root takes the form — r— . If in the original equation we put a= 0, it becomes bx + c~0, so tluit T. A. 13 194 QUADRATIC EQUATIONS AND EXPRESSIONS. x = — ; and we m, q, k are any rational quantities. 1 4. Shew that if ax^ +hx + c^O and a'x^ + h'x + c' = have a common root, then (ofc - ac'Y = (a'b - ah') (b'c — ch). 2x — 7 15. If flj be real, prove that 7^-^ — ~ can have no real ^ 2x - 2x-5 value between — and 1. 16. If p he greater than unity, then for all real vpJues of x ^ ^^ ^ 73' 7j — 2 the expression — ; r^ —, lies between and X' + zx + 2^ 2^ + 1 p + l SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 107 XXIII. SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 31 G. "We v.ill now give some examples of simultaneous equa- tions ^yllere one or more of the equations may be of a degree higher than the first ; various artifices are employed, the proper application of which must be learned by experience. (1) Suppose x^ — 2ij' = 71, X + 1/ = 20. From the second equation y = 20 — a: ; substitute in the first, thus x'-2('20-xy=7l; therefore - re' + 80x - 800 = 7 1 , therefore a;"— 80x = — S71. From this quadratic we shall obtain a: =13 or 67; then from the equation y=20 — x we obtain the corresponding values of y, namely, y = 1 or — 47. (2) Suppose X' + y" — 25, xy = \'2. Here x' + 7f = 25, 2xy = 24 j therefore, by addition, a;=' + 2a;?/ + 2/' = 25 + 24 = 49; that is, (x + y)' = 49 ; therefore a; + y = =fc 7. Similarly, by subtraction, {x-yy = 25-2i = l; therefore x - y = ^\. We have now four cases to consider ; namely, x + y = 7, x-y^ 1; x + y^-7, x-y= 1; x + y=:7, x-y=--l-y a; + y=-7, x-i/=---l. 198 SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. By solving these simple equations we obtain finally a; = ifc 3, 7/ = ± 4 ; or a; = ± 4, y = ± 3. (3) Suppose 2?/' - ixy + 3a;= - 1 7, 7/ - a;' = 1 6. Let ?/ = vx, and substitute in both equations ; thus x' {2v- - 4z; + 3) = 1 7, x' {v' -l) = \(3 ; therefore, by division, 2^- -4^ + 3 17 therefore 322;= - G4v + 48 = 1 7t;' - 1 7 ; therefore 1 5^'' - G4v + G5 = 0. 5 13 Prom tliis quadratic we shall obtain v = -^ or ■ — . Take the 3 5 1 /? former value of v ; then x" = -. — - = 9 : therefore a; = ± 3 ; and y = vx = ^^. Again, taking the second value of v we have 3 25 5 , 13 ^ -~q'} therefore, a; = ± - ; and 3/ = =t — . The artifice here used may be adopted conveniently when the terms involving the unkno^Ti quantities in each equation consti- tute an expression which is homogeneous and of the second degree; see Art. 24. (4) Suppose x" ■¥ xy - Qy" = 24, x^ + 3xy - 10^^ = 32. Let y = vx'j substitute in both equations, and divide ; thus 1+3^-107;^ 32 _4^ l+v-Qv' ~24~3' therefore Qv^ — ^v+1 = 0. From this quadi-atic we shall obtain v = q or ^. The value 2j = - we shall find to be inapplicable ; for it leads to the inad- missible result a;^ X = 24. In fact the equations from which the values of v were obtained may be written thus, x' (1 - 2v) (1 + Zv) = 24, x' (1 - 2v) (1 + bv) = 32 ; SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 199 and hence we see that the vakie of v found from 1 - 2v - is 1 + 'Sv 24 inapplicable, and that we can only have = - - which 1 gives «? = 3 . Then a;'^^!- I) (1 + 1) = 24; therefore »" = 36 ; therefore ic = =fc G ; and y = =fc 2. (5) Suppose X + y - a^ x^ + y^ = h^, By division, — = — ; ^ x+y a ' that is, X* - x^y + x^y^ - xy^ + y* = - ; ' o o o b' or ic* + ?/* — xy {x^ + y") + x'y^ ^ — . CL Now since x -\- y = a, X' + y^ = a' — 2xy ; therefore x* + y* + 2x^y^ = {or — 'Ixyf = a* — ia^xy + 4a;V^ ; therefore a* + y* = a* — ia'xy + 2x~y'. By substituting the values of x^ + y* and a;" + y"- we obtain ^' a* — iccxy + 2ic"2/^ ~ ^2/ (<^' ~ ^^''^2/) + ^V^ — ~ > (X b' that is, 5x*"3/" — 5a'xy = — — a^. We may obtain this result also in another way. It may be shewn that a'^ = x^ + y^ + 5xy {x^ + y"") + \()x~y' {x + y) ; thus a^-h^ = f)xy {x^ + y^) + 1 Oaxy' ; and a^ = x^ + y^ + Zxy {x + y) = x^ ■¥ y^ + Zaxy : therefore ct* — 6* - 5x'7/ (a^ — Zaxy) + lOax^y", or ^ax~y^ - 5a^xy = 6' — a*. 200 EXA^IPLES. XXIII. From this quadratic we can find two values of xy ; let c denote one of these values, then we have X + y = a, xy - c \ tlius {x. ^- y)' — ixy = «^ — 4c, that is, [x — yY = a' — ic ; therefore a; — ?/ = ± J{a^ — 4 c). Thus since x + y and x — y are known, we can find immediately the values of x and y. Or we may proceed thus. Assume x — y = z, then since X + y = a, we ol^tain x = -^{a + z), y = -{a-2). >Substitute in the second of the given equations : thus {a + zy + {a-zy = 32b', therefore 5az' + 1 Oa'z' = 16b' - a'. From this quadratic we may find z^, and hence z^ that is, X - y \ and hence finally x and y. More examples will be found in Chapter liv. EXAMPLES OF SIMLTLTANEOUS EQUATIONS INVOLVING QUADRATICS. 1. 4aj= + 7/ = U8, Zx'-y'=W. 2. a; + ?/ = 100, .r^ = 2400. o. . A "111 6. £c + 2/ = 4, - +- = 1. 4. iB + 2/ = 7, a;' + 2/=34. 5. x-y-\1^ ic^ + ?/^ = 74. ,. x — y, x+^y ^ 2 ' -^ x + '2 7. a;=+2/' = C5, a'^ = 28. 8. .T2/ = l, 3^'-5y=2. 9. - + - = 2, a; + 2/ = 2. X y ' ^ EXAMPLES. XXIII. 201 10. x' + xy+2y' = 1\, 2a:' + 2^y + y'- 73. 1 1 14 11. 2a: + 3v=37, t + ^.=tl, 12. x^ ■¥ 3xy = 54, xy + 4y' = 115. 1 3. XT + xy = 1 5, xy — y'^ = 2. 14. x' + xy + Ay^=G, 3x' + 8y'=li. 15. a;'' + a;y = 12, xy-2y' = I. 16. x'-xy + y^ = 2l, y' — 2xy + 15 = 0. 17. x'-i2/ = 9, xy + 2y' = 3. 18. 7x^-Sxy = ld9, 5x + 2y^7. 19. x'-2xy-y'=l, x + y=2. „^ x + y x-y 10 , - 20. — -' + ^ = --- , a;' + y = 45. x — y X + y 6 21. ^±Z + ^Zl_5 «;' + ^» = 20. X — y x + y 1 22. 'ly + '12dx = y — X, y - '5x = '7Dxy—Sx. 23. '3x4- -125?/ = 3a; -7/, 3a;- -52/ = 2-25a;?/+ 3y. 24. y'~ 4xy+20x'+ 3y - 264a; = 0,) 5y'-3Sxy+ x^ - 12y+ 1056a; = O.j' 25. x + y = x', 3y - X = y^. 26. x^ + y^^-^xy, x-y=-xy. 27. ^ + 2y + — -16, 3x + y + ~=23. ^ y y 28. 4 (a; -t- ?/) = 3xy, x+ y ^x' + y'' ^- 26. 29. x-y = 2, x'-y' = S. 30. a; + 2/ = 5, a;' + ?/" = 65. 31. a; + ?/ = ll, a;' + 2/'= 1001. 32. a:^/ (a; + ?/) = 30, x' + y' ^ 35. 202 EXAMPLES. XXIII. 2 3 33. -+^ = 18, x + 7j = 12. U. .T + ?/=18, a;' + 2/' = 4914. on ^" y" ^ 113 35. — + ^ = 9, ^- + - = y . y X X y A 3G. a;- (.« + ?/) - 80, x" {2x - 3y) - 80. 115 37. a;V + 2/'^' = 20, - + - = v • 38. x^ + ?/' = 7 + ajy, o;^ + 7/^ = 6xy - 1. 39. a;= + / = 8, -o + i=i. •^ ' X- y^ 2 40. a; + 2/ -4, a;' + y* = 82. 41. a;'-y=3093, aj-2/=3. 42. f3--^Y+ (3 + ^^82, xy~-2. \ x + yJ \ x-yJ 43. x^ - x-f + 2/" ^ 1 9, X - xy ^- y = ^:. 44. x^ -xy ^-y^ = 1, aj* + a^y + 2/''= 133. 45. a;^ + a.'2/ + 2/' = 49, a;* + a:;y + 2/*= 931. 46. a;'' -a;' + 2/' -2/' = 84, a;=' + a^y + 2/' = 49. 47. x{l2-xy) = y{xy-2>), xy {y + ix-xy) = l2{:c + y -Z). 48. a; + 2/ + Ay(^2/) = 14, x- + y^ + xy=^i. 49. x + y - J{xy) = 7, x^ ^y^ + xy = 133. 50. a; + 2/ = 72, ^a; + 4/2/ = 6. 51. a; + 7(a;' - 2/') = 8, x-2/=l. EXAMPLES. XXIII. 203 54. Jx-Jy=^2 J{xy), x + y = 10. 55. J(x + y) + 2 J(x ~ y) = - , { , ^ ^ — . 56. ^(3 + o;^) + 2y = 8, 2a;' + Jipy'' + 4a:^) = 9. tz^ X y ^ ah 58. x^ -y^ = a', xy = 6^ 59. x + y ^a, o;^ -f ?/* = 6*. 60. a;"'4-7/*= 14xy, x + y = a. Dl. + -. ^=1, x + 7/^a + o. a + X + y y+0 x+a 2 a o 63. x-y~a, x^-y^ = b\ 64. J{x' + y') + J{x'-f) = 2y, x'-y'^a\ 65 . 2a6 {a-\-h)x-\-y^ = abx^ + 2a62/, a6x + {a + b)y ~ xy. 66. 2VC^'-2/0 + a:y = l, ^-^ = «. y X 67. x + y = aj{xy), x-y = c /'^ . 68. 7(a; + y) + ^(a; - y) = ^a, ^(rc' + y") + ^/(a;^' - ?/') = h. \y -b' a - xy \y' + b^ a + x J 70. a;^ + 2/* - (a; + ?/) = a, a* + ?/* + a; + 7/ - 2 (a;'' + t/'^) = 6. 71. yz = hc, -+f-l, ^ + - = 1. 72. 1+1 + ^^9, ? + ^ = 13, 8a; + 3y = 5. X y z X y OQ4. EXAMPLES. XXIII. 1 1 1 73. 2/ + ^ = ^' -^^ = y^ x + y^^. 7 i. xyz = a^y + z) = b' {z + x) = c' {x + y). 75. X' + yz = y^ + zx= c, z' 4 xy = a. X + y + z = 15. "^ i("?)4('n')-J 77. a? + y + 2! = ::, + -+-= 9, xt/z=i. 1 1 1 xyz 78. x^ + y^ + z^ = x^ + y' + z' = X + y + z = 1. 79. a: (a: + y 4- ;:;) = a^ y[x + y + z) = b'y z(x + y + z) = c' 80. xy + xz + yz = 26, ^ iry (ic + 2/) + 2/^ (y + ^;) + ^a: (;^ + ic) = 1 62, V xy {x' + 2/^) + yz [y' + z') + xz {x' + ^s") = 538. j XXIY. PEOBLEMS WHICH LEAD TO QUADRATIC EQUATIONS. 347. We sliall now solve and discuss some problems which lead to quadratic equations. A man buys a horse which he sells again for £24:; he finds that he thus loses as much per cent, as the horse cost; required the i^rice of the horse. Let X denote the price in pounds ; then the man loses x per X XT cent, and thus his total loss is r-j— x x, that is, — — ; but this loss is also ic - 24 ; thus -— - a: - 24 • 100" ' PROBLEMS WHICH LEAD TO QUADRATIC EQUATIONS. 205 therefore jc" - 100a; = - 2400, and x' - 100a; + (50)' = 2500 - 2400 = 100 ; hence x - 50 -^^10, and ic := GO or 40. Thus all we can infer is, that the price was either £Q0 or ^40, for each of these values satisfies all the conditions of the problem. 348. Divide the number 10 into two parts, such that their product shall be 24. Let X denote one part, and therefore 1 — a; the other part ; then a;(10-a;) = 24; therefore «* - lOaj = - 24, and a;' - lOx + 5- = 25 - 24 = 1 ; hence a;-5=±l, and a; = 4 or G. Here although x may have either of two values, yet there is only one mode of dividing 10, so that the product of the two parts shall be 24 ; one part must be 4 and the other 6. 349. A person bought a cei-tain number of oxen for £80 ; if he had bouo^ht 4 more for the same sum each ox would have cost .£1 less ; find the number of oxen and the price of each. 80 Let X denote the number of oxen, then — is the price of each X in pounds ; if the person had bought 4 more, the price of each in 80 pounds would have been - — j : thus, by suj^position, a^ T "X 80 _ SO _ X + 4: X therefore 80a; = 80 (a; + 4) - a;" - 4a:, 206 PROBLEMS WHICH LEAD TO therefore x'' + ix = 320, and x' + 4:X + 2'=320 + i = 324 ; hence a; + 2=±18, and - £C = 16 or — 20. Only the positive value of aj is admissible, and thus the niunber of oxen is 1 6, and the price of each ox is £5. In solving problems, as in the proposed example, results will sometimes be obtained which do not apply to the question actually proposed. The reason appears to be that the algebraical mode of expression is more general than ordinary language, and thus the equation, which is a proper representation of the conditions of the problem, will also apply to other conditions. Experience will convince the student that he vrill always be able to select the result which belongs to the problem he is solving, and that it will be sometimes possible, by suitable changes In the enunciation of the original problem, to form a new problem, corresponding to any result which was inapplicable to the original problem. Thus in the present case we may propose the following modification of the original problem : a person sold a certain number of oxen for £80 ; if he had sold 4 fewer for the same sum, the price of each ox would have been £1 more ; find the number of oxen and the price of each. Let X represent the number ; then by the question we shall have 80 80 , = — + 1. X— 4 X The roots of this quadratic will be found to be 20 and — 16; thus the number 20 which appeared with a negative sign as a result in the former case, and was then inapplicable, is here the admissible result. 350. Find a number such that twice its square increased by three times the number itself may amount to 65. QUADRATIC EQUATIONS. 207 Let X denote the number ; then, by the question, 2x'-\-3x = Q5. 13 The roots of this quadratic will be found to be 5 and — — ; A tlie first value satisfies the conditions of the question. In order to interpret the second value, we obseiwe, that if we write — x for x in the equation, it becomes 2x' - 3a; = 65 ; 13 and the roots of the latter equation are -— and — 5, as will be 13 . found on trial, or may be known from Ai-t. 340. Hence — is the A answer to a new question, namely : find a number such that twice its square diminished by three times the number itseir may amount to 65. 351. Divide a given line into two parts, such that twice the square on one part may be equal to the rectangle contained by the whole line and the other part. Let a denote the length of the line, and x the length of one j)art, then a — x is the length of the other part ; thus, by the question, 2a;^ — a{ci — x) ', therefore 2x^ -^ ax = a", „ ax a^ and X' + -Tr- = o" , 12 A , g ax /a\^ a' a^ 9a^ and ,,+_+/_j =_+_ = _; , a Sa hence aj + - = ± — - 4 4 and x = -x ov - a. A a Here - is the required length. The negative answer sug- A gests the following problem : produce a given line, so that twice the square on the part produced may be equal to the rectangle 208 PROBLEMS WHICH LEAD TO QUADRATIC EQUATIONS. contained by the given line, and the line made up of the given line and the part produced ; the result is, that the part produced must be equal to the given line. 352. In the examples hithei-to given, both roots of the quad- ratic equation have applied to the actual problem, or to an allied problem which was easily formed. Frequently, however, it will be found that only one root applies to the problem proposed, and that no obvious interpretation occurs for the other. 353. Problems may be proj^osed which involve more than one unknown quantity, and thus lead to simultaneous equatioiis ; we will give an example. Two men A and B sell a quantity of wheat for X28. 85. B sells four quarters more than A, and if he had sold the quan- tity A sold, would have received XI for it; while A would have received 16 guineas for what B sold. Find the quantity sold by each, and the rates at which they sold it. Let 33 denote the number of quarters which A sold, and there- fore 03 + 4: the number which B sold ; and suppose that A sold his wheat at y shillings per quarter, and that B sold his at z shillings per quarter. Then sioce the value of the wheat sold is 568 shil- lings, we have xy + {x + i)z=bQd> (1). If B had sold the quantity A sold, he would have received 200 shillings ; thus a:;^ = 200 (2). Similarly, (a; + 4)?/ = 336 (3). From (3) we have ajy = 336-4?/; by substitution in (1) we have 336-4y + 200 + 4;^ = 568; therefore 4 (« - 3/) = 32, and z-y = ^ (4). EXA^Q'LES. XXIV. 200 From (2) wo liave 200 X = z ' and from (3) we have 336 , X — 4 ; y 200 33G ^ thus - = 4, z y and — = — -1 (5). z y We may now find y and z from (4) and (5). Substitute in (5) the value of z from (4) ; thus 50 8f -, . y+« y therefore 50y = 84 (y + 8) - (2/' + 8^/), hence ?/*-26?/-672 = 0. From this quadratic we shall find 2/ = 42 or - 16, The former is the only admissible result ; thus z=60\ and a; = 4. EXAMPLES OP PROBLEMS. 1. Find two numbers such that their sum may be 39, and the sum of theii- cubes 17199. 2. A certain number is formed by the product of three con- secutive numbers, and if it be divided by each of them in turn, the sum of the quotients is 47. Find the number. 3. The length of a rectangular field exceeds the breadth by one yard, and the area is three acres : find the length of the sides, 4. A boat's crew row 3J miles do^vn a liver and back again in 1 hour and 40 minutes : supposing the river to have a cuirent of 2 miles per hour, find the rate at which the crew would row in still water. T. A. U 210 EXA^IPLES. XXIV. 5. A farmer wishes to enclose a rectangular piece of land to contain 1 acre 32 perches with 176 hurdles, each two yards long; how many hurdles must he place in each side of the rectangle ? 6. A person rents a certain number of acres of land for £84 ; he cultivates 4 acres himself, and letting the rest for IO5. an acre more than he pays for it, receives for this portion the whole rent, £84. Find the number of acres. 7. A person purchased a cei-tain number of sheep for £35 : after losing two of them he sold the rest at 10 shillings a head more than he gave for them, and by so doing gained £1 by the transaction. Find the number of sheep he purchased. 8. A line of given length is bisected and j^roduced : find the length of the produced part so that the rectangle contained by half the line and the line made up of the half and the produced 2:)art may be equal to the square on the produced part. 9. The product of two numbers is 750, and the quotient when one is divided by the other is 3^ : find the numbers. 10. A gentleman sends a lad into the market to buy a shil- ling's worth of oranges. The lad having eaten a couple, the gentleman pays at the rate of a penny for fifteen more than the market-price ; how many did the gentleman get for his shilling ? 11. What are eggs a dozen when two more in a shilling's worth lowers the price one penny per dozen? 12. A shilling's worth of Bavarian kreuzers is more nume- rous by G than a shilling's v,^orth of Austrian kreuzers ; and 1 5 Austrian kreuzers are worth Id. more than 15 Bavarian kreuzers. How many Austrian and Bavarian kreuzers respectively make a shillino: 1 13. Find two numbers whose sum is nine times their differ- ence, and whose product diminished by the greater number is equal to twelve times the gi-eater number divided by the less. 14. Two workmen were employed at different wages, and paid at the end of a certain time. The first received £4. I65., EXA^IPLES. XXIV. 211 and the second, who had worked for 6 days less, received £2. 14^. If the second had worked all the time and the first had omitted 6 days, they would have received the same sum. How many days did each work, and Avhat were the wages of each ? 15. A party at a tavern spent a certain sum of money. If there had been five more in the party, and each person had spent a shilling more, the bill would have been <£G. If there had been three less in the party, and each person had spent eightpence less, the bill would have been £2. 12s. Of how many did the party consist, and what did each person spend ? 16. A j)erson bought a number of ^20 railway shares when they were at a cei-tain rate j^er cent, discount for XI 500; and afterwards when they were at the same rate per cent, premium sold them all but GO for £1000, How many did he buy, and what did he give for each of them 1 17. Fiiid that number whose square added to its cube is nine times the next higher number. 18. A person has <;^1300, which he divides mto two portions and lends at different rates of interest, so that the two portions produce equal returns. If the fii-st portion had been lent at the second rate of interest it would have produced £36 ; and if the second portion had been lent at the first rate of interest it would have produced £19. Find the rates of interest. 19. A person having travelled 5Q miles on a railroad and the rest of his journey by a coach, observed that in the train he had performed a qiiarter of his whole journey in the time the coach took to ofo 5 miles, and that at the instant he anives at home the train must have reached a point 35 miles further than he was from the station at which it left him. Compare the rates of the coach and the train, and find the number of milos in the rest of the journey. 20. A sets off from London to York, and B at the same time from York to London, and tliey travel uniformly; A reaches York 16 houi-s, and B reaches London 36 hours, after they have 11—2 212 EXAJSIPLES. XXIV. met on the road. Find in what time each has performed the journey. 21. A courier j^roceeds from one place P to another place Q in 14 hours; a second courier starts at the same time as the first from a place 10 miles behind P, and arrives at Q at the same time as the first couiier. The second courier finds that he takes half an hour less than the first to accomplish 20 miles. Find the dis- tance of Q from P. 22. Two travellers A and B set out at the same time from two places P and Q respectively, and travel so as to meet. "When they meet it is found that A has travelled 30 miles more than B^ and that A Avill reach (^ in 4 days, and B will reach P in 9 days, after they meet. Find the distance between P and Q. 23. A vessel can be filled with water by two pipes ; by one of these pipes alone the vessel would be filled 2 hours sooner than by the other ; also the vessel can be filled by both pipes together in 1| hours. Find the time vvhich each pipe alone would take to fill the vessel. 24. A vessel is to be filled with water by two pipes. The first pipe is kept open during three-fifths of the time which the second woiild take to fill the vessel ; then the firet pipe is closed and the second is opened. If the two pipes had both been kej^t open together the vessel would have been filled 6 hours sooner, and the first pipe would have brought in two-thirds of the quantity of water which the second pipe really brought in. How long would each pipe alone take to fill the vessel ? 25. A certain number of workmen can move a heap of stones in 8 hours from one place to another. If there had been 8 more workmen, and each workman had carried 5 lbs. less at a time, the whole work would have occupied J hours. If however there had been 8 fewer workmen, and each workm.an had carried 1 1 lbs. more at a time, the work would have occupied 9 hours. Find the number of workmen and the weight which each carried at a time. IMAGINARY EXPRESSIONS. 213 XXY. IMAGINARY EXPRESSIONS. 354. Although the square root of a negative quantity is the symbol of an impossible operation, yet these square roots are fre- quently of use in Mathematical investigations in consequence of a few conventions which we shall now explain. 355. Let a denote any real quantity; then the square roots of the negative quantity —a^ are expressed in ordinary notation by =fc^(-a^). Now ~a^ may be considered as the product of a^ and - 1 ; so if we suppose that the square roots of this product can be formed, in the same manner as if both factors were posi- tive, by multiplying together the square roots of the factors, the square roots of —a^ will be expressed by ^a J[-\). We may therefore agree that the expressions ± J{- a') and ± a J(- 1) shall be considered equivalent. Thus we shall only have to use one imaginary expression in our investigation*, namely, J(—l). 356. Suppose we have such an expression as a + /3 J(—l), where a and (3 are real quantities. This expression may be said to consist of a real part a and an imaginary part /?^(- 1) ; or on account of the presence of the latter term we may speak of the whole expression as imaginary. Wlien /B is zero, the tenn y3y(— 1) is considered to vanish; this may be regarded then as another convention. If a and /? are both zero, the whole expres- sion vanishes, and not otherwise. 357. By means of the conventions already made, and tlio additional convention that such terms as ^ ^/(— 1) shall be subject to the ordinary rules which hold in Algebraical transfonuations, we may establish some propositions, as will now be seen. 358. Ill order that two imaginary expressioTis may he eqiud, it is necessary and sufficient that the real parts should he eqiml, and that the copfficients of J{— 1) shoidd he equal. 211? IMAGINARY EXPRESSIONS. For suppose a + /3 J{- 1 ) = y + 8 J(- 1) ; then, by transposition, a - y + (^ - S) ^(— 1 ) = ; thus, by Art. 35 G, a - y = 0, and /5 - 8 = ; that is, a — 7) ai^*-^ ft = 8. Thus the equation may be considered as a symbolical mode of assei-ting the two equalities a = >• and ^ = 8 in 07ie statement. 359. Take now two imaginary expressions a + /?^/(— 1) and y + S/s/(~l)) ''^^^'-^ form their sum, difference, product, and quotient. Their sum is a + y + (/? + S)V(-i). If the second expression be taken from the first, the re- mainder is a-y + (/3-S)^/(-l). Tlieir product is {a + ^ J(- 1)1 ly + S V(- 1)1 = ay - /3S + (aS + /3y) ^(- 1 ) ; for J(- 1) X J(- 1) is, by supposition, - 1. The quotient obtained by dividing the first expression by the second is a + /?s/(-l) This may be put in another form by multiplying both numerator and denominator by y — 8 ^/(- 1). The new numerator is thus ay + /5S + (/3y-aS)^/(-l); and the new denominator is y^ + 8" ; therefore a + ;gV(- l)_ ay + ^8 /?y-a8 360. We will now give an example of the way in which imaginary expressions occur in Algebra. Suppose we have to solve the equation x^=l. We may write the equation thus, a/- 1 = 0; or in factors, {x - I) {x^ + x + I) = 0. IMAGINARY EXPRESSIONS. 21 Thus we satisfy the proposed equation either by putting ic - 1 = 0, or by putting x^ + x + 1 =0. The first gives a; = 1 ; the second may be written x^ + X = -l, therefore x^ + x+ l-^j ~ i~ ^ ''^~ a' therefore x+ ^^ = ^/(- ^ - ± ^ J{-'^)y and ^=._Jivf;^/(_l). ^ .J Thus we conclude that if either of the imaginary expressions last written be cubed, the result will be unity. This we may verify ; take the upper sign for example, then .3(--y{fv(-i)}V{f./(-i)r Now (-!)=-§' ' (- 2)' f ^(- ') = 4 f ^'(- ^) - -f -'(- ^)' {^v'(-i)f={^v(-i)y^^/(-i) Thus the residt is unity. If cc^=l, we have x = {l)^ ; it appeai-s then that there are 1 ^/3 three cube roots of unity, namelv, 1 and - ^ ^ \ Ji~ ^)- 216 I^IAGINARY EXPRESSIONS. 3G1. We liave seen in Art. 337, that the quadratic expression ax' + hx + c is always identical with a (x —j)) {x — q), where p and q are the roots of the equation ax^ + 6a; + c — 0. If the roots are imaginary, ^; and q will be of the forms a^^ ^{—1) -, thus we have then ax- + hx+ c = a{x-a- p J{- 1)} {a; - a + /? J{- 1)}. This will present no difficulty when we remember the conven- tion that the usual algebraical operations are to be applicable to the term /? ^/(— 1). For the second side of the asserted iden- tity is a {{x - a)- + B'^}, that is, a [x^ - 2aoz ■¥ or + (i'}, and from the values of a and /? we have 2a = , and a" + B" = ; a a thus the second side coincides with the first. 362. Two imaginary expressions are said to be conjugate when they dilTer only in the sign of the coefficient of J{^— 1). Thus a + y8 Ji^— 1) and a — /?>/(— 1) are conjugate. Hence the sum of two conjugate imaginary expressions is real, and so also is their product. In the above example the sum is 2a, and the product is a^ + j8'. 363. The positive value of the square root of a^ + /3" is called the viodulus of each of the expressions a + PJ{-l) and a-/5^/(--l). From this definition it follows that the modulus of a real quantity is the numerical value of that quantity taken positively. In order that the modulus J(a' + fi') may vanish, it is neces- sary that a ^ and ^ = ; in this case the expressions a+PJ{-l) and a-l3J{-l) vanish. And conversely, if these expressions vanish, then a = and /3 = 0, and thus the modulus vanishes. IMAGINARY EXPRESSIONS. 21? 364. If two imaginaiy expressions are equal, their moduli are equal. It is not however necessarily tiiie, that the expressions are equal if the moduli are equal. 365. The modulus of the product of a + ^;^/(— 1) and 7 + 8 ^/(-l) is J{{a.y - phf + (^y + a8)=} ; (see Ait. 3 j9). But (ay - ^Zy + (/3y + aS)^' = {a' + j^') {y' + ^) ; thus the modulus is Hence the modulus of the product of two imaginary expres- sions is equal to the product of their moduli. Therefore the product of two imaginary expressions cannot vanish if neither factor vanishes. It will follow from this that the modulus of the quotient of two imaginary expressions is the quotient of their moduli. This can also be shewn by forming the modulus of the expression for the quotient given in Aii:.. 359. 366. It is often necessary to consider the powers of ^^(-1). We may form them by successive multiplication ; thus, y(-i)r=v(-i). y(-i)f=-i. {J{- l)r = {J{- 1)}-' X v/(- 1) = - ^/(- 1), { s/(- 1}}* = 1- If we proceed to obtain liigher powers we shall have a re- currence of the results J{-'^), -1, - J(-^)y 1- ^^^® may then express all the powers by four formulae. For every whole number must be of one of the four foi-ms in, in + l, in + 2, in + 3, according as it is exactly divisible by 4, or leaves, when divided by 4, a remainder 1, 2, 3, respectively. And {s/(-i)r=i, {^/(-l)r*'=^/(-l), ■ {s/(-i)r"=-i, {v/(-i)r"=-N/(-i)- 218 IMAGINARY EXPRESSIONS. 3G7. The square root of an imaginary expression of the form a + p J(— 1) vunj be expressed in a similar form. For suppose J {a + /? J{- 1 )} - ic + 2/ J{- 1 ) ; tlien a + 13 J{- 1) = {x + y ^/(- 1)}^ = x'-f + 2xy J(- 1). Hence, by Art. 358, x--y'^ = a (1), 2xy = (3 (2); therefore from (1) and (2) {x^ + yj = a^ + l3^ thus x' + r = J{o.' + f^') (3). From (1) and (3) we obtain hence x = ^ ^^ f^ j , 2/ = ± ^^— j-^ j . Since the values of x and y are supposed real, x' + y^ is j^osi- tive, and thus the positive sign must be ascribed to the quantity ;^(a' + ^^). And since the values of x and y must satisfy the equation 2xy — (3, they must have the same sign if yS be positive, and different signs if P be negative. On account of the double sign in the values of a; and y, we see that a-{-(3 J{—\) has two square roots which differ only in sign. 368. "We may obtain the square roots of ± J{— 1) by suj> posing that a = and /? = ± 1 in the results of the preceding Ai-ticle. Thus we shall obtain If we suppose that z* = — \, we deduce z^-'^J{-\)] thus 2; = ± ^{± J{- 1)}, And since z* = -\, we have 2; = (- 1) . Thus there are four fourth roots of — 1, namely, the four expressions EXAMPLES. XXV. 219 contained in ± ~J2' ^^^®^'® ^^'® ^^^^ ^°^^^* ^^^^'^^ ^'^^^^ of 1, since if we put z* = I, we find z' = ±l, and z = J= Jl or ^ = =*= v(- !)• Similarly there are eight eightli roots of 1 or - 1, and so on. mSCELLAXEOUS EXAMPLKS. 1. Simplify 4- — ~ + (a-b)(a-c) {b-c){b-a) (c-a)(c-b)' T^ ct — b c— d - , - it , 7 + t; — — , = 0, shew that \ + ab 1 + ctZ a-d b-c ^ a + c b + d and I + ad I +bc I — ac I — bd' 3. Shew that a'4-b' + c'-3abc = 1 {(a - by +{b- cf + (c - af\ {a + 6 + c}, a=' + &' + c' + 24a5c = (a + 6 + c)^ - 3 [a (6 - c)- + Z/ (c - of ^c{a- 6)*}, (a + 6 +c)^-27a6c = J {(a + 6 + 7c) (a - 6)- + (6 + c + 7rt) (5 - c)" + (c + « + lb) (c - a)*}, 9(a^ + ^'-' + c^)-(a + 6 + c)^ = {ia+4.b + c)(a-by-+(4:b + 4:C + a)(b-cy+(-ic + ia + b){c-ay. 4. Shew that if a + b + c is zero the following expression is also zero, b' c' , + ..., + 7^, r - 1. 2«" + be 2b^ + ca 2c- + ab 5. If the square root of the product of two quantities is rational, shew that the square root of the quotient obtained by dividing one by the other is also rational. 220 EXAMPLES. XXV. G. Extract tlic square root of [l + x] [l + x^ + 2 (I - x^) Jx]. 7. Express in the form of the sum of two simple surds the roots of the equation x* - 2ax^ + h' = 0. 8. Express in the form of the sum of two simple surds the roots of the equation ix* —4(1 + n') a'x^ + n^a* = 0. 9. By performing the operation for extracting the square root, find a value of x which will make x^ + Qix^ + 11 03^ +3^; + 31 a perfect square. 1 0. Shew that if x'^ + ax^ + bx' + ex + d be a perfect square, the coefficients satisfy the relations 8c = a{-ib-a') and {ib-ay=Ud. 11. If the values of x, y, x, y be all possible, and \+xx+ yy = 7(1 + X' + y~) J{1 + a;" + y^), shew that x = x and y = y\ 12. Shew that the equation a'b' {x - x'Y + a'b' {y - yj + (6 V + a\f - a'b') {b'x" + a'y"- a%') = is equivalent to the two a^b^ — a^yy — b^xx' = and xy'— xy = 0. 13. A man sells a horse for .£24. 125., and loses 18 per cent. on what the horse cost him : find the original cost. 14. Divide the number 16 into three such parts that the dif- ference of the two less shall be the square root of the greatest, and the difference of the two greater shall be the square of the least. 1 5. Shew that i+N/(-3)r , f -i-V(-3) r —2 1 + 1 2 J is equal to 2 if ?^ be a multiple of 3, and equal to - 1 if n be any- other integer. EXA]\IPLES. XXV. 221 Solve the following equations : -. x+l x+ 2 ^x + 3 16. + ^ = 2 ^, X — I X — 2 x — o 17. -2 ^=-^ +X''-X. ''■ (»'-x)(^-3(--S=(''-i)(--2)(-3)- 19. x*-8x'' + 12x'+ lGa;-16 = 0. 20. J{2x - 1) + J{3x - 2) ---- J{ix - 3) + J(rjx - 4). 21. 2h[J{x + a) -h] + 2c{J{x-a) + c] = a. 22. {J{a -\-x)- Ja] {J{a -x) + Ja] = nx. 23. X + y = a ■¥ h, - + - = 2. X y oi ax hi/ (a + h)c 24:. + r-^^ = -^ — 7-^ , x+y^c. a + X + y a + + c ^ \y x) 2Q. X {be - xy) = y {xy - ac), xy {ay + hx- xy) = ahc {x + y- c). 27. (x-3y + -){x + z) = 6, (x+^)-^d, 1+1+-=^. \ !^/ \ z) y X y z 2 28. (?; + a:) (y + 2;) = 6 + c - «, (?; + 2/) (;^ + a;) = c + a — 6, {v ^- z) {x ■¥ y) = a ^h — c, v' + a;' + ?/' + ;s' = 3 (a + 6 + c). 25. 6 f - - ^ ) = 5 = 6 f- + ^ \2/ «^/ \aJ y 222 RATIO. XXVI. RATIO. 369. Ratio is the relation which one quantity bears to another with respect to magnitude, the comparison being made by considering what multijjtle, part, or parts, the first quantity is of the second. Thus in comparing 6 with 3, wc observe that 6 has a certain magnitude with respect to 3, which it contains twice j again, in comjiaring 6 with 2, we see that 6 has now a different relative magnitude, for it contains 2 three times ; or 6 is greater when compared with 2 than it is when compared with 3. 370. The ratio of « to 6 is usually expressed by two points placed between tliem, thus, a '. b ] and a is called the antecedent of the ratio, and h the consequent of the ratio. 371. A ratio is measured by the fraction which has for its numerator the antecedent of the ratio, and for its denominator the consequent of the ratio. Thus the ratio of a to 6 is measured by J- ; then for shortness we may say that the ratio of a. to h is , a . a equal to r^ or is r- . 372. Hence we may say that the ratio of a to 6 is equal to 1 . - ^ , a G the ratio oi c to cL when - = - , b a 373. If the terms of a ratio be niultiplied or divided by the same quantity the ratio is not alteredo ^ a ma , . , i oe\ For - =— -, Art. 135). 6 mo ^ 374. We may compare two or more ratios by reducing the fractions which measure these ratios to a common denominator. RATIO. 223 Thus suppose one ratio to be that of a to 6, and another ratio to be that of c to d: then the first ratio -r = , ,, and the second bd c he ratio ~7 = f—i' Hence the firet ratio is greater than, equal to, or less than, the second ratio, according as ad is greater than, equal to, or less than be. 375. A ratio is called a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to, the consequent. 376. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding any quantity to both terms oj" the ratio. Let the ratio be -- , and let a new ratio be formed by adding ft -\~ cc X to both terms of the original ratio : then y is greater or less C/ I Co ,, a b(a + x) . «(^+«^) XT. X than Y , according as -r^. ( is greater or less than -r-^ ( : that b ° 6 (6 + a;) ° b{b+x)' is, according as b{a + x) is greater or less than a{b + x) ; that is, according as xb is greater or less than xa j that is, according as b is greater or less than a. 377. A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking from both terms of the ratio any quantity which is less than each of those terms. Let the ratio be y , and let a new ratio be formed by taking df — cc X from both terms of the orijnnal ratio : then . is gi'cater or b -X less than ^ » according as -j—-. : is greater or less than r— i x J 6 ' ^ b{b-x) ^ b{b-x)* that is, according as 6 (x :a is nearly equal to the ratio {a + xY ; a^, and a ¥ ix : a is nearly equal to the ratio {a + x)* : a*. These results may be generaKsed by the student when he is acquainted with the Binomial Theorem. 384. We will place here a theorem respecting ratios which is often of use. ft P Suppose that - - - :^ -, then each of these ratios is equal to v-yi, — -jn — ^j"* "where p, q, r, n are any quantities whatever. T. A. 15 226 ii^VTio. Jb'or lot k = ^ =2= -? } ^^^^^ kb = a, kd = c, kf= e ; therefore p {khf + q {kdf + r (kf)" = pa" + qc" + re" ; 7» pa" + qc" + re" , , /pa" + qc" + re" \l therefore k - ;^,.^ ^^»^ ,^-,. , -^l k= (^^^^„__^,^^-^J . The sanie mode of demonstration may be applied, and a similar a c e result obtained, Avhen there are more than three ratios - , - -^ given equal. It may be observed that p, q, r, n are not neces- Barily positive quantities. As a particular example we may suppose n = \, then we see that if ? = ^ = - each of these ratios is equal to ^ , ^, -.-, h d j pb + qd + rf and then as a special case we may suppose p = q=r, so that each , . a + c + e of the given equal ratios is equal to , — j—^ . 385. Suppose that we have three unknown quantities x, y, z connected by the two equations ax-^hy+cz=^0, ax + h'y + c's; = ; these equations are not sufficient to determine the unknown quan- tities, but they will determine the ratios subsisting between them. For multiply the first equation by c', and the second by c, and subtract: thus . (ac' - a'c) oj + (5c' - 6'c) y = ; X y therefore t-t — tt - 'TD — ZC. • DC — be ca — c a Again, multiply the first equation by b', and the second by b, and subtract : thus Ave shall obtain X _ z bc'-b'c^ ab' -ab' Hence we may write the results in this form : X y _ z he — b'c ca' — c'a ah' — ah EXAIVIPLES. XXVI. 227 These results are very important, and should be carefully re- membered ; the second denominator may be derived from the fii-st, and the third from the second, in the manner explained in Ai-t. 211. Denote the common value of these fractions by k} then x = k {be- h'c), y=k {ca - c'a)^ z = k{ah' - a'b). Now suj)pose that we have also a third equation connecting the unknown quantities x^ y, z] then by substituting in it for X, y, z the expressions just given, we shall obtain an equation which will determine h : thus the values of x, y^ z become known. • Suppose, for example, the third equation is Ix^ + my^ + nz' — 1, then k is determined by k' {Uho' - h'cy + m{ca- ca)' + n {ah' - a'hy] = 1. EXAMPLES OF RATIO. 1. Write down the duplicate ratio of 2 : 3, and the sub- duplicate ratio of 100 : 14-4. 2. Write down the ratio which is compounded of the ratios 3 : 5 and 7 : 9. 3. Two numbers are in the ratio of 2 to 3, and if 9 be added to each they are in the ratio of 3 to 4. Find the numbers. 4. Shew that the ratio a : 6 is the duplicate of the ratio a + c : 6 + c if c^ = a6. 5. There are two roads from A to B, one of them 14 miles longer than the other, and two roads from B to C, one of them 8 miles longer than the other. The distances from A to B and from B to C along the shorter roads are in the ratio of 1 to 2, and the distances along the longer roads are in the ratio of 2 to 3 Determine the distances. 13— L' 228 EXAMPLES. XXVI. 6. Solve the equations ax + hy cz + ax by + c^ ' = — = -^ = X A- y + z. cz at/ ax 7 Prove that if -^ ^ = -2 ^ = — , ejich of these 1 + .'^ ratios is equal to -— ^ , suj^posing «, + a, + rt^ not to be zero. ^-a-6 h — c c- a a+h + c ,. . ^ 8. If ^ = :. = = 7 , then each of ay + ox 0Z + ex oy + az ax + by + cz these ratios = , supposing a + h + c not to be zero. .^ay-bx cx-az bz—cy^, x y z 9. Shewthat if -^— = j — = ^ ^, then ="^ ^ -. c a a c 10. If , ,,— 77 — TT/ = ^r 77 J then each of these ratios a —a 0—0 c - c ab'—a'b bc—b'c ca—c'a a + 6 + c— (a' + 6' + c') ao —a c — a c ca —c a a + o+c— (a+o+c) 1 1 . Solve the e'juations 2x + y-2z==0, 7a; + 6y- 9:^ = 0, x^ + y^ + z^^ 1728. 12. Solve the equations ax + 6y + c^ = 0, 6ca; + cay + a5;2; = 0, xyz -f- a6c (a^o; + b^y + c^;2;) = 0. XXYII. PKOPOPvTION. 386. Four quantities are said to be proportionals when the first is the same multiple, part, or parts, of the second, as the third is of the fourth ; that is, when - —. -, the four quantities U CI/ * a, b, c, d, are called proportionals. Tliis is usually expressed by saying, a is to 6 as c is to d, and is represented thus, a :b '.:c : d, or thus, a :b = c:d. PROPORTION. 229 The terms a and d are called the extremes, and the terms h and c are called the means. 387. When four quantities are proportionals, the product of the extremes is equal to the product of the vieans. Let a, h, c, d be the four quantities; then since they are pro- portionals y- = -^ (Ai't. 386); and by multiplying both sides of the equation by hd, we have ad = be. Hence if the first be to the second as the second is to the third, the product of the extremes is equal to the square of the mean. 388. If any three terms in a proportion are given, the fourth may be determined from the equation ad = be. 389. If the product oj two quantities be equal to the product of two others, the four are proportionals; the terms of either product being taken for the means, and the terms of the other product for the extremes. X b Let xy = ab; divide by ay, thus, - = - ; or X \ a :\ b '. y (Art. 386). 390. li a : b :: c : d, and c '. d v. e \ f then a \h w e \f Because 7- = -7 and -■, — -., therefore r = ■> J b d d J b f or a \b v. e \f. 391. If four quantities be propiortionals, they are proportionals when taken inversely. If a : b \: c \ d, then b : a :: d : c. For - = • divide unity by each of these equal quantities, thus - = - ; or 6 : a :: c/ : c. a c 230 PROPORTION. 392. If four quantities he projyortionals, they are j^roportionals when taken alternately. If a : h :. G : dj then a : c :: h : d. For r = -7 ; multiply by - ; thus - = -,; b d' ^ -^ -^ c c d' or a : c :: b : d. Unless the four quantities are of the same kind the alter- nation cannot take place ; because this operation supposes the first to be some multiple, part, or parts, of the third. One line may have to another line the same ratio as one weight has to another weight, but there is no relation, with respect to magni- tude, between a line and a weight. In such cases, however, if the four quantities be represented by numbers, or by other quantities ■which are all of the same kind, the alternation may take place. 393. When four quantities are proportionals, the first together with the second is to the second as the third together with the fourth is to the fourth. If a : b :: c : d, then a + b : b :: c + d : d. ft ft For :,=-', add unity to both sides : thus b d -^ a ^ c - ,, , . a+ b c + d 7 + 1 = -7 + 1 : that IS, -^ — = — z- ; a a or a + b \ b '.: c + d : d. This operation is called componendo. 394. Also the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. For - = - ; subtract unity from both sides ; thus h d a - c - ., ,. a — h c — d - — 1 = - — 1 ; that IS, b d ' ' h d or a — b'.b::c — d:d. This operation is called dividendo. PROPORTION. 2:U 395. Also the first is to the excess of the first above the second as the third is to the excess of the third above the Jour th, a — b c — d By the last Ai*ticle, also b d ' b _d_ a c ' a-b b c-d d a—b c- d , oi" = - c a c therefore , x — — — x , or — - , baa or a — b : a :: c — d : Cy and inversely, a : a — b :: c : c — d. This operation is called convertendo. 396. WJien four quantities are propo7'tionals, the sum of th^ first and second is to their difference as the suvi of the third aiul fourth is to their difference. If a : b :: c : d, then a + b : a — b :: c + d : c — d. By Art. 393, "^^ ^^^ and by Art. 394, therefore that is. b d ' a—b c-d b d ' a + b a—b c+d c-d b • b ~ d ' a + b cA-d d a-b c —d' or a + b : a — b :: c + d : c — d. 397. W/ien any number of quantities are proportionals^ as one antecedent is to its consequent, so is the sum of all tite antecedents to the sum of all the consequents. Let a '. b w G : d '.: e : f y then a '. b :: a ■¥ c + e \ b "Vd -\- f. 232 PROPORTION. For ad -he, and aj —he, (Ai-t. 386), also ah — ha ; hence ah + ad + a/- ha + he + he ; that is, a{h + d + /) = h{a + e + e). Hence, by Ai-t. 389, a : h :: a-\- e + e : h + d + /. Similarly the proposition may be established ^vhen more quan- tities are taken. 398. WJien four quantities are proportionals, if the first and fecond he multiplied, or divided, hy any quantity, as also the third and fourth, the resulting quantities will he pi'oportionals. Let a : h :: c : d, then 771a : mh :: nc : nd. For , = -, , therefore — -. = — •. a mo nd or ma : mh :: no : nd. 399. If the first and third he multiplied, or divided, hy any quantity, and also the second and fourth, the resulting quantities imll he 2)yoporlionals. Let a '. h '.'. c '. dj then ma : 7i6 :: mc : nd. ^ a c . ,. ma mc . ma mc For 7- = -7 i therefore —j- =^ —r- , and , = — - ; a d no nd or ' ma : nh :: mc : nd. 400. In tv:o ranks of proportionals, if the con'esponding terms he multiplied together, the products will he proportionals. Let . a : h :: c : d, and e : f :: g : h, ^ then ae : hf :: eg : dh. „ a C ^ ^ 9 ,^ r- ttC CO i^ or , = -, and -^ = ; : therefore 7-. ^ „ ; d J II oj dk or ae : hf \\ eg \ dh. PROPORTION. 233 Tliis is called compounding tlie proportions. Tlie proposition is true if applied to any number of proportions. 401. If four quantilies he jjrojyortionals, the like powers, or roots, of these quantities will he projjortionals. -Let a : h '.'. c : d, then a" : 6" : : c" : cZ". -T or - = - , therefore — = ^ , where n may be whole or frac- tional ; thus a : :: c : a . 402. If a : 6 :: 6 : c, then a : c :: a^ : h^. ^ a h IX- 1 1 ^ ,1 a a a h Jb or T = -^ multiply by - , thus , x - = , x , be ^ ■^ -^ b b b b c , , , . a' a that IS, — . = - • b' c or a : c :: a' : 6'. The three quantities a, b, c are iii this case said to be in continued 2^roportion ; and b is said to be a mean proportional between a and c. 403. Similarly we may shew that if a : b :: h : c :: c : d, then a : d :: a^ : b^. Here the four quantities a, b, c, d are said to be in continued proportion. 404. It is obvious from the preceding Articles, that if four quantities are proportionals, we may derive from them many other proportions. "VVe will give another example. If a : 6 :: c : d, then ma + nb : pa + qh :: mc + nd : pc + qd. _ a c ,, ^ 7na vie For , ^--,, therelore — -- : a b a 234 PRoroRTiON. add n to both sides ; thus ma + nh mc + ncl b " d • <-,. ., , pa + qh iJC-^-qd Sinularlv t^- = r . b a ^ Dia + nb jja + qb mc + nd pc + qd Hence -^ ^ - — ^— =. -_^— H- ^ ^ ; ,T , . 7^1 a + n6 ??ic + nd tliat IS, ^ ; jt?a + qb -pa + 5'a or ma + n6 : j^a + qb w mc + nd : ^jc + (^t/. 405. In the definition of Proportion it is supposed that one quantity is some determinate multiple, part, or parts, of another ; or that the fraction formed by taking one of the quantities as a numerator, and the other as a denominator, is a determinate fraction. This will be the case whenever the two quantities have any common measure whatever. For let cc be a common measure of a and b, and let a — mx and b ~nx ] then a mx m b 7ix n ' where m and n are whole numbers. 406. But it sometimes happens that quantities are incom- TiiensurabUj that is, admit of no common measure whatever. If, for example, one line is the side of a square, and another line is the diagonal of the same square, these lines are incommensurable. In such cases the value of -z- cannot be expressed by any fraction m — where m and n are whole numbers ; yet a fraction of this kind may be found which will express the value of y to any required degree of accuracy. PROPORTION. 235 For let h - nx, where n m an integer , also let a be greater than 7nx but less than (in + l)x; then , is greater than — , but less than . Thus tlie difference between -, n n b ^ m . . .1 ana — is less than - . And since nx = h, when x Ls diminished n n ^ n is increased and - is diminished. Hence by takinc: x small n Jo enough, - can be made less than any assigned fraction, and n therefore the difference between — and y can be made less than n any assigned fraction. 407. If c and d as well as a and h are incommensurable, and if when y lies between — and , then -. also lies be- 71 n a m , on + 1 . , ' , . _ tween — and however the num!:;ers tji and n are increased, n n Y is equal to -. . b ^ 'l = -^- , where 7/i is constant, A is said to vary inversely B as^. 417. One quantity is said to vary as two others jointly when, if the former is changed in any manner, the product of the other two is changed in the same proportion. Or if A =mBCj where m is constant, A is said to vaiy jointly as B and C. 418. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the second and the reciprocal of the third. p Or if ^ = —sp , where m is constant^ A is said to vary directly as B and inversely as C. 419. 7/* A cc B, and B cc C, then A «= C. For let A — niB, and B = nC, where m and n are constants j then A = mnC ; and, as mn is constant, Aoz C. 420. If Ace C, and B cc C, then A ± B « C, and J{A'B) cc C. For let A = m(7, and B = nC, where m and n are constants ; then A + B = (in + 7i) C, and A — B — {ni — n)C ; therefore A^ B cc — , Simi- m (J C larly Coc — . 422. // A oc B, and C oc D, tJien AC cc BD. For let A^mB, and C = nDj then AC=mnBD', therefore AC<:^BD. 423. 7/' A cc B, ^Aen A" 03 B". For let A=mB, then ^'' = m"^" ; therefore A" cc i?^ 424. //" A oc B, ^7i€?i APccBP, v:here P {5 a?i?/ quantity variable or invariable. For let ^ = viB, then .4P = inBP; therefore AP az BP. 42o. 7/ A cc B loAen C ^5 invariable, and Ace C when B is invariable, then will A cc BC when both B and C are variable. The variation of ^ depends upon the variations of the two quantities B and (7; let the variations of the latter quantities take place separately, and when B is changed to h, let A be A B changed to a' ; then, by supposition, — = — . Now let C bo changed to c, and in consequence let a be changed to « ; tlien, by supposition, — = — . Thus a c A a' ^, X — : a a BC " he ' that is, A a BC " 'he ' therefore A cc BC. A very good example of this proposition is furnished in Geometry. It can be proved that the area of a triangle varies as the base when the height is invariable, and that the area varies as the height when the base is invariable. Hence when both the T. A. IG 242 EXAMPLES. XXVIII. b. 2. If a varies as h and a = 15 when 6 = 3, find the equation between a and h. 3. Giv^en that z varies jointly as x and y, and that z = \ when x~\ and 2/ = I5 ^^'^ ^^^^ value of z when ic = 2 and y = 2. 4. If z varies as px + ?/, and if ;2; = 3 when x = \ and y = 2, and z— 5 when a; = 2 and y = 3, find j)- 5. If cc varies directly as y when z is constant, and inversely as ,-:: when y is constant, tlien if y and z both vary, x will vary y as -. z 6. If 3, 2, 1, be simultaneous values of x, y, z in the pre- ceding Example, determine the value of x when y = 2 and z = ^. 7. The wages of 5 men for 6 weeks being XI 4. 55., how many weeks will 4 men work for £191 (Apply Example 5.) 8. If the square of x vary as the cube of y, and cc = 2 when 2/ = 3, find the equation between x and y. EXAMPLES. XXVIIL 213 9. Given that y varies as the sum of two quantities, one of which varies as x directly, the other as x invei-sely, and that y = ^i when a; = 1, and y = 5 when a; = 2, find the equation be- tween X and y. 10. If one quantity vary directly as another, and the foimer be \ when the latter is |, find what the latter will be when the former is 9. 11. If one quantity vary as the sum of two others when their difference is constant, and also vary as theii' difference when their sum is constant, shew that when these two quantities vary independently, the first quantity will vary as the difference of their squares. 12. Given that the volinne of a sphere varies as the cube of its radius, prove that the volume of a sphere whose radius is 6 inches is equal to the sum of the volumes of three spheres whose radii are 3, 4, 5 inches. (13. y Two circular gold plates, each an inch thick, the diame- ters o£ which are 6 inches and 8 inches respectively, are melted and formed into a single circular plate one inch thick. Find its diameter, having given that the area of a circle vanes as the square of its diameter. 1-4. There are two globes of gold whose radii are r and r'; they are melted and formed into a single globe. Find its radius. 15. If cc, y, z be variable quantities such that y + z — x is constant, and that {x -\- y - z) [x -{■ z - y) varies as yz^ prove that jc + 2/ + « varies as yz. 16. A point moves with a speed which is different in different miles, but invariable in the same mile, and its speed in any uiilo varies inversely as the number of miles travelled before it com- mences this mile. If the second mile be described in 2 liours, find the time occupied in describing the n^^ mile. 17. Suj^pose that y varies as a quantity which is the sum of three quantities, the first of which is constant, the second varies IG— 2 244 SCALES OF NOTATION. as X, and the third as x*. And suppose that when a; = a, y = 0, ■when aj = 2a, y = «, and when x = 3a, y = ia. Shew that when X = Tia, y={n- \y a. 18. Assuming that the quantity of work done varies as the cube root of the number of agents when the time is the same, and varies as the square root of the time when the number of agents is the same ; find how long 3 men would take to do one-fifth of the work which 24 men can do in 25 houi^. (See Art. 425.) XXIX. SCALES OF NOTATION. 427. The student v.dll of course have learned from Arith- metic that in the ordinary method of expressing whole numbers by figures, the number represented by each particular figure is always some multiple of some power of ten. Thus in 347 the 3 represents 3 hundreds, that is, 3 times 10^; the 4 represents 4 tens, that is, 4 times 10^; and the 7 which represents 7 units, may be said to represent 7 times 10°. This mode of representing numbers is called the common scale of notation, and 10 is said to be the base or radix of the common scale. 428. "We shall now prove that any positive integer greater than unity may be used instead of 10 for the radix, and shall shew how to express a number in any proposed scale. "We shall then add some miscellaneous proj)ositions connected with this subject. The figures by means of which a number is expressed are called digits. "VVlien we speak in future of any radix we shall always mean that this radix is some positive integer greater than unity. 429. To sliew that any lolwle number may be expressed in terms of any radix. Let N denote the whole number, r the radix. Suppose that r" is the highest power of r which is not greater than N ) divide SCALES OF NOTATION. 245 N by r", and let p^^ be the quotient and N^ tlie remainder ; thus Here, by supposition, 2^,^ is less than r ] also N^ is less than r". Next divide N^ by r"~', and let 2^,^.^ be the quotient and N^ the remainder; thus ^■=^'„-,'-"""+-^;- Proceed in this way until the remainder is less than r ; thus we find N expressed in the manner indicated by the equation Each of the digits p^, i^„_,; 2^^^ Po ^^ ^®^^ than r, and any one or more of them after the first may be zero. The best practical mode of determining the digits is given in the next Article, 430. To ex2)ress a given whole numher in any proposed scale. By a given whole number we mean a whole number expressed in words or else expressed by digits in some assigned scale. If no scale is mentioned, we understand the common scale to be intended. Let iV be the given number, r the radix of the scale in which it is to be expressed. Suppose ^9^, Pj, 2^n ^^ ^® ^^® required digits by which N is expressed in the new scale, beginning with that on the right hand ; then ^" = P. f +?,.-/"' + +i^/+iv+i^o; we have now to find the value of each digit. Divide N by r, and let Q^ denote the quotient; then it is obvious that Qx=p,f"-^Pn-x'>'^~'' + +p/+;'p and that the remainder is p^. Hence p^ is found by this rule ; divide the given numher hy the proposed radixy and the reniaitider is the first of the required digits. 24G SCALES OF NOTATION. Again, divide Q^ by r, and let Q^ denote the quotient ; then it is obvious that ^.=;v''"'+iv/''"'+ +;'.» and that the remainder is 2^1' Hence the second of the required digits is fouu'l. By proceeding in this way we shall find in succession all the required digits. 431. For example, transform 43751 into the scale of which 6 is the radix. The division may be performed and the remainders noted thus : 6^4 3 7 5 1 6^7 2 9 1 5 6J 1 2 1 5 1 6y)202 3 6^3 3 4 5. 3 Thus 43751 = 5 . 6' + 3 . 6' + 4 . 6' + 3. 6' + 1 . 6 + 5, so that the number is expressed in the new scale thus, 534315. 432. Again, transform 43751 into the scale of which 12 is the radix. 12;4 3 7 5 1 12^3 6 4 5 11 12; 3 3 9 12; 2 5 3 2 1 Thus 43751=2.12^ + 1.12^ + 3.12^ + 9.12 4-11. In expressing the number in the new scale we shall requii-e a single symbol for eleven ; let it be e ; then the number is ex- pressed in the new scale thus, 2139e. SCALES OF NOTATION. 247 We cannot of course use 1 1 to express eleven in the new scale, because 1 1 now represents 1.12 + 1, that is, thirteen. 433. We will now consider an example in which a number is given, not in the common scale. A number is denoted by t^\le in the scale of which twelve is the radix, it is required to express it in the scale of which eleven is the radix. Here t stands for ten, and e for eleven. eJtS 4:7 e c2 7 3 2 The process of division by eleven is performed thus. First e is not contained in t, for eleven is not contained in ten, so we ask how often is e contained in ^3 ? here t stands for ten times twelve, that is one hundred and twenty, so that the question is, how often is eleven contained in one hundred and twenty-three ] the answer is eleven times, with two over. Next we ask how often is e contained in 24 ; that is, how often is eleven contained in twenty-eight 1 the answer is t^vice, with six over. Then how often is e contained in 67 ; that is, how often is eleven contained in seventy-nine 1 the answer is seven times, with two over. Last, how often is e contained in 2e ; that is, how often is eleven contained in thirty-five ? the answer is three times, with two over. Hence 2 is the first of the required digits. The remainder of the process we will indicate ; the student should carefully work it for himself, and then compare his result with that here given. e J e 27 3 «; 1 2 t 1 «; 1 1 4 2 e J\ 2 6 1 3 243 SCALES OF NOTATION. Hence the given numbei' is equal to that is, it is expressed in the scale with radix eleven thus, 136212. 434. The process of transforming from one scale to another may be effected also in another manner. Suppose for exam2)le that we have to transfonn to the common scale 24613 which is in the scale of seven. We have in fact to calculate the value of 2x' + 4:X^ + 6x' + x + 3, when 03 = 7. We may adopt the method which is explained in the Theory of Equations, Art. 11. 2+ 4+ 6+1+3 14 + 126 + 924 + 6475 18 + 132 + 925 + 6478 The result is 6478. This method is advantacreous when we o have to transfonn from any otlier scale into the common scale. 435. It will be easy to form an unlimited number of self- verifying examples. TIius, take two numbers expressed in the common scale and obtain their product, then transform this pro- duct into any proposed scale ; next transform the two numbers into the proposed scale, and obtain their product in this scale ; the result should of course agree with that already obtained. Or, take any number, square it, transform this square into any pro- posed scale, and extract the square root in this scale ; then trans- form the last result back to the original scale. 436. Next let it be required to transform a gi\ei\ fraction from one scale to another. This may be effected by transforming separately the numerator and denominator of the given fraction by the method of Art. 430. Thus we obtain a fraction identical with the proposed fraction, having its numerator and denominator expressed in the noAv scale. 437. We stated in Art. 427, that in the common scale of, notation, each digit which occurs in the expression of any integer SCALES OF NOTATION. 24^ by figures rciDresents some muUijyIe of some jyowcr of ten. This statement may be extended, and we may assert tliat if a number be expressed in the common scale, and the number be an integer, or a decimal fraction^ or 'partly an integer and jyartly a decimal fraction, then each digit represents some midtiple of some p>ower of ten. Thus in 347 "958 the 3, the 4, and the 7, have the values 9 assigned to them in Art. 427 ; the 9 represents r-^ , that is, 5 9 times 10 ' ; the 5 represents tjj:, that is, 5 times 10~^; and g the 8 represents ; that is, 8 times 10"^. It may therefore naturally occur to us to consider the follow- ing problem : requii'ed to express a given fraction by a series of fractions in any 2'»roposed scale analogous to decimal fractions in the cojnmon scale. "We will speak of such fractions as radix- fractions. 438. Required to express a give7i fractiori hy a series of radix- fractions in any proposed scale. By a given fraction we mean a fraction expressed in words or expressed by figures in any given scale. - Let F denote the given fraction, r the radix of the pro- posed scale. Suppose t^, t^, ... the numerators of the required radixfractions beginning from the left hand ; thus 7^ tx i, K r r^ r^ ' where f,, t^, t^, are to be foirad. Multiply both members of the equation by r ; thus Fr = f^+^-^+k+ 1 r r^ The right-hand member consists of an integer ^, and an additional fractional pai-t. Let I^ denote the integi-al part of Fr, and jPj the fractional remainder ; then we must have i,^t„ F,^^:+^ + 250 SCALES OF NOTATION. Thus, to obtain the first numerator, ti, of the series of radix- fractious, we have this rule ; niultiply the given fraction hy tlie proposed radix ; then the greatest integer in tJie product is the first of tlie required numerators. Again, multiply Fi hj r ; let /j be the integral part of the product, and F^ the fractional remainder ; then T, = t,, i^, = -'+-S Hence t , the second of the required numerators, is ascertained. By proceeding in this way we shall determine the required nu- merators in succession. If one of the products which occur on the left-hand side of the equations be an exact integer, the process then terminates, and the proposed fraction is expressed by a finite series of radix-fractions. If no integi-al product occur, the process never temiinates, and the proposed fraction can only be expressed by an infinite series of the required radix-fractions; the numera- tors of the radix-fractions will recur like a recurring decimal. 123 439. For example, express y-^ by a series of radix-fractions in the scale 8. 123 .123 .11 MultiiDly -=-;- by 8 ; thus we obtain -^V » ^^^^^ is 7 + ^ . '■ "^ 128 "^ 16 lo Multiply tt: by 8 ; thus we obtain -— , that is 5 4- ~ . ^ •' 16 *^^ 2 ' 2 Multiply -- by 8; thus we obtain 4. 123 7 5 4 440. We may remark that the radix ten is not only the base of the common mode of expressing numbers by figures, but is in fact assumed as the base of our language for numbers. This will be seen by observing at what stage in counting upwards from unity new words are introduced. For example, all numbers between twenty-one and twenty-nine, both inclusive, are expressed SCALES OF NOTATION". 2ol by means of words that have already occuiTed in counting up to twenty; then a new word occurs, namely thirty j and we can count on without an additional new word as far as thirty-nine ; and so on. The number ten has only two divisors different from itself and unity, namely 2 and 5j the number twelve has four divisors, namely 2, 3, 4, and G. On this account twelve would have been more convenient than ten as a radix. This may be illustrated by reference to the case of a shilling; since a shilling is equivalent to twelve pence, the half, the third, the fourth, and the sixth of a shilling, each contains an exact number of pence; if the shilling were equivalent to ten pence, the half and fifth of a shilling would be the only submultiples of a shilling containing an exact number of pence. Similarly, the mode of measuring lengths by feet and inches may be noticed. 441. We may observe that if two be adopted as the radix of a scale, the operations of Arithmetic are in some respects much simplified. In this scale the only Jigures which occur are and 1, so that each separate step of a series of arithmetical operations would be an addition of 1, or a subtraction of 1, or a multiplica- tion by 1, or a division by 1. The simplicity of each operation is however counterbalanced by the disadvantage arising from the increased number of such operations. "We give in the following two Articles two problems connected with the present subject. 442. Determine which of the series of weights 1 lb., 2 lbs., 2^ lbs., 2^ lbs., 2Mbs,, must be used to balance a given weight of A^ lbs., not more than one weight of each kind being used. It is obvious that this question is the same as the foUo^ving; express the number A in the scale of which the radix is 2. Hence it follows from Art. 429 that the problem can always be solved. 443. Suppose it required to determine which of the weights 1 lb., 3 lbs., 3Mbs., 3Mbs.,... must be selected to weigh Albs., not 252 SCALES OF NOTATIOX. more than one of each kind being used, hut in eitJcer scale that may be necessary. Divide N by 3, then the remainder must be zero, or one, or two. Let iVj denote the quotient ; then in the first case we have iV=3iVj, in the second case iV"=3iVj + l, and in the third case N=^N + 2. In the first or second case divide iV^ by 3; in the third case we may write i\r= 3 (iV^ + 1) - 1, then we should divide N^ + 1 by 3. Proceed thus, and we shall finally have a result of the following form, ^-gn3"+^„_i3'^-^+ +^,3 + ^„ where each of the quantities g^j, q^, q^ is either zero, or + 1, or — 1. Thus the problem is solved. 444. In a scale of notation of which the o'adix is r, the sum of the digits of any whole number divided hy v — 1, or by any factor of r — 1, will leave the same rem^ainder respectively as the whole number divided by r — 1 or by the factor ofr—l. Let N denote the whole number, p^, p^, p^^ the digits be- ginning with that in the units' place; then ■N' = Po + 2\r+ +;^r" + p{r-l)+p,{r'-l)+ +;7„(r'^-l)j therefore =^-^ — ^ — ^-^— ^~ r— 1 r— 1 r"- 1 +^, +i;,(r+i) + +Pn^^rr[' r"- 1 . . But - — ~ is an integer whatever positive integer n may be; thus ^ • X Po-^P\-^ +Pn = some mteger +^— — — ~ ^^. r-1 ^ r-1 Next let ^ be a factor of r — 1, say that r — \=pq. Then multiplying the last result by q we have — = some inteerer + — — — ^ . P P This establishes the proposition. SCALES OF NOTATION. 253 445. In a scale of notation in wJiich the radix is v let any whole number he divided by r + 1 ; and let the difference between the sum of the digits in tlie odd 'places and the sum of the digits in the even places he divided by r + 1 j then either the remainders will he equal or their sum will he r + 1 . Witli the same notation as in the preceding proposition we have + 2h{r + 1) j,p^{r'-l)+p,(r'+ 1) + ... +^„{r" - (- 1)"}. Thus -^ = some integer ^hzPl±Pl^J^^:i^lzRP, ^ r + 1 r+ I Fii-st, suppose p^-Pi+Pi- ••' + {-'^Yp,x to bo positive, and denote ithj D ] then N . D r- = some integer H : r+l ^ r + 1 thus when N' and D are divided by r + 1 the remainders are equal. Secondly, suppose 2\~ Pi'^ P-2~ •-• '^ {~^"Pn to be negative, and denote ithj—D; then = some inteirer — r+l ° r+l' .1 X . N D that IS- r- H — — - = some mtesrer : ' r+\ r+i ^ ' thus when JS/" and D are divided by r + 1 the sum of the remain- ders must be r+l, unless either remainder is zero, and then the other remainder also is zero. For example, supposes =10 and iV=263419. Hero 9-l + 4-3 + G-2 = 13 = i>; and iVand D when divided by 11 each leave the remainder 2. Again, suppose r = 10 and i\^=G15372. Hero 2-7 + 3-5 + l-G=-12 = -/>; and iV and D when divided by 11 leave the remaindci-s 10 and 1 respectively. 2o^ EXAMPLES. XXIX. 44:G. It appears from Art. 444 that a number is divisible by 9 when the sum of its digits is divisible by 9 ; and that when any number is divided by 9, the remainder is the same as if the sum of the digits of that number were divided by 9. And as 3 is a factor of 9 a number is divisible by 3 when the sum of its digits is divisible by 3 ; and when any number is divided by 3 the re- mainder is the same as if the sum of the digits of that number were divided by 3. It appears from Art. 445 that a number is divisible by 11 when the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by 11. 447. From the property of the number 9, mentioned in the preceding Article, a rule may be deduced which will sometimes detect an error in the multiplication of two numbers. Let 9a + x denote the multiplicand, and db +y the multiplier; then the product is Slab + 9bx + 9ay + xy. If then the sum of the digits in the multiplicand be divided by 9, the remainder is x ; if the sum of the digits in the multiplier be divided by 9, the remainder is y ; and if the sum of the digits in the product be divided by 9, the remainder ought to be the same as when xy is divided bv 9, and will be if there be no mistake in the operation. EXAMPLES ON SCALES OF NOTATION. Transform the follow^ing sixteen numbers from the scales in which they are given to the scales in which they are required : 1. 123456 from the scale of ten to the scale of seven. 2. 1357531 from the scale of ten to the scale of five. 3. 357234 from the scale of ten to the scale of seven. 4. 333310 from the scale of ten to the scale of eleven. 5. 545 from the scale of six to the scale of ten. EXAilPLES. XXIX. 255 6. 4444 from the scale of five to the scale of ten. 7. 3413 from the scale of six to the scale of seven. 8. 40234 from the scale of five to the scale of twelve. 9. 64520 from the scale of seven to the scale of eleven. 10. 15951 from the scale of eleven to the scale often. 11. 15*75 from the scale of ten to the scale of eight. 12. 314G2-125 from the scale often to the scale of eight. 13. 221-248 from the scale of ten to the scale of five. 14. 444*44 from the scale of five to the scale often. 15. 1845-3125 from the scale of ten to the scale of twelve. 16. 3065-263 from the scale of eight to the scale often. 17. Express in the scale of seven the numbers which are expressed in the scale of ten bv 231 and 452 ; multiply the num- bei"S together in the scale of seven, and reduce to the scale of ten. 18. Divide 17832126 by 4685 in the scale of nine. 19. Extract the square root of 33224 in the scale of six. 20. Extract the square root of 123454321 in the scale of six. 21. Extract the square root of 3445-44 in the scale of six, and reduce the residt to the scale of three. 22. Subtract 20404020 from 103050301 in the scale of eight, and extract the square root of the result. 23. Extract the squai-e root of 1 1000000100001 in the binary scale. 24. Extract the square root of 67556^21 in the scale of tAvelve. 25. Express |— - in a series of radix-fractions in tlie sc}d« of twelve. 26. Find in what scale 95 is denoted by 137. 27. Find in what scale 2704 is denoted by 20304. 28. Find in what scale 1331 is denoted by 1000. 256 EXAMPLES. XXIX. 29. Find in what scale IGOOO is denoted by 1003000. 30. A number is represented in the denary scale by 35f and in another scale by 55-5, find the radix of the latter scale. 31. Find in what scale of notation sixteen hundred and sixty- four ten- thousandths of unity is represented by "0404. 32. Shew that 12345654321 is divisible by 12321 in any scale ; the radix being supposed greater than six. 33. Shew that 144 is a perfect square in any scale ; the radix being supposed greater than four. 34. Shew that 1331 is a perfect cube in any scale; the radix being supposed greater than three. 35. Find which of the weights 1, 2, 4, 8, 2" pounds must be selected to weigh 1719 pounds. 36. Find which of the weights lib., 3 lbs., 3^ lbs., must be selected to weigh 1027 lbs., not more than one of each kind being used, but in either scale that is necessary. 37. Find which of the same weights must be selected to weigh 716 lbs. 38. Find which of the same weights must be selected to weigh 475 lbs. 39. Find by operation in the scale of twelve what is the height of a parallelepiped which contains 94 cubic feet 235 cubic inches, and whose base is 24 square feet 5 square inches. 40. Express 2 feet 10 j inches linear measure, and 5 feet 73- inches square measure, m the scale of twelve as feet and duodecimals, of a foot; and the latter quantity being the area of a rectangle, one of whose sides is the former, find its other side by dividing in the scale of twelve. 41. li p , ]) , 2^oi - ^Q "t^^® digits of a number beginning with the units, prove that the number itself is divisible by eight if /? -f- 2/>j + 4p, is divisible by eight. ARITHMETICAL PROGRESSION. 257 42. Prove that the diflference of two numbers consisting of the same figures is divisible by nine. 43. Find the greatest and least numbers ^vith a given number of digits in any proposed scale. 44. Prove that if in any scale of notation the sum of two numbers is a multiple of the radix, then (1) the digits in which the squares of the numbers terminate are the same, and (2) the sum of this digit and of the digit in which the product of the numbers terminates is equal to the radix. 45. A certain number when represented in the scale two has each of its last three digits (counting from left to right) zero, and the next digit diflferent from zero ; when rej^resented in either of the scales three, five, the last digit is zero, and the last but one difierent from zero ; and in every other scale (twelve scales ex- cepted) the last digit is different from zero. What are these twelve scales, and what is the number ? XXX. ARITHMETICAL PROGPESSIOi^. 448. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common difference. Tlius the following series are in Arithmetical Progression : 1, o, D, I, J, 40, 36, 32, 28, 21, a, a + h, a + 2b, a + 3b, a, a-b, a - 2b, a - Sb, In the first example the common difference is 2, in tlio second -4, in the third b, in the fouilh -b. 449. Let a denote the first term of an Arithmetical Progres- sion, b the common difference ; then the second term is a + 6, the third term is a + 26, the fourth term is a + 36, and so on. Thus the w**" term is a + (n - 1 ) b. T. A. 17 258 ARITHMETICAL PROGRESSION. 450. To find the sum of a given number of quantities in Arith- metical Pro(jression, the first term and the common difference being supjiosed known. Let a denote the first terai, b tlie common diiSerence, n tlie number of terms, I tlie last term, s the sum of the terms. Then s-a+(a + b) + {a + 2b) + + 1. And, by writing the series in the reverse order, we have also s = l+(l-b) + {l-2b)+ +a. Therefore, by addition, 2s = (I + a) + (I + a) + to n terms = oi(l + a) ; n therefore s = -^(l + a) ( 1 ). Also l = a+{n-l)b (2), n thus 8 = -{2a + {n-l)b] (3). The equation (3) gives the value of s in terms of the quan- tities which were supposed known. Equation (1) also gives a con- venient expression for 5, and furnishes the following rule : the sum of any number of terms in Arithmetical Progression! is equal to the product of the number of the terms into half the sum of the first and last terms. 451. In an Arithmetical Progression the sum of any two terms equidistant from the beginning and the end is equal to the sum of the first and last terms. The truth of this has already been seen in the course of the preceding demonstration ; it may be shewn fonnally thus : Let a be the first term, b the common difierence, I the last term ; then the r^^ term from the beginning is a + (r — 1 ) 6 and the r^^ term from the end is Z — (r — 1)&, and the sum of these terms is therefore I -\- a. ARITHMETICAL PROGRESSIOX. 259 452. To insert a given number of arithmetical means between two given terms. Let a and c be tlie two given terms, n the number of terms to be inserted. Then the meaning of the problem is, that we are to find n + 2 terms in Aiitlimetical Progression, a being the fii-st term, and c the last term. Let b denote the common difference ; then c = a + (n + l)b : therefore h = ^ . This finds b, and tho ^ ' ' n+\ ' n required terms are a + b, a+ 2b, a -hZb, ... ... a + nb. 453. In Art. 450 we have five quantities occun-ing, namely, a, b, I, n, s, and these are connected by the equations (1) and (2), or (2) and (3) there established. The student will find that if any three of these five quantities are given, the other two can be found ; this will fiu-nish some useful exercises. We give one as an example. 454. Given the sum of an Arithmetical Progression, the first term, and the common difference ; required tJie number of teims, tyt Here s = ^{2a + {n-l)b} j therefore 2s - n% + (2a — b) n. By solving this quadratic in n we obtain b-2a^J{{2a-bY+m] ^^ 26 • 455. It will be seen that two values are found for n in the preceding Ai-ticle ; in some cases both values are applicable, as will appear from the following example. Suppose a = ll, 6 = -2, 5 = 27; we obtain n=Z or 9. The arithmeticid progression is 11, 9, 7, 5, 3, 1, -1,-3,-5, etc., and it is obvious that the sum of the firat three terms is th3 same as the sum of the first nine terms. 17—2 2G0 ARITHMETICAL PROGRESSION. 456. Agaiu, suppose a = 4:, b = 2, s -IS ; we obtain n = 3 or — G. The sum of three terms beginning with 4 is 4 + + 8 or IS. If we put on terms hefore 4 we obtain the series -2 + + 2 + 4 + + 8, and the sum of these six terms is also 18. From this exam2:)le we may conjecture that when there is a negative integral value for the number of terms as well as a j^ositive integral value, the following statement will be true : begin from the last term of the series which is fiu-nislied by the positive value, and count backwards for as many terms as the negative value indicates, then the result will be the given sum. The truth of this conjec- ture may be shewn in the following manner. The quadratic equation in n obtained in Art. 454 is 2s = ?r5 + (2a - &) 71 (1). Suppose a series in which the first term is h — a, the common difference 6, the number of tenns 7?i, and the sum s ; then 25 = m'6 + (26-2f*-6)m (2). The roots of (1) and (2) are of equal values but of opposite signs (Art. 340); so that if the roots of (1) are denoted by n^ and — TZj, those of (2) will be n^ and — n^. Hence n.-^ terms of a series which begins with h — a and has the common difference 6, will amount to the given sum 5. The last tenn of the series which begins with a and extends to n^ tenns is a+ (?^l - 1)5; we have therefore to shew* that if we begin with this term and count backwards for w^ terms, we arrive at h — a. This amounts to shewing that a-^{n^-\)h-{ii.-\)h=-'b-a\ that is, that a + (?^^ - n^ h^h-a. 2a ~h Now n^ -n. = - ^—j — , (Art. 335) ; therefore a + {n^ - n^ h = a- (2a -h) -b-a. ARITHMETICAL PROGRESSION. 2G1 457. Another point may be noticed in connexion witli a negative integral value of n. Let -n^ be a negative integral value of n which satisfies the equation 77 s^-^{2a + {n-\)h}; then 6- = _ —i {2a - njj - h). Therefore - 5 = '^ {2 (a - &) + {n^ -^){-h)\. Tliis shews that if we count hacJcwards n^ terms beginning with a^h, the sum so obtained will be — s. For example, taking the case in Art. 456, by beginning at 2 iind counting backwards for six terms we obtain 2 + 0-2-4-6-8, that is, — 18. 458. In some cases, however, only one of the values of n found in Art. 454 is an integer. Suppose a=ll, 6 = — 3, s^24; we obtain n=^ or 5^. The value 5^ suggests to us that of the two numbers 5 and 6, one will correspond to a sum greater than 24, and the other to a sum less than 24. In fact the sum of 5 teiTQS is 25, and the sum of 6 terms is 21. We may notice the following point in connexion with a frac- tional value of 71. Suppose - a fractional value of n which satisfies the equation q s=l[2a + (n-\)h\; 262 ARITHMETICAL PROGRESSIOX. This shews that s is equal to the sum of p terms of an Arith- 1 ' 1 1 n .ah h metical Progression m which the first term is — "^9"^ ^ . h the common difference ls -^ , In the example given above ^ = 5J = -^ ; so that ^ -^ 16 and q 6 y = 3. And q Yq'^2q'~Z^2 6 ' q'~ 3* thus 24 is the sum of IG terms of an Arithmetical Progression in which the first term is 4 and the common difference is - tt. 459. The results in the following two simple examples are worthy of notice. To find the sum qfn terms of the series 1, 2, 3, 4,... Here the n^^ term is ?^ ; thus, by Art. 450, To find the sum o/i\ terms of the series 1, 3, 5, 7, ... Here a=l, 6 = 2; thus, by Art. 450, s = ^{2 + 2(n-l)} = 7'x2?i = 7i^ "We add two similar questions which lead to important results, although not very closely connected with the present subject. 460. To find the sum of the squares of the first n natural numbers. Let s denote the required sum; then 5 = r + 2'+3'+ +n\ A ^.^^ ^t, + n(n + l)(2n+l) and we shall prove that s = p . ARITHMETICAL PROGRESSION. 2G3 We liave {n-rf-{n-2Y^3(n-iy-3{n-l) + \, {n-2f-(7i-3y = 3{n-2y -3(71-2) + 1, 7.1 3 3^-2^=3. 3^^-3.3+1, 2'-r=3. 2^-3. 2 + 1, ^ \ " r-0^=3. l-'-3. 1 + 1. Hence, by addition, »i^=3{r+2^+ + n'}-3{l + 2 + + 7z}+n, . , ^ . 3 o 3?^ (?^ + 1 ) that IS, /i = o5 ~^ ' + n. Therefore 3.- = n^ + ?^i%^^ - » = "('' + 1)(2»^1) _ and ' ^__^l(!LJL^)(2!i+l) and 5- 2^ . 461. Tb ^Mc/ ih^ sum of the cubes of the Jirst n natural number's. Let s denote tlic required sum ; tlien s = V+2' + 3^+ +n', and we shall prove that s = - — ^— x — / . K ^ ) We have n* - ()i -iy = in' - Gn' + 471-1, (^_l)*_(^_2)^=4(H-l)^-6(n-l)-+4(u-l)-l, {n-2y-{7i-3y = 4:{n-2y-6{n-2y+i{n-2)-], 3* - 2* = 4 . 3' - G . 3' + 4 . 3 - 1, 2'-r-4.2'-G.2=' + 4.2- 1, r_0*=4.1='-G.r + 4.1-l. 264 ARITHMETICAL PROGRESSION. Hence, by addition, n* = 4{l''4-2^+ + 71"} -6 {I- + 2'+ +n'} + 4 { 1 + 2 + +n]-n; that is, n* = 4. 2 13 7. Sum to 21 terms - , - , kt , • • • ^ o Ji 1 2 8. Sum to 50 terms - , -, 1, ... o o 9. Sum to 30 terms 116, 108, 100,... 10. Sum to n terms 9, 11, 13, 15,... 5 2 11. Sum to n terms 1, ^ , „ , ... EXAMPLES. XXX. 265 12. Find an a. p. such tliat the sum of the first five terms is one-fourth the sum of the following five teiTns, the first tei^m being unity. 13. The first tei-m of a series being 2, and the fifth term being 7, find how many terms must be taken that the sum may be 63. 1-1. Given a =.16, b = i, s = SS, find n. ] 5. If the sum of m terms of an A. p. be always to the sum of n terms in the ratio of m^ to n^, and the first term be unity, find the n^^'term. 16. The sum of a certain number of terms of the series 21 + 19 + 17 + is 120: find the last term and the number of terms. 17. What is the common difierence when the first term ib 1, the last 50, and the sum 204: ] 18. Insert 6 arithmetical means between 1 and 21). 19. If 2)1 + 1 terms of the series 1, 3, 5, 7, 9, be taken, then the sum of the alternate terms 1, 5, 9, will be to the sum of the remaining terms 3, 7, 11, as ?i + 1 to n. 20. Find the sum of the first n numbers of the form 4r + 1. 2 1 . Find how many tenns of 1 + 3 + 5 + 7+ amount to 1231321. 22. Find liow many terms of 16 + 2-1+32 + 40-1- amount to 1840. 23. On the gi'ound are placed 7i stones ; the distance be- tween the first and second is one yard, between the second and third three yards, between the third and fourth five yards, and so on. How far will a person have to tra^vel who shall bring them, one by one, to a basket placed at the first stone 1 24. The 14th, 134th, and last terms of an a. p. are 06, 666, and 6666 respectively : find the first -^erm and the number of terms. 266 EXAJMPLES. XXX. 25. Find a series of aritlimetical means between 1 and 21, such that tlieir sum has to the sum of the two greatest of them the ratio of 11 to 4. 2G. The sum of the terms of an A. P. is 28^, the first term is — 12, the common difference is |. Find the number of terms. 27. Find how many terms of the series 3, 4, 5, must be taken to make 25. 28. Find how many terms of the series 5, 4, 3, must be taken to make 14. 29. Shew that a certain number of terms of an A. p. may be found of which the algebraical sum is equal to zero, provided twice the first term be divisible by the common difference, and the series ascending or descending according as the first term is negative or positive. 30. If the 711^^ term of an A. p. be n and the qi^^ term m, of how many terms will the sum be ^ (m + n) (m + h — 1), and what will be the last of them ? 31. If 5 = 72, a =24:, h = - 4, find oi. 32. If 5 = p?i + qn" whatever be the value of n, find the m*'' term. 33. If >S'^ represent the sum of n of the natural numbers beginning with a, prove that ^3a+„_i = 3^„. 34. Prove that the squares of x^ -~ 2x — 1, ic* + 1, and a;^ + 2a; — 1 are in a. p. 35. The common difference of an A. P. is equal to the differ- ence of the squares of the first and last terms divided by twice the sum of all the terms diminished by the first and last term. 36. The sum of m terms of an A. p. is n, and the sum of n terms with the same first term and the same common difference is m. Shew that the sum of vi + n terms is — (m + 7i) and the sum of m — n terms is (t?^ — n) ( 1 + — ) . EXAMPLES. XXX. 267 37. Find the number of arithmetical means between 1 and 19 when the second mean is to the last as 1 to G. 38. How many terms of the natural numbei'S commencing with 4 give a sum of 5350 ? 39. In a series consisting of an odd number of terms, the sum of the odd temis (the first, third, ttc.) is 44, and the sum of the even terms (the second, foiu'th, ^• 45. The j/^, q^^, and r^^ teiTQS of an A. P. are x, y, c, re- spectively j prove that if x, i/, z be positive integers, there is an A. P. whose ic^**, y*^, ^^ terms are p, q, r, respectively ; and that the product of the common differences of the progressions is unity. 46. Tlie interior angles of a rectilinear figure are in A. P. ; the least angle is 120° and the common difference 5". Requii-ed the number of sides. 47. Find the sum to 71 terms of 1 .2 + 2. 3 + 3.4 + 4.5 + ... 48. If the second terai of an A. p. be a mean i)ropoi-tional between the first and the fourth, shew that the sixth term will be a mean proportional between the fourth and the ninth. 49. If (^ {n) be the sum of n terms of an A. p., find <^ {n) in terms of n and the first two terms. Also shew that {n + 3.) - 3<^ {n + 2) + 3^ miles the second hour, 4 miles the third hour, and so on. Find in how many hours B will overtake A. 54. A number of persons were engaged to do a j^iece of work, which would have occupied them m hours if they had commenced at the same time ; but instead of doing so they commenced at equal intervals, and then continued to work till the whole was finished : the payment being proportional to the work done by each, the first comer received r times as much as the last. Find the time occupied. 55. A number of three digits is equal to 26 times the sum of its digits; the digits are in arithmetical progression; if 396 be added to the number the diji^its are reversed : find the number. "&' 56. Shew that the sum of any 2,'z + 1 consecutive integers is divisible by 2/1 + 1. XXXI. GEOMETRICAL PROGRESSIOX. 462. Quantities are said to be in Geometrical Progression when each is equal to the product of the preceding and some constant factor. The constant factor is called the common ratio of the series, or more shortly, the ratio. Thus the following series GEOMETRICAL PROGRESSION. 269 are in Geometrical Progression : 1, 2, 4, 8, 16, 1 1 2_ _1 ' 3' 9' 27' 81' a, aVy ar^, ar^, ar*, In the first example the common ratio is 2, in the second J, in the third r. 4G3. Let a denote the first term of a Geometrical Progression, r the common ratio, then the second term is ar, the third tenn is ar^, the fourth term is ar^, and so on. Thus the n*^ term is ar""\ 464. To find the sinn of a given nuniber of quantities in Geometrical Frogression, the first term and the common ratio being supposed hnowii. Let a denote the first term, r the common I'atio, n the number of terms, s the sum of the terms. Then s = a ^ ar-^ ar^ + ar^ + + a?*"~* ; therefore sr = ar + ar^ + ar^ + + fir""' + a?'". Hence, by subtraction, sr - s = ar" - a ; air"-!) ,,. therefore 8 - —^ =-^ (1 ). r - 1 If I denote the last term, we have l = ar"-' (2), hence 8^ (3). r- 1 ^ Equation (1) gives the value of s in terms of the quantities which are supposed known. Equation (3) is sometimes a con- venient form. 270 GEOMETRICAL PROGRESSION. 4G5. We may write tlie value of s thus, 0(1-0 "- 1-r • Now suppose r less than unity ; then the larger n is the smaller will ?'" be, and by taking n large enough r" can be made as small as we please. If then 71 be taken so large that r" may be neglected in comparison with unity, the value of 5 reduces to . We may enunciate the result thus : hi/ taking n large enough, the sum of n terms of the Geometrical Progression can he made to differ as little as we i^lease from -z . This statement is sometimes abbreviated into the following : the sum of an infinite number of terms of the Geomet^'ical Progression is -z ; but it must be remembered that it is to be considered as nothing more than an abbreviation of the preceding statement. The preceding remarks suppose that r is less than unity. In future, both in the text and in the examples, when we speak of an infinite Geometrical Progression we shall always sujipose that r is less than unity. We may apply the preceding remarks to an example. Con- sider the series 1, ^, J, |, ; here a = l, r = ^; thus the sum of 71 terms is ^j ^ (1 - ^-n ) ? that is, 2 - ^j;:^ . Now by taking n large enough, 2""^ can be made as large as we please, and 1 therefore -^-^zti as small as we please. Hence we may say that hy taking n large enough, the sum of n tei'ms of the series can he made to differ from 2 hy as small a quantity as we please. This is abbreviated into the following : the sum of an infinite number of terms of this series is 2. 466. In a geometrical progression continued to infinity each term hears a constant 7'atio to the sum of all which follow it; the common ratio heing supposed less than unity. GEOMETRICAL PROGRESSION. 271 Let the series be a + ar + ar^ + ar^ + . . . ; tlieii the rt** term is ar""'^ ; the sum of all the terms which follow this = ar" (1 + r + ?'^ + . . .) = . i — r The ratio of the r^^ tenn to the sum of all which follow it is ar"-^-- , 1 — r 1-r that is . This is constant whatever n may be. If we wish to determine r so that this ratio may have a given 1-r 1 value %> we put = w ; therefore r = , — - . r 1 +p 467. Recurring decimals are cases of what are called infi- nite Geometrical Progressions. Thus, for example, -2343434 1 . 2 34 34 34 tt .. . ^2 denotes tt: + T73 + r^ + 77-^+ Here the terms alter — constitute a Geometrical Progression, of which the first term is 34 1 —r^ , and the common ratio is -r-—, • Hence we may say that the 10 10 sum of an infi.nite number of terms of this series is - 3 ~=" ] ^ ~ i nz r > 34 . . 2 34 that is, —rr- . Therefore the value of the decimal is y^ + ^r-— . We will now investigate a general rule for such examples. 4G8. To find the value of a recurring decimal. Let P denote the figures which do not recur, and suj^pose them p in number ; let Q denote the figures which do recur, and suppose them q in number. Let s denote the value of the recurring decimal; then s='PQQQ , Ws = F-QQQ , W^'s = rQ-QQQ ; hv subtraction, (IC^' - lO^^ = PQ- P. 272 GEOMETRICAL PHOGRESSIOX. Now 10''^^ - 10'' = (10' - 1) lOP; and 10' - 1 when expressed by jSgures in the usual way will consist of q nines. Hence we deduce the usual rule for finding the value of a recurring decimal : subtract the integral number consisting of the non-recurring figures from the integral number consisting of the non-recurring and recurring figures, and divide by a number consisting of as many aiines as there are recurring figures followed by as many cyphers as there are non-recurring figures. 4G9. To insert a given number of Geometrical means between tioo given terms. Let a and c be the two given terms, n the number of terms to be inserted. Then the meaning of the problem is that we are to fi.nd n+ 2 terms in Geometrical Progression, a being the first term and c the last. Let r denote the common ratio ; then c = ar"""^^ ; thus r — (-y'^^. Tliis fiLnds r, and the required terms are ar, ar^y ar^, ar". 470. In Art. 464 we have five quantities occurring, namely, a, r, I, n, s; and these are connected by the equations (1) and (2)? or (2) and (3), there given. We might therefore propose to find any two of these five quantities when the other three are given ; it will however be seen that some of the cases of this problem are too difiicult to be solved. The following four cases present no difficulty : (1) given a, r, n; (2) given a, n, I; (3) given r, n, I ; (4) given r, n, s. 471. Suppose, however, that a, s, n are given, and therefore r and I are to be found. Then r would have to be found fi'om the equation s{r-l)==a(r"-l); we may divide both sides by r— 1, and then we shall have an equation of the (ri — 1)*^ degree in the unknown quantity r, which therefore cannot be solved by any method yet given, if n be greater than 3. Similar remarks will hold in the case where I, s, n are given, and therefore a and r are to be found. GEOMETRICAL PllOCKESSlOX. 273 472. Four cases of the problem remain, namely, those four in which n is one of the quantities to be found. Sujiposc a, r, I given, and therefore s and n are to be found. Here n would have to be found from the equation I — ar"~\ where the unknown quan- tity n occurs as an exponent; nothing has been said hitherto as to the solution of such an equation. 473. To find the sum ofn terms of the following series; a, {a + 6}r, {a + 26}r^, {a + W^r^^ Let s denote the sum; then s = a + [a + h]r + [a + 2h}r' -¥ -it[a + {n-\)h]r'"\ rs= ar + {a+ b]r^ + + {a + {n - 2) b] r'"^ + {a + (n-l)b]r\ By subtraction s{l-r) = a + br + br'+ +br''-' -{a + {n-l)b]r'' = a + ^''^l-_'p -{a + {^z-l)b]r^, therefore s = ' — --^ '—^ — + — ^ -,— ^ . 1-r (!-'') EXAMPLES OF GEOMETRICAL PUCGHLSSION. 1 c . • . 8 8 40 1. bum to SIX terms ^ + - + -^ + O J 2. Sum to ten temis 2 - 2" + 2' - 2* + 3. Sum to n terms 3 + 2 + ^+ 2 1 3 4. Sum to n terms .^ + o + o "^ o Z o . 2 4 8 5. Sum to infinity -x + „ + ,« + "3 i) 2/ 4 3 6. Sum to infinity - + 1 + -i- T. A. 18 274 EXAilPLES. XXXI. .... 1 1 1 1 7. Sum to intimty 9 + 7 "^ o + 1 7^ + 4 8. Sum to infiiiity 3 + 2+,. + .... 12 3G 9. Sum to mfimty 4 + -^ + ^ . + 10. Sum to infinity 1 +-+ — + Ill 11. Sum to infinity ^ - 9 + ^q - 2qq "^ ...,111 1 2. Sum to mnmty 1-^+t-o+ •^ 2 4 8 ..•32 8 13. Sum to mfimty 9 - o + 97 ~ • ... 1 1 1 14. Sum to mfimty ^ - ^5 +125" . 1111 15. Sum to infimty - - - + ^ — yr. + 16. Sum to infinity ^, ^ _ . + » _ ^^ "*" 2 ^ 2 3 2 3 1 7. Sum to infinity ^ + —3 + ^3+^+ 0000 1 8. Sum to n terms r + 2r" + 3r^ + 4?'" + 2 3 4 1 9. Sum to n terms 1 + - + —^ + 93+ '^ Zi ^ 3 5 7 20. Sum to n temis l + o+y + o + J 4 o 3 5 7 21. Sum to 71 terms 1— -+ — — -r + 2 4b 22. Find tlie sum of any number of terms in g. P. whose first and third terms are given. EXAAIPLES. XXXI. 275 23. If the common ratio of a g. p. is - 3, find the common ratio of the series obtained by talcing every fourth term of the original series. 24. The sum of £700 was divided among four persons, wliose shares were in €. p. ; and the difference between the gi'eatest and least was to the difference between the means as 37 to 12. Find their respective shares. 25. Sum to n teims the series whose m^^ term is (— Vfa*"*. 26. If P be the sum of the series 1 + r'' + ?*"^+?'"^4- ad inf., and Q be the sum of the series 1 + r'^ + r^' + r'^' + ad inf., prove that F'{Q-\Y = Q'{P - l)'. 27. Shew that J{-iU ) ^-^ -^(j^ 28. A person who saved every year half as mucli again as he saved the previous year had in seven yeai'S saved £102. 19^. How much did he save the first year ] 29. In a G. p. shew that the product of any two terms equi- distant from a given term is always the same. 30. In a G. p. shew that if each term be subtracted from the succeeding, the successive differences are also in g. p. 31. The square of the arithmetical mean of two quantities is equal to the arithmetical mean of the arithmetical and geometrical means of the squares of the same two quantities. 32. Find a g. p. contmued to infinity, in which each term is ten times the sum of all the terms which follow it. 33. If S represent the sum of ?i terms of a given g. p., find the sum oi S^ + S,^ + S^+ + '^,' 34. If n geometrical means be found between two quantities n a and c, their product will be (ok have occasion to use a method of proof which is called mathematical induction or demonstrative induction, and we shall now exemplify the method. 481. Suppose the following assertion made : the sum of n terms of the series 1, 3, 5, 7, is ?i^ This assei'tion we can 282 MATHEMATICAL INDUCTION. see to be true in some cases ; for example, the sum of two terms is 1 + 3 or 4, that is, 2" ; the sum of three terms is 1 + 3+5 or 9, that is, 3^; we wish however to prove the theorem universally. Suppose the theorem w^ere known to be true for a certain vaUie of 01 • that is, suppose for this value of n tliat 1 + 3 + 5+ +{2n-l) = n'; add 2n + l to both sides; then 1 + 3 + 5 + +{2n-l) + (2n + l) = n' + 27i+l = (n+l)\ Thus, if the sum of n terms of the series = ?i^, the sum of 91 + 1 terms will = (n + l)^ In other w^ords, if the theorem is true when we take a certain number of terms, w^hatever that number may be, it is true Avhen we increase that number by one. But we see by trial that the theorem is true when 3 terms are taken, it is therefore true when 4 terms are taken, it is therefore ti*ue when 5 terms are taken, and so on. Hence the theorem must be universally true. 482. We will now take another example ; w^e propose to establish the truth of the following formula : "We can easily ascertain by trial that this formula holds in simple cases, for example, when ?i = 1, or 2, or 3 ; we wish, how- ever, to establish it universally. Suppose the theorem w^ere known to be true for a certain value of n; add {n + iy to both sides; then 12 02 Q' 2 / i\2 n(n+l) (271 + 1) F + 2^+ 3-+ +n + (?i + 1) = P + {n+ 1). But ^^-(^'^^)^^^^'^ 1) , /., , i\2__ /.. , i\ r^(2?^+ 1) / i\2 / i\ (n(2?i+ 1) _ ) + (?l + 1)^ = {71 + 1) } -^—pi + w + 1 > * = -'^~{27i' + 77i+6} 91 + 1 . _. ,_ _, m(7n + 1) (27?i +1) — ^ {n + 2) {271 + 3) = — ^ ^ , where m = 7i + l. MATHEMATICAL INDUCTION. 283 Tlius we obtain the same fonnula for the sum of n+\ terms of the series P, 2^, 3" as was supposed to hold for n terms. In other words, if the fonnula holds when we take a eei-tain number of terms, whatever that number may be, it holds when we increase that number by one. But the formula does hold when 3 terms are taken, therefore it holds when 4 terms are taken, therefore it holds when 5 terms are taken, and so on. Hence the formula must hold universally. 483. The two theorems which we have proved by the method of induction may be established otherwise. The first theorem is an example of an Arithmetical Progression, and the second has been investigated in Art. 460. There are many other theorems which are capable of easy proof by the method of induction ; for example, that in Art. 4G1. The theorems asserted in Art. 69, respecting the divisibility of a;" ± a" by x ± a, may be proved by induction. For -X -T — ' : x — a x — a hence x" — a" is divisible hj x- a when a;"~^ — a"~^ is so. Now we see that x — a is divisible by x — a, therefore x^ — a* is divisible by x — a, therefore again x^ — a^ is divisible hj x — a, and so on ; hence x^ — a" is always divisible by x — a when ?i is a positive in- teger. Similarly the other cases may be established. As anotlier example the student may consider the theorems in Art. 225. 484. The method of mathe'niatical induction may be thus described : "VVe prove that if a theorem is true in one case, what- ever that case may be, it is true in another case which we may call the next case; we prove by trial that the theorem is tnie in a certain case ; hence it is tiiie in the next case, and hence in the next to that, and so on; hence it must be true in every case after that with which we began. 485. It is possible that this method of proof may be le.ss satisfactory to the student than a more direct proceeding; it may 284 MATHESUTICAL INDUCTION. appear to him that he is rather compelled to believe propositions so proved than shewn why they hold. But as in some cases this is the only method of 2:)roof which can be used, the student must accustom himself to it, and should not pass over it when it occurs until he is satisfied of its validity. 486. We may remark that the student of natural philosophy will find the word induction used in a different sense in that sub- ject; the word is there applied to the assumption or conjecture that some law holds generally which is found to be true in certain cases that have been examined. There, however, we cannot be sure that the law holds for any cases except those which we have examined, and can never arrive at the conclusion that it is a necessary truth. In fact, induction, as used in natural philosophy, is never absolutely demonstrative, often far from it ; whereas the method of mathematical induction is as rigid as any other process in mathematics. MISCELLANEOUS EXAMPLES. 1. Transform 221*342 from the scale with radix ten to the scale with radix five. 2. If the radix of a scale be im + 2 the square of any num- ber whose last digit is 2m + \ or 2m + 2 ^\dll terminate with that digit. 3. A digit is written down once, twice, thrice, uf) to n times respectively, so as to form qi numbere consisting of one, two three, n, places of figures respectively. If a be the fii'st and b the last of the numbers, and r the radix of the scale, the simi of rh — na the numbers is r-l 4. If m, n be any two numbers, g their geometrical mean, a^ , Aj the arithmetical and harmonical means between m and g, and CTg, h^ the arithmetical and harmonical means between g and n, prove that aji^ = 9' = <^^.h^ • EXAMPLES. XXXIII. 285 5. If between h and a there be inserted n aritlimetical means, and between a and b there be insei'ted n harmonical means, the sum of the series composed of the products of the corresponding terms of the two series is (n + 2) ah. 6. If n harmonical means are inserted between the two posi- tive quantities a and 6, shew that the difference between the first and the last bears to the difference between a and b a less ratio than that of n — 1 to n + 1. 7. A sets out from a certain place and travels one mile the first day, two miles the second day, three the third, four the foui-th, and so on. B sets out five days after ^.i and travels the same road at the rate of 1 2 miles a day. How far will A travel before he is overtaken by i> 1 8. From 256 gallons of wine a certain number are dra'WTi and replaced ^\^.th water ; this is done a second, a thii-d, and a fourth time, and 8 1 gallons of wine are then left. How much was drawn out each time ] 9. A and Jj have made a bet, the amount of the stakes being £90, and the sum staked by each being inversely proportional to all the money he has. If A wins he will then have five times what B has left; if B wins he will then have double what A has left. Wliat sum of money had each ? 10. li (a + b + c)(a + b + cl) = {c + d + a) {c + d + b), prove that each of these quantities is equal to (a-c) { a-d){b-c ) (b-d) {a + b-c- dy 11. If the roots of ax^ + 2bx + c = be ])0ssible and different, those of (a + c) (ax' + 2bx -^c) = 2{ac- 6') {x' + 1) will be impossi- ble ; and vice versd. 12. If a + b + c^O, x + ij + z-i-w^ 0, then the two equations J (ax) + J {by) + J{cz) ^ 0, J{bx) - J {ay) + J{cw) = 0, ai'C deilucible the one from the other. 28G PERMUTATIONS AND COMBINATIONS. XXXIY. PERMUTATIONS AND COMBINATIONS. 487. Tlie different orders in which any things can be ar- ranged are called their permutations. Thus the permutations of the letters a, h, c, taken two at a time are ab, ha, ac, ca, be, cb. 488. The combinations of things are the different collections that can be fonned out of them, without regarding the order in vrhich the things are placed. Thus the combinations of the letters a, b, c, taken two at a time are ab, ac, be; ah and ba though different permutations foiming the same combination. 489. We may observe that a difference of language occurs in books on this subject; what we hare C2l\Qdi permutations are called variations or arrangements by some writers, and they restrict the word permutations to the case in which all the things are used at once; thus they speak of the variations or arrangements of four letters taken two at a time, or three at a time, but of the permuta- tions of them taken all toG^ether. 490. To find the number of i^er mutations of n things talcen r at a time. Suppose there to be n letters a, b, c, d, ; we shall first find the number of permutations of them taken two at a time. Put a before each of the other letters ; we thus obtain n—\ permutations in which a stands first. Next put 6 before each of the other letters ; we thus obtain n—\ permutations in which h stands first. Similarly there are n-1 permutations in which c stands first ; and so on. Thus, on the whole, there are n {n — \) permutations of n letters taken two at a time. We shall now find the number of permutations of the n letters taken three at a time. It has just been shewn that out of n letters PERMUTATIONS AND COMBINATIONS. 287 we can form n{n-~l) peniiutations each of two letters; lience out of the n—1 letters b, c, d, we can form (n — 1 ) (n — 2) per- mutations each of two lettei's ; put a before each of these and we have {n—l){n—2) permutations each of three letters in which a stands first. Similarly there are (?i-l)(n — 2) permutations each of three letters in wliich h stands first. Similarly tliere are as many in which c stands first ; and so on. Thus on the whole there are n{n—l)(7i — 2) permutations of oi letters taken three at a time. From these cases ifc might be conjectured that the number of permutations of n letters taken r at a time is n{}i-l){n-2) (n-r+l), and we shall prove that this is the case. For suppose it true that the number of permutations of n letters taken r — 1 at a time is 7i(n-l) {n-(r-l) + l}, we shrJl shew that a similar formula will give tlie nimiber of per- mutations of the letters taken r at a time. For out of the n — 1 letters h, c, d, we can form (71-1) (n- 2) {^_l_(r_l) + i} permutations each of r — 1 letters; put a before each of these, and we obtain as many permutations each of r lettei-s in which a stands first. Similarly we liave as many in which b stands firet, as many in which c stands first, and so on. Thus on the whole tliere are n{n-l)(n-2) (n-r+l) permutations of n letters taken r at a time. If then the formula holds when the letters are taken r — 1 at a time, it ^\^.ll hold when they are taken r at a time; ; but it has been proved to hold when they are taken three at a time, therefore it holds when they are taken four at a time, therefore it holds when they are taken five at a time, and so on; thus it hoMs universally. 288 rEllMUTATlONS A^D COMBINATIONS. 491. Hence the number of pennutations of n things taken all together is n (?i - 1) (n - 2) 1. For the sake of brevity n {n - 1) {ji - 2) 1 is often denoted by 1 71 ; thus In denotes the product of the natural numbers from 1 to 71 inclusive. The spnbol \n may be v'esid, factoi'ial n. 492. The formula for the number of permutations of n things taken r at a time may also be obtained in another manner. Let P denote the number of permutations of n letters taken r — 1 at a time. To form the permutations of n letters taken r at a time we may proceed thus : take any one of the P pennuta- tions, and place at the end of it any one of the n-r + 1 letters which it does not involve. Thus the whole number of the per- mutations of the n letters taken r at a time will be {n-r+l) P. Now the number of the permutations of ?i letters taken one at a time is n ; therefore the number taken two at a time is 71(01 -1); therefore the number taken three at a time is 71(71 -1) [n- 2); and so on. 493. Any combination of r things will produce \r pennuta- tions. For, by Ai^ticle 491, the ?• things which form the given combination can be arranged in |r different ways. 494. To fiiul the number of comhinatioiis of n things take7i r at a tinie. The number of combinations of 71 things taken ?• at a time is 7i(7i-l)(7i-2) Qi-r + l) \I For the number of peiTQutations of 7i things taken r at a time is 71 {n - 1) (71 - 2) (?^-r+l), by Art. 490; and each combination produces r permutations, by Ai't. 493 j hence the number of combinations must be 71 (?^ - 1) (n - 2) (n-r+l) \r PERMUTATIONS AND COMBINATIONS. 289 If we multiply both numerator and denominator of this ex- pression by \n — r it becomes -, — f^ . ' [r \n — r 495. The number of comhinations of n tilings taken r oi a t'mie is the same as the number of them taken n — r at a time. The number of combinations of n things taken n-r at a time is n{n-l){n-2) { n - (h - r) + 1 } n — r that is, n (n-l)(n~2) (. + !) _ I n — r Multiply both numerator and denominator by \r and we ob- t^ . ~ . tani — j which, by Art. 494, is the number of combinations \r\n — r ' -^ ' of n things taken r at a time. The proposition which we have thus demonstrated will Ije evident too if we observe that for every combination of r things which we take out of n things, we leave one combination of n — r things. Hence every combination of r things coiTesponds to a combination oi n — r things which contains the remaining things. Such combinations are called complementary. 496. To fiiul for ichat value of r the number of combinations of n things taken r at a time is greatest. Let (n)^ denote the number of combinations of n tilings taken r at a time, (^)r-i "th® number of combinations of n things taken r — 1 at a time, , . n-r + \ , . then ('Or = : (>i)r-i . n — r ■\- \ , ., 714-1 . ,., , The factor may be written — -1, which shews r T that it decreases as r increases. By giving to r in succession the T. A. l'*^ 290 PERMUTATIONS AND COMBINATIONS. values 1, 2, 3, the number of combinations is continually increaae-1 so long as 1 is greater than unity. First suppose n even and = 2m, then - — -^ 1 is greater than 1 until r = wi inclusive, and when ?- = m + 1 it is less than 1. Hence the greatest number of combinations is obtained when the things are taken m at a time, that is, ^ at a time. 2^+1+1 ^ . Next suppose n odd and = 2m + 1, then ^ 1 is equal to unity when r = m-hl. Hence the greatest number of combinations is obtained when they are taken m at a time or m + 1 at a time, the result being the same in these two cases, n — 1 . '^^ + 1 . X- that is, when they are taken — -— at a time, or — ^ at a time. 497. To find the number of permutations ofn things taken all together which are not all different. Let there be n letters ; and suppose p of them to be a, q of them to be h, r of them to be c, and the rest to be unlike ; the number of permutations of them taken all together will be ■ • . J^ For let N represent the required number of permutations. If in any one of the permutations the p letters a were changed into p new letters different from any of the rest, then wdthout altering the situation of any of the remaining letters, we could from the single permutation produce \p different permutations ; and so if the p letters a were changed into p) different letters, the whole number of permutations would be iVx !jt?. Similarly, if the q letters h were also changed into q new letters different from any of the rest, the whole number of permutations we could now ob- tam would be iV x j^ x l^' ; and if the r letters c were also changed, the whole number would be ^xj^jx[£x|r. But this number must be equal to the number of permutations of n dissimilar things PERMUTATIONS AND COMBINATIONS. 291 taken all together, that is, to \n. Thus iV"x[^x[£x[r = [ii; [n therefore iV = j — , — — . [p[q{r Aiid similarly any other case may be treated. 498. There is another mode in which the result of the pre- ceding Aiiiicle may be obtained which will be instructive for the student. We will explain it for simplicity by the aid of a par- ticular example ; but the reasoning is perfectly general in cha- racter. Su23pose we have 10 letters; suppose 2 of them to be a, 3 of them to be b, and 5 of them to be c : required the number of permutations of the 1 letters taken all together. "We may consider that we have 10 places which are to be occupied by the 10 letters. Choose any 2 of the places and put a 10. 9 in each ; this can be done in ., ' , wavs. Choose any 3 of the remaining 8 places, and put b in each ; tliis can be done in 8 7 fi ■■ ' ' vrays. Then put c in each of the remaining 5 places ; 5 4.3 2.1 this can be done in 1 way : and 1 = i no / - • Now the -^ ' 1.2.3.4.0 product of the results thus obtained will obviously give the total |10 number of permutations : this number therefore is , -^ , ^ . . 499. If there be n things not all different, and we require the number of permutations or of combinations of them taken r at a time, the operation will be more complex; we will exemplify the method in the following case : There are n things of v:hlch p are alike and the rest inilike ; required the number of combinations of them taken r at a time. We shall suppose r less than n - ]), and put n-p^q. Con- sider first the number of combinations that can be formed witliout 19-2 292 PERAIUTATIONS AND COISIBINATIONS. using any of the p like things ; this ls the number of combinations \q of q tilings taken r at a time, that is, "■_ . Next take one of the p things and r - 1 of the q things ; the number of ways in wliich combinations can thus be formed is the same as the num- ber of combinations of q things taken r - 1 at a time, that is, I? — . Kext take two of the ;; things and combine them \ r-\\q - r+\ Avith ?• - 2 of the q things ; this can be done m j — — ^i — _ o ways. Proceed thus, and add the number of combinations so obtained together, which will give the whole number of combi- nations. If however r is not less than q we should consider first the case in which r -q things are taken from the 2^ like things, and q things are taken from the q unlike things ; thLs can be done in only one way. Next take r-q+l things from the p things, and ^ - 1 from the q things ; this can be done in q ways. And so on. If the number of permutations be required, we have only to observ'e that each combination of r things in which s are alike and I ^ the rest unlike, will produce ^=^ permutations (Ai-t. 497), and thus the whole number of permutations may be found. 500. By the following method the formula for the number of combinations of n things taken r at a time may be found without assuming the formula for the number of permutations. Let {7i)r denote the number of combinations of n things taken r at a time. Suppose n letters a, h, c, d, ; among the com- binations of these r at a time, the number of those which contain the letter a is obviously equal to the number of combinations of the remaining n-l letters r— 1 at a time, that is, to {n- l)^_i. • The number of combinations wliich contain the letter h is also (ji~l\_^, and so for each of the letters. But if we form, first all ■ PEllMUTATIONS AND COMBINATIONS. 293 the combinations which contain a, then all the comhinationa which contain h, and so on, each particular combination will ap- pear r times ; for if r = 3, for example, the combination abc will occur among those containing a, among those containing 6, and among those containing c. Hence In this formida change n and r first into ?i - 1 and r - 1 respectively, then into n - 2 and r - 2 respectively, and so on ; thus (n-l),_x = ^(n-2),.„ 7 — 4 / o\ n-r + 2 (n-r + 2)^ = ^ {7i~r + l)^. IMultiply, and cancel like terms, and we obtain , n{n — \) (?i — r + 2) (n-r+ 1) for (71 — r + l)j = w — r + 1. 501. To find the lohole number of permutations of n things when each may occur once, twice, thrice, ^ip to r times. Let there be 71 letters a, h, c, First take them one at a time ; this gives the number n. Next take them two at a time ; here a may stand before a, or before any one of the remaining letters; similarly b may stand before b, or before any one of the remaining letters ; and so on ; thus there are n* different per- mutations of the letters taken two at a time. Similarly by put- ting successively a, b, c, before each of the permutations of the letters taken two at a time, we obtain n^ permutations of the letters taken three at a time. Thus the whole number of permu- tations when the letters are taken r at a time will be n'. 294 EXAMPLES. XXXIV. 502. Since the number of combinations of oi things taken r at a time must be some integer, the expression n [n — V) (n — r + 1) must be an integer. Hence we see that the product of any r successive integei'S must be divisible by [r. We shall give a more direct proof of this proposition in the Chapter on the theory of numbers. EXAMPLES OF PERMUTATIONS AND COMBINATIONS. 1. How many different permutations may be made of the letters in the word Caraccas taken all together ? 2. How many of the letters in the word Heliopolis ? 3. How many of the letters in the word Ecclesiastical ? 4. How many of the letters in the word Mississippi f 5. If the number of permutations of n things taken 4 toge- ther is equal to twelve times the number of permutations of n things taken 2 together ; find n. 6. In how many v/ays can 2 sixes, 3 fives, and 5 twos bo thrown with 1 dice ? 7. If there are twenty pears at three a penny, how many different selections can be made in buying six-pennyworth ] In how many of these will a particular pear occur ? 8. From a company of soldiers mustering 96, a picket of 10 is to be selected ; determine in how many ways it can be done, (1) so as always to include a particular man, (2) so as always to exclude the same man. 9. How many parties of 12 men each can be formed from a company of 60 men ] 10. If the number of combinations of n things r — r' toge- ther be equal to the number of combinations of n things r + r' together, find n. EXAMPLES. XXXIV. 295 11. In how many ways can a party of six take their places at a round table t 1 2. In how many diflfercnt ways may n persons form a ring ? 13. How many diflferent numbers can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 ; each of these digits occun-ing onco and only once in each number ? How many with the digits 1, 2, 3, 4:, 5, 6, 7, 8, 9, 0, on the same supposition 1 14. Out of 12 conservatives and 16 reformers how many different committees could be formed each consisting of 3 con- servatives and 4 reformers ? 15. If there be x things to be given to n persons, shew that n' will represent the whole number of different v/ays in which they may be given. 16. Suppose the number of combinations of oi things taken r together to be equal to the number taken r + 1 together, and that each of these equal numbers is to the number of com- binations of n things taken r—1 together as 5 is to 4, find the values of n and r. 17. Given 7ii things of one kind, and n things of a second kind, find the number of j)ermutations that can be formed con- taining r of the first and s of the second. 18- Find how many different rectangular parallelepipeds there are satisfying the conditions that each edge shall be equal to some one of n given straight lines all of different lengths j and that no face of a parallelepiped shall be a square. 19. The ratio of the number of combinations of 4;i things taken 2?^ together, to that of 2>i thmgs taken 7i together is 1.3.5 (4n-l) {1.3.5 (2/1-1)}'' 20. Out of 17 consonants and 5 vowels, how many words can be formed, each containing two consonants and one vowel ? 21. Out of 10 consonants and 4 vowels, how many words can be formed each containing 3 consonants and 2 vowels ] 29G KXAT^IPLES. XXXIV. 22. Find the number of words wliicli can le formed out of 7 letters taken all together, each word being such that 3 given letters ai'e never separated. 23. AVitli 10 flags representing the 10 numerals how many- signals can be made, each re2:>resenting a number and consisting of not more than 4 flags ? 24. How many words of two consonants and one vowel can 1)9 formed from 6 consonants and 3 vowels, the vowel being the middle letter of each word ] 25. How many words of 6 letters may be formed with 3 vowels nnd 3 consonants, the vowels always having the even places 1 26. A boat's crew consists of 8 men, 3 of whom can only row on one side and 2 only on the other. Find the number of ways in which the crew can be arranged. 27. A telegraph has on arms, and each arm is capable of n distinct positions : find the total number of signals which call be made with the telegraph, supposing that all the arms are to be used to form a signal. 28. A pack of cards consists of 52 cards marked differently: in how many different ways can the cards be arranged in four sets, each set containinsr 1 3 cards ? 29. How many triangles can be formed by joining the angular points of a decagon, that is, each triangle having three of the angular points of the decagon for Us angular points 1 30. There are n points in a plane, no three of which are in the same straight line with the excej^tion of ^9, which are all in the same straight line: find the number of straight lines which result ii*om joining them. 31. Find the number of triangles which can be formed by joining the points in the preceding Example. 32. There are n points in space, of which 2^ ^^'^ in one plane, and there is no other plane which contains more than three of them: how many planes are there, each of which contains three of the points ] EXA^IPLES. XXXIV. 297 33. If n points in a ])liine be joined in all jwssible ways by indefinite straight lines, and if no two of the straight lines be coincident or parallel, and no three pass thi'ough the same point (with the exception of the n original points), then tire number of points of intersection, exclusive of the n points, will be n {n-\){n-2){n-Z) 8 34. There are fifteen boat-clubs ; two of the clubs have eacli three boats on the river, five others have two, and the remaining eight have one : find an expression for the number of ways in which a list can be formed of the order of the 24 boats, observin^^ that the second boat of a club cannot be above the first. 35. A shelf contains 20 books, of which 4 are single volumes, and the others form sets of 8, 5, and 3 volumes respectively : find in how many ways the books may be arranged on the shelf, the volumes of each set being in their due order. 36. Find the number of the permutations of the letters in the word examination taken 4 at a time. 37. Find the number of the combinations of the letters in the word ijroportion taken 6 at a time. 38. There are n—1 sets containing 2a, 3a, na things respectively : shew that the number of combinations which can l>e formed by taking a out of the first, 2a out of the second, and \na so on for each combination, is ~~. 39. Find the sum of all the numbers which can be formed with all the digits 1, 2, 3, 4, 5, in the scale of 10. 40. The sum of all numbers that are exi)rcssed by the same diirits is divisible bv the sum of tlio diirits. 298 BINOMIAL THEOREM. XXXV. BIXO:^IIAL THEOREM. POSITIVE INTEGRAL EXPONENT. 503. We have already seen that {x + cif = x' ->r 2xa + a^ , and that {x + af = x^+ 3x^a + Sxa'^ + a^ ; the object of the present Chapter is to find an expression equal to {x + a)" where n is any- positive integer. 504. By ordinary multiplication we obtain (a; + rtj) (x + «J =^x^ + (ttj + a,^)x + d^a.,', {x + ai) {x + a^ (x + a^) = x^ + (a, + «, + a^x' + (a^a^ + a/i^ + a^a^)x 4- a^a^a.^, {x + rti) {x + a.,) (x + a.^ {x + a^ =^x^ + (a^ + a., + a3 + ai)x^ + {a^a^ + a^a^ + a^a^ + a./t.^ + a./i^ + a.^a^ x' + (a^a./i.^ + a^^a./i^ + aya^a^ + a./i^a_^x + a^a^a.fj^. Now in these results we see that the following laws hold : I. The number of terms on the right-hand side is one more than the number of the binomial factors which are multiplied together. II. The exponent of x in the first term is the same as the number of binomial factors, and in the succeeding teims each exponent is less than that of the preceding term by unity. III. The coefficient of the first term is unity ; the coefficient of the second term is the sum of the second terms of the binomial factors ; the coefficient of the thii'd term is the sum of the pro- ducts of the second terms of the binomial factors taken two at a time; the coefficient of the fourth term is the sum of the pro- ducts of the second terms of the binomial factors taken three at a time ; and so on ; the last term is the product of all the second terms of the binomial factors. We shall now prove that these laws always hold whatever be the number of binomial factors. Suppose the laws to hold when n — \ factors are multiplied together; that is, suppose BINOMIAL THEOREM. POSITIVE INTEGRAL EXPONENT. 299 where », = the sum of the terms a , a a ,, p^ = the sum of the products of these tei-ms taken two at a time, p^ = the sum of the products of these tei-ms taken three at a time, 2>„^i = the product of all these terais. Multiply both sides of this identity by another factor x + a^; thus (x + a,){x + a,) (x + «„) = ic«+ {p^+ a„) re""' + (p., +p,a,) a:""" + (p3+;;, ^^^^ so on. Hence by Art. 513 the total number of ways of taking n things is 2"- 1. This theorem was obtained by the early writers on Algebra Ijeforo the Binomial Theorem was kno^vn ; the proof is a simple example of mathe- matical induction which is deserving of notice. We have to T. A. -'0 306 EX.N3IPLES. XXXV. shew that if unity be added to the total number of \rays of taking n things, tlie residt is 2". Suppose we have four letters a, h, c, d; form all the possible selections and prefix unity to tliem. Thus we have 1, a, h, c, d, ah, ac, ad, he, hd, cd, ahc, ahd, acd, hcd, ahcd. Here the total number of symbols is 16, that is, 2*. Now take an additional letter e; the corresponding set of symbols will consist of those already given, and those which can be formed from them by afiixing e to each of them. The number will there- fore be doubled; that is, it will be 2^ The mode of reasoning is general, and shews that if the theorem is true for n things, it is true for ?? + 1 thin-:]:s. EXAMPLES OF THE BINOMIAL THEOREM. 1. Write down the 3'^'^ term of {a + 5)'\ 2. "Write down the 49*^ term of {a-xy\ 3. Write down the 5*^ term of (a' - hy\ i. — 4. AVrite down the 200P' term of (a^ + x'')''''. 5. Write down all the terms of (5 — 4ic)*. G. AYrite down the 5^ term of {3x^' - Ay-)\ 7. Write down the 6'^ term of {2a^ - h^-'f, 8. Write down all the terms of ( 5 — ^ ) . 9. Write down the middle term of (a + xy*. EXAMPLES. XXXV. 307 1 0. Write down the two middle terms of {a + a;)'. 1 L Expand [a + J {a' - 1)}" + {a - J {a" -\)Y in powers of a. 1 2. Write down the coefficient of y in the expansion of 1 3. If xi be the sum of the odd terms and B the sum of the even terms in the expansion of {x + a)", prove that A'-B'={x'-ay. 14. Prove that the difference between the coefficients of a;*"*^* and a*" in the expansion of (1 -^-x)"'^^ is equal to the differ- ence between the coefficients of oj*"^^ and x'""^ in the expansion of (1 + x)\ 15. Shew that the middle term in the expansion of (1 + xf 1.3.5 ... (2n-l) , \n 16. Find the binomial expansion of which foui' consecutive terms are 2916, 48G0, 4320, 21G0. 17. Prove that if the tei-m a;*" occurs in the expansion of / iN" . k* I a; + - I the coefficient of the term = xj \\{n-r) ' 4 jn + r) ' 18. Wiite down the coefficient of a;*'"^' in the expansion of 1 \ 2/1+1 X- X 19. Find the r^^ term from the beginning, the r^^ term from the end, and the middle term of f a; — - j . 20. If tg, t^, f^, t^, rejiresent the terms of the expansion of (a + a;)", shew that (^0-^.-^^- )'+(^.-^+^- y={a'+xy. 20-2 308 BINOMIAL THEOREM. ANY EXPONENT. XXXYI. BIN^OMIAL THEOREM. ANY EXPONENT. 516. We liave seen that wlien w is a positive integer /I \n 1 W (n - 1) 2 (1 +a:)"= 1 +71X+ -A. — o-^ic% 1 . ^ We now proceed to shew that this relation holds when 7i has any value positive or negative^ integral or fractional, that is, we shall prove the Binomial Theorem for an7/ exponent. We shall make some observations on the proof after giving it in the usual form. 517. Suppose m and n are positive integers; then we have (i+^r=i+«u.^^;i^>^'-.'"^'"-,l><"'-^)«.-+ (1), /I \-. 1 n (n-l) ^ n (n - 1) In - 2) „ ,^. {\+xY= 1 +nx+ ^ ^ ' x' + -^ -L^ ' x^+ (2). But (1 + xY X (1 + xy = (1 + xf-"', hence the product of the series which form the right-hand mem- bers of (1) and (2) must = (1 + x)"*^"; that is, T / , (m + n) (m + ?z - 1) „ I + {m + n)x + — ^-^-^ x^ 1.2 {jn + n){m + 7i — l)(m + 7i — 2) 3 f- m(m — 1) , 7n (m — 1) (m — 2) , ) =|i+^»+_^__i^-+^j — \t — ^^^ 1 .|l^^ + 'L^)^^^»(''-|)("-^)^3^ J (3). Equation (3) has been proved on the supposition that m and n are positive integers ; but the product of the two series which occur on the right-hand side of (3) must be of the same form whatever BINOMIAL THEOREM. ANY EXPONENT. 309 m and n may be ; we therefore infer that (3) must be true what- ever m and n may be. We shall now use a notation that will enable us to express (3) briefly. Let f{m) denote the series - 771 (m — 1 ) - m Im — 1) (m — 2) , 1 + mo; + — ^ — K— ^ x^ + — ^^ ttP ^• 1.2 "^ ' 13 ^ "^ whatever m may be ; then f{n) will denote what the series becomes when n is put for m; and f{m + n) will denote what the series becomes when m + ?i is put for m. And when ni is any positive integer f{m) = (1 + ic)"* j also /(O) = 1. Thus (3) may be written f (m + 71) =f{m) x/(7t) (4). Similarly, /{^i + n + p) =f{pi + n) -k f{2^) =f(jn)xf{n)xf{p). Proceeding in this way we may shew that f{m + n + p + q+ )=/W x/W ^f{p) x/fo)^ (•5)- s Now let m — n=p = q= =-, where s and r are positive integers, and suppose the number of terms to be rj then (5) becomes /<-)={/e)p therefore {-^Wi'^/G)' But since 5 is a positive integer f{s) = {\ + »)', and therefore therefore ( 1 + x)- =j^r^j = l + s ■ ^ , -ic+ — ; — z — ^ + r i . .i This proves the Binomial Theorem when the exponent is any positive quantity. 310 BINOMIAL THEOREM. ANY EXPONENT. Again, in (4) put — n for m; thus /(-n)x/(/i)^/(0)=l; therefore __^=/(_^). But \£n be any positive quantity, f{n) = (1 + a;)"; hence that is, (1 + x)-» :^\+{-n)x+ ^~ ''\^~!^ ~^K '+ This proves the Binomial Theorem when the exponent is any negative quantity. 518. The proof of the Binomial Theorem for any exponent contained in the preceding Article was first given by Euler ; although difficult and not altogether satisfactory, it is a valuable exercise for the student. We shall now ofier some remarks upon it. The first point we have to notice is the mode of proving that y(m + ?z) =/(w) x/{n). The student should for an exercise write down three or four terms of the series for /(m), and also of the series for /(7i), and multiply them together; if the pro- duct be arranged according to powers of x, it will be found that so far as it has been completely formed, it will agree with the series for /(m + n). But from knov»dng what /(m) and f(n) represent when m and n are positive integers, we infer without the trouble of actual multi^^lication, that the law which is expressed ^J f {ni + n) = f (vi) X f (n) must hold. The mode of establishing this law in the simple case in v/hich m and n are j)Ositive integers is a valuable and important algebraical artifice. But the way in which we infer that f {ni + n) =/(m) x/(n)j ichatever m and n may he, is still more important. The princii^le is merely this : the form of any algebraical product is the same whether the factors represent whole numbers or fractions, positive or negative numbers; thus, for example, {a +b) {a + c) = a'' + {b + c) a+bc BIxVOMIAL THEOREM. ANY EXPONENT. 311 is true whatever a, h, and c may be. Hence we infer that f{m) x/(?i) will have the same form in all cases, whether m and n be positive integers or not. The student may also notice the proof of this result which is given in the Tlieory of Equations, Chapter xxiv. 519. Tlie most difficult point however to he considered is the meaning of the sign =^in the assertion /■f \« 1 n(n-\) . ,,. (1 +xY= 1 +7ix^ \ ' x^ ^ (1). 1. . Li Suppose, for example, that ?i = - 1, then the above becomes (1 +a:)~*= l-x + x--x^ + (2). Isow we know that the sum of r terms of the series \ - X + X' - x^ -\- is — r-^ — -- : hence when x is numerically \ + X '' less than "u^itY, by taking enough terms of the series, we can obtain a result differing as little a^ loe please from and thus 1 4- a;' we can in this case understand the assertion in (2). But when X is numerically greater than unity, there is no such numerical approximation to the value of :j obtained by taking a large number of terms of the series 1 — x -^ x' — x^ ■¥ "We shall see in the Chapter on the Convergence of Series, that when X is numerically less than unity, we can form a definite conception of the series on the right of (1) whatever n may be. In this case there is no difficulty in the assertion f{m + n)=f{m)xf(ii)', each of the three series which it involves is arithmetically intelli- gible. But when x is numerically greater than unity, we cannot give an arithmetical meaning to the senes or to the assertion; all we ought to say is, that if we form the product of the fii-st r terms of /(m) and the first r terms of /(n), the first r terms of the result wiU agree with the first r terms of /(7/i + n); but this will 312 BINOMIAL THEOREJkL ANY EXPONENT. not justify us in writing / (m + n) = / (m) x/(n). The case in which X is numerically equal to unity would require special in- vestigation which would be out of place here. See Art 777. On the whole then we may conclude that the Binomial Theo- rem for the expansion of (1 + ic)" gives a result which is arithme- tically intelligible and true v/hen x is numerically less than unity; in what sense the result is true when x is numerically greater than unity has not yet been explained in an elementary manner. The subject of the expansion of expressions is however properly a portion of the Differential Calculus, to which the student must be referred for a fuller consideration of the difficulties. 520. To Ji7id the numerically greatest term in the expansion of (i + xy. We consider x as positive. I. Suppose n a positive integer. The (r + l)^*" term may be formed by multiplying the r^ term by- X, that is, by ( — ——Ijx; and this multipKer di- minishes as r increases. Put /n+1 \ {n + l)x { 1 ]x=l, thereiore p = ^r— • \ 2^ J ^ x+l If p be an integer, two terms of the expansion are equal, namely, the p^^ and the (p+ 1)*^, and these are greater than any other term. If p be not an integer, suppose q the integral part of p, then the (q + l)**" term is the greatest. II. Suppose n positive but not integral. As before, the (r + ly^ term may be formed by multiplying the r^^ term by ( — Ijx. If then X be greater than unity, there is no greatest term ; for the above multiplier can, by increasing r, be made as near to — a; as we please; that is, each term from and after some fixed term can be made as nearly as we please nuvierically x times the pre- ceding term, and thus the terms increase without limit. BINOMIAL THE011E.U ANY EXPONENT. 313 But if X be not greater than unity tliere will he a greatest term ; {ti -\- \\x for if 2^ = ^^-; — r— , then as long as r is less than p the multiplier is greater than unity, and the terms go on increasing ; but when r is greater than p the multiplier is less than unity, and so long as it continues positive it diminishes as r increases; and when the multiplier becomes negative it is still numerically less than unitv • so that each tenn after r has passed the value p is nimierically less than the preceding term. Hence, as in the first case, if p be an integer, the ^^ term is equal to the (p + lf" term, and these are greater than any other term ; if j) be not an integer, sui)pose q the integral part of 2^, then the (q + l)**" term is the greatest. III. Suppose n negative. Let m = - n, so that m is positive. The numerical value of the (r+ 1)"' term may be obtained by multiplying that of the r'^^ term by i J X, that is, by f — — + \\x. If X be greater than unity we may shew, as in the second case, that there is no greatest teitn. If X be less than unity, put fm-\ \ ^ , „ (m- \)x I hi ]x—\, therefore ?J = — , — . \ p J i-x If 2^ be a positive integer, the j»y^ term is equal to the {p + l)*"* term, and these are greater than any other term. If p be positive but not an integer, suppose q the integral part of p, then the {q + l)*** term is the greatest, li p be negative, then m is less than unity; in this case each term is less than the preceding, and the fii"st term, that is, unity, is the gi'eatest. If X be equal to unity, then when m is gi'eat^r than unity the terms continually increase and there is no greatest terra, when in is equal to unity the terms are all equal, and when m is less than unity the terms continually decrease so that the firet is the gi-eatest. We have supposed throughout that x is jiositive ; if x be nega- tive, put y = — Xj so that y is positive; then find the numerically 314 BINOMIAL THEOREM. ANY EXPONENT. greatest term of (1 + y)", and this will also be the numerically greatest term of (1 + x)". 521. The fii'st term of the expansion of (1 + x)" is unity; any other term is kno^^^l since the [r + If^ term is n { n-\) (n-r + 1) r This expression is called the general term, because by putting 1, 2, 3, successively for r, it gives us in succession the- 2"**, 3"*, 4''', terms; that is, we can obtain from it any teiTo. after the fii'st. The expression for the general term may be modified in particular cases, and sometimes simplified, as will be seen in the following examples : (1 + a:)""*. Here n = — vi ] the general term becomes (— T)i) {- m — \) {— m — r + \) ^ ■ X , [I. which may be %\Titten m(m + l) (m + r-l) ^ ^^ LT (1 + x)-. Here n = ^; the numerator of the coeflicient of ic" is K^OG-) G-'-)= if ?• is not less than 2, this may be written 1.3.5.7 (2r-3) 2'" (-1)'-; hence in the expansion of (1 +x)-, the first term is 1, the second is ^x, and arti/ suhsequent term may be found by putting for the (r + 1)*^ teiTQ 1.3.5 ^;-^^^^(-ir- 2',r (l + o:)"''. This is a jmrticular case of (1 + a:)""*. The co- efficient of x^ is 2.3.4 (2+r-n -^^ \-ir, that is, (r4-l)(-ir. BINOMIAL THEOREM. ANY EXPONENT. 315 (1 — a)"'. By the preceding example the (r + 1)'^ term is (r + 1) (- IX (- a:)^ that is, {r+l)x\ (1 + a;)~^ This is a particular case of (1 + ic)"*". The cocfB- cient of x"" is (1 — 5c)~\ By the preceding example the (r + 1)''* term is (!:±I)i!l±^(_ !).(_,), that is, (il±ilf±l)»f. If a; and n are ppsitive it will be found that in the expansion of (1 + a;)~" the terms are alternately positive and negative; and in the expansion of (1 — a;) ~" the terms are all positive. If x and n are positive, and n not an integer, it will be found that in the expansion of (1 + x)" the terms begin by being positive, and eventually become alternately positive and negative ; and in the expansion of (1 — x)"' the terms begin by being alternately positive and negative, and eventually become all of one sign. 522. A Multinomial expression may be raised to any power by repeated use of the Binomial Theorem ; thus, for example, {a + b -^ cY = {a + {h + c)Y = a' + 3a' (b + c)+ da{b + cY + {b + cy ; if we now expand (b + cf and {b + cf and put the resulting ex- pansions in the place of these quantities respectively, we shall obtain the expansion of [a + b + c]^. Similarly, [a + b + c + df = {a + {b+c + d)Y = a^ + 3a' (5 + c + d) + 3a (5 + c + df +{b + c + dy ; the expansion may then be completed by finding the expansion oi (Jy+c + df and of (6 + c + df in the manner just exemplified. Or we may proceed thus, {a + b +c + dY ={{a-rb) + {c + d)Y =-{a + bf + 3 (a + by (c + d) + 2>{a + b) (c + dy + (c + dy ; the expansion may then be completed by expanding (a + 6)', {a + by, (c + dy, and (c + dy, and effecting the requisite multipU- cations. 316 BINOMIAL THEOREM. ANY EXPONENT. 523. To find the mmiber of homogeneous 'products ofr dimen- sions that can he funned out of n letters a, b, c, and tlieir powers. By common division, or by the Binomial Theorem, :; = 1 +aa; + aV + aV+ ... \—ax \ — bx 1 \ — ex -1 + CX + c"x^ + c x^ + . . . Thus 1 1 \ — ax' \ — bx' \ — ex = \l+ax-\- a^x^ + aV + ) X |l + 5a; + ^>V + 6V+ I x |l -^ cx-\- c^x^ ^ c'^x^ + / = 1 + aS'iCc + S^ + S^ + suppose. Here /S'j = a + 6 + c + , iS'2 = a^ + a6 + 6Vac + , that is, ^j is equal to the sum of the quantities a, 5, c, ; >S'g is equal to the sum of all the products, each of two dimensions, that can be formed of «, 5, c, and their powers ; S^ is equal to the sum of all the products, each of three dimensions, that can be formed ; and so on. To find the nuTnber of products in any one of these sets of products, we put a, b, c, each = 1 ; thus 111 V !,,,-„ Ti . , J—. - — ■ — becomes — ; ;r- or (l-x) . 1-ax I - bx I - ex (1 - ^) BINOMIAL THEOREZvI. ANY EXPONENT. 317 Henc9 in this case S^ is the coefficient of x' in the expansion of (1 -x)~" ; that is, _n(?i + l) (n+r- 1) r Tliis is therefore the number of homogeneous products of r dimen- sions that can be formed out of a, h, c, and their j^owers. 524. To jind the ninnher of terms in the expansion of any multinomial, the exponent heing a positive integer. The number of terms in the expansion of (^a^ + a^ + a^ + ... + a^)" is the same as the number of homogeneous products of n dimen- sions that can be formed out of a^, a^, a^^ a^, and their powers. Hence, by the preceding Article, it is r(r + l)(r+2) {-r + n-l) ■ \n • 525. Tlie Binomial Theorem may be applied to extract the roots of numbers approximately. Let X be a number whoso rC-^ root is required, and sujDpose ^ = a" + 6 ; then N''^={ar + hf = a (l -h -^y = a(l + x)\ where x — — . If now aj be a small fraction, the terms in the a" 1 expansion of (1 -f- xY diminish rapidly, and we may obtain an ap- proximate value of (1 -f- a-)", and therefore of iV", by retaining only a few of these terms. It will therefore be convenient to take a so that a" may differ as little as possible from iV, and thus h may be as small as possible. Sometimes it will be better to suppose JSf = a""—!}. 526. We will close this Chapter with six examples which will illustrate the use of the Biiiomial Theorem. (1) The ratio {a + a;)" : a" is nearly equal to the ratio a + nx \ a when nx is small compared with a. This holds whether x be positive or negative, and for values of n integi*al or fractional, positive or negative. See Art. 383. 31 S BINOMIAL THEOREM. ANY EXPONENT. ^2) Expand in a series of ascending powers of x. a -\-hx a+hx lia.,x)(l.fy, expand ( 1 + — ) by the Binomial Theorem ; thus we have a + bx 1, 7 V /-, qx ^-V q^x^ p + qx p^ ^ \ p p' p a X f , ciq\ q^ + -6-^-^(6--^) + P P\ pj p)' ^(^-f) Or Ave may proceed thus, a ■\-hx p \ p J a X fj aq\ ( ^ _ qx" I) A- qx p^qx p + qx p p\ 1^ J \ P J v2^2 ^3 3 = %^(5- \ pj\ p P' f J' V V and thus we obtain the same result as before. Tliis example frequently occurs in mathematics, especially ui cases where x is so small that its square and higher powers may be neglected; Ave have then approximately a + hx a X , ^ =- +- U p + qx p p)\ p }' (3) Required approximate values of the roots of the quad- ratic equation aQ<^ + 5x + c = 0, when ac is very small compared with h\ The roots are ^-^ , 2a And by the Binomial Theorem, JiTy^ - iac) = h\\ — j^\ _ , I 1 iac 1 /4ac\' 1 /4:ac\' | " { 2T'"'8\V)'~IQ\V)~ j* BINOMIAL THEOREM. ANY EXPONENT. 319 Thus for the root with tlie upper sign we get G ac^ la^c^ b b' b' and for the root with the lower sign we get b c ac^ 2a^c^ a b b^ b^ If a be very small, while b and c are not small, the former root c does not differ much from — , and the latter root is numerically very large. See Art. 342. It is deserving of notice that the approximate value of the root in the former case coincides with what we shall obtain in the following way. Write the equation thus, bx + c = — ax'^. For an approximate result neglect the term ax^ as small ; thus c we obtain x = - j ' Then substitute this approximate value of 6 X in the term ax^ ; thus we obtain , ac"* , . G ac^ that IS, X = -Y — r^ . ' b b^ Again, substitute this new approximate value of x in the term ax'f and preserve the terms involving a and a^; thus we obtain bx + G = 77 — b' b' ' ,, , . c ac 2a c thatis. ^ = _______, and so on. (4) To prove that if n be any positive integer the integral part of (2 -^sJ^T is ^^ ^^^^ number. The meaning of this proposition will be easily seen by taking some simple cases; thus 2 + tJ2> lies between 3 and 4 in value, so that the integral part of it is the odd number 3 ; (2 + ^/3)* will be found to lie between 13 and 14 in value, so that the integral p.irt cf it is the odd number 1 3. 320 BINOMIAL THEOREM. ANY EXPONENT. Suppose tlieii / to denote the integral part of (2 + a/B)", and I -Jt F its complete value, so that -F is a proper fraction. "We have by the Binomial Theorem I+F=T + n1-'Z'^+ \ ./^ 2"-'^3^+...... + 3^ (1). I . '^ Now 2 — ^3 is a proper fraction, thei-efore also so is (2 — ^/B)"; denote it by F' ', then F'= 2"-n2"-^3U '' ^'""/^ T-'Z^- +(-l)"3^ (2). Now add (1) and (2); the irrational terms on the right dis- appear, and we have I + F + r='2,{2'' + '^^-^-^T-'^^ 7^(r^-l)(7^-2)(.^-3) ^„_ 1 ) ^ E '" / ■ = an even inte^'er. But F and F' are proper fractions : we miist therefore have F + F' = 1, and / = an odd integer. A similar result holds for (a + Jb)" if a is the integer next greater than Jb, so that a — Jb is a proper fraction. (5) Bequired the sum of the coefficients of the first r +\ terms of the expansion of (1 - a;)~". AYe have /I \-„ 1 n{n+l) 2 n{n-¥\) ...{n + r-\) ^ (\-x) " = 1 + Tia; + \ ^, ^ x + . . . + — ^ — , ' o;*^ + . . . 1.2 [r {X-xy^^l+x + x- + x^+ Therefore (1 - x) ^""^^^ is equal to the product of the two series. Now if we multiply the series together, we see that the coefficient of ic*" in the product is 1 . ., . ^(^^+1) , . n{n + \) (n + r-1) "^''■^ 1.2 ^ ^ \^ ' EXAMPLES, xxxvr. 321 we may naturally assume then that this must be equal to the co- efficient of x"" in the expansion of (1 — a;)"'""^'^; that is, to (yi+1) (n + 2) {n + r) Li ~ ' thus the required summation is effected. (6) The Binomial Theorem may be applied in the manner just shewn to establish numerous algebraical identities; we will give one more example. T . I / . m (m - 1) (m - 2) . . . (m - r + 1) Let 6 Im, r) = — ^ ^— r-^ ^^ ' ; it is required to shew that -^ (n, 0) cj, (n, r)- (n, 1) cj> (n- 1, r - I) + cj, {n, 2) <^ (ri- 2, r- 2) -<^{n, 3)?\ ft y ^ ascending powers of .u Write down the coefficient of x*'' and of a;^'"^\ 39. Shew that the ril^ coefficient in the expansion of (1 - ic)"" is always the double of the (n — 1)'''. 40. Shew that if t^ denote the middle term in the expansion of (1 + a;)% then ^^ + ^^ + ^, + = (1 - 4a;)-^. 41. "Write down the sum of ,11.31.3.5 ... ^^'4-"4T8-'478n2'- ""^'"'J' 43. Find the sum of the squares of the coefficients in the expansion of (1 + a;)", where ?^ is a positive ip.teger. ,, ^^ 1.3.5 (2r-l) , , 43. If ;?_ = — r — - — ^ — - — - , prove that '' z . 4 . b 2r P2.^X+PlP2.+P2P2n-, + + Pn-X Pn^2 + PnPn^^ = k' 44. Shew that if m and ?i are positive integers the coefficient of x"" in the expansion of j- —^ is equal to the coefficient of x" in the expansion of (l-x)""^'' 45. Find the coefficient of a;*" in the expansion of (1 + 2a: + 3a;' + 4a;' + ad inf.)\ 21-2 ( 324 ) XXXVII. THE MULTINOMIAL THEOREM. 527. We have in tlie preceding Chapter given some examples of the expansion of a multinomial ; we now proceed to consider this point more fully. We propose to find an expression for the general tenn in the expansion of (a^-\- a^x -{• a^x' + a^x^ + )". The number of terms in the series a^, a^ a^, may be any whatever, and n may be positive or negative, integral or frac- tional. Put 5 J for a^x+ ajx' + a^x^ + , then we have to expand [a +hY J the general term of the expansion is /x being a positive integer. Put h^ for a^ A- a^ •\- , then h^ = {a^x + Jj)'^ ; since /x is a 2^ositive integer the general term of the expansion of {a^x + h^^ may be denoted either by '- {a^xf-^\\ or by J^ {a.xybj^-^ ; we will adopt the latter form as more convenient for our purpose. Combining this with the former result, we see that the general term of the proposed expansion may be written \q \i^-q Again, put h^ for a^x^ + ajjc* + , then h/-'^ = {a^x^ + b^~% and the general term of the expansion of this will be \r \fji, — q-r ^2/3 Hence the general tenn of the proposed expansion may be written \q\r\ix-q — r ° \)/\2/3 fHE MULTINOMIAL THEOREM. 325 Proceeding in tliis way we shall obtain for the requii-ed general term n(n — l)(n—2) .(n—a + l) where q + r +s + t + 1 2 3 4 If we suppose n - /x—p, we may write the general term in the form [£[r[s[_^ 12 3 4 where p-rq + r + s + t+ =n. ThvLB the expansion of the proposed multinomial consists of a series of terms of which that just given may be taken as the general type. It should be observed that q, r, s, t, v...... are always positive integers, but p is not a positive integer unless n he sl positive integer. When /? is a positive integer, we may, by multiplying both numerator and denominator by '/>, write the factor n(7i-l){n-2) {p + ^) \i\z\i\l in the more symmetrical iform \n 1/^ i£ k [_£ [^ . In the above expression for the general term we may regard the multiplier of a3'+="-^=^+«+-- as the coefficient of the term. Some- times however the word coefficient is applied to the ftictor n(n — l) (p + ^) \ — j — i — r. ; this is usually the meanin«r of the word [^[rl^l^ ' -^ ^ in the cases in which x has been put equal to unity, as in the Examples 25... 3 2 at the end of this Chapter. 32G THE MULTINOMIAL THEOEEJM. 528. Suppose we require the coefficient of an assigned power of X in the expansion of (a^ + a^x + a^x^ + )", for example^ that of a;*". We have then q+2r + 3s + 4:t + p+ q + r + s + t + = w, = n. We must find by trial all the positive integral values of q, r, s, t, which satisfy the first of these equations ; then from the second equation ]) can be found. The required coeffi- cient is then the sum of the corresponding values of the ex- pression n(n-l)(n-2) (p + 1) \l\zil\A When w is a positive integer, then p must be so too, and we may use the more symmetrical form \n \p'.q\^\s\t a^a^a^a^al- V Q 529. For example, find the coefficient of x' in the expansion of {\+2x + 3x^ + ixy. Here q + 2r + 3^ = 7, p-\-q + r + s = ^. Begin with the greatest admissible value of 8 ; this is s = 2, with which we have r = 0,q=\, p=l. Next try s = 1 ; with this we may have r = 2, 5' = 0, p—\; also we may have r = 1, q=2, p = 0. Next try s = ; with this we may have r = 3, ^'=1,^3 = 0. These are all the so- lutions ; they are collected in the annexed table. Also «^ = 1 , ^1 = 2, 0,^ = 3, a^ = 4. Thus the requii'ed coeffi- cient is 1 1 2 2 1 1 2 1 1 1 3 that is. 384 + 432 -h 576 + 216; that is, 1608. THE MULTINOMIAL THEOREM. Again ; find the coefficient of x^ in the expansion of (l + 2a;+3x' + 4a;='+ f. Here q -\- 2r + '^s + = 3, 327 p-\- q + r + s + 1 2* All the solutions are collected in the an- nexed table, and the required coefficient is p q r 8 1 1 1 i 2 3 1 1 i 2 5 3 1 i 2 G)-®(-^) --!!■.»■♦ that is, 3 1 2 - ^ + -^ ; that is, 1. And In this case, since l + 2ic+3^' + 4a;'+ ={\-x)-\ the proposed expression is {(l-a;)""}^, that is, {\-x)~ (1 - x)~^ = \+x + x^ + x^ + ; thus we see that the coefficient of x^ ought to be 1 ; and the student may exercise himself by applying the multinomial theo- rem to find the coefficients of other powers of x : for example, the coefficient of x* will be found to be 530. The form of the coefficient in the Multinomial Theorem in the case in which the ex2:)onent is a positive integer might be obtained in another way. Suppose, for example, that we have to expand (a+/!? + y)'°. When the multiplication is effected every term in the result is a product formed by taking one letter out of each of the 10 trinomial factors. Thus if we require the tenn which involves a"^^y' we must take a out of any two of the 10 trinomial factors, fi out of any three of the remaining 8 trinomial factors, and y out of the remaining 5 trinomial factors. The num- 328 EXAMPLES. XXXVII. ber of ways iii wliicli this can be done is .^ ,^ ,, , by Art. 498 : [2 [3 |5 '' |10 thus the required term is . TT a^^S^y". Hence it follows that if we have to expand (a + /3fl5 + yx^Y^ the tenn which involves a^/3^y* is 110 110 j^ a\PxY {yxy, that is ^ a^^Vo.--. Similarly any other case might be treated. Thus we could give the investigation of the Multinomial Theorem in the following manner : Begin by establishing in the way just exemplified the form of the coefficient in the case in which the exponent is a positive integer. Then su2:)pose we have to find the general term in the expansion of {a^ + a^x + a^ + a^ +...)", where n is not a positive integer. Put h for a^x-V a^ + a^ + . . . ; then we have to expand {a^ + 6)" ; the general term of this expansion is - ^ a, h . and as /x is a positive integer the general term in the expansion of (ttjic + a^x' + a^ + )'^ is I I I I . 1*1 1*2 ^Z ^4 '■'' \i\l\i\1. Hence the required general term is \q\r\s\t_ ° 12 3 4 EXAMPLES OF THE MULTINOMIAL THEOREM. Find the coefficients of the specified powers of a; in the expan- sions in the following 24 Examples : 1. a;* in (1 + x-^x^Y. 2. x' in (1 - a; + x^. EXAMPLES. XXXVII. 329 3. a;« in (1 -2x+ 3x' -ix')*. 4. a;** in (1 + x + x^ + x^ + x* + x^f. 5. x' in {2-3x- ix')'. 6. a' in (1 - aj + 2a;^'^ 7. x^ in (2-5a;-7a;y. 8. a« in (l-2a;'+4a;V- 9. X* in (l4-a; + a;')-^ 10. x' in (l + 2aj-£cV^. 11. ..i.(i_J,J)-= 12. x' in (l + 2x-4a;'-2a;^)-^. 13. a;' in (1 - 2a; + a;*)*. 13 8 7 14. X* in (1 + x- + a;^ + a; — a;^)*. 15. X* in (1 +x + xy. 16. x* in (l + 3a; + 5aj' + 7a;' + 9x"+ y. 17. aj'-in (l + a; + a;'+ )'. 18. a;« in (1 + 2a; + 3:c')". 19. x' in (l + 2a; + 3a;' + 4a;'+ yK 20. x^' in (1 + a^x + a^x^ + a^x^f. 21. a^ in {a^ + a^x-^ a^x^y. 22. a;« in (1 - x' + a;" - x')\ 23. a;^ in (1 + ^i.+i)- 35. Expand (1 — 26a; + ic^)"^ as far as a;*. 36. Expand {a + hx + cx^)~^ as far as ct*. 37. Expand {\—x — x^ — x^f as far as x^, 38. In the expansion of (1 + a; + x^ + + a*")", where oi is a positive integer, shew that (1) the coefficients of the terms equidistant from the beginning and the end are equal ; (2) the coefficient of the middle term, or of the two middle terms, according as nr is even or odd, is greater than any other coefficient ; (3) the coefficients continually increase from the first up to the greatest. 39. If a^, «j, a^, a.^, be the coefficients in order of the expansion oi {\-\-x + x' + + x^'Y, prove that (1) a^+a^ + a,+ + «„,= {r + 1)" ; (2) «j + 2^3 + 3^3 + + nra,,^ = \nr (r + 1)". 40. If a^, a^, ^2' *^'3' ^® ^^® coefficients in order of the expansion of (1 + a; + o;^)", prove that »;-<+»;-»/+ +(-ir'<-,+i(-i)x'=i».- LOGARITHMS. 831 XXXYIII. LOGARITHMS. 531. Suppose a' =w, then x 13 called the logarithm ofnto the base a ; thus the logarithm of a number to a given base is the index of the jDower to which the base must be raised to be equal to the number. The logarithm of n to the base a is written log„ n ; thus log^ n = X expresses the same relation as a' = n. 532. For example, 3* = 81; thus 4 is the logarithm of 81 to the base 3. If we wish to find the logarithms of the numbers 1, 2, 3, to a given base 10, for example, we have to solve a series of equa- tions 10'= 1, 10' = 2, 10' = 3, We shall see in the next Chapter that this can be done approximately, that is, for example, although we cannot find such a value of x as will make 10'= 2 exactly, yet we can find such a value of x as will make 10' difier from 2 by as small a quantity as we j^lease. "VYe shall now prove some of the properties of logarithms. 533. The logarithm of I is whatever the base may be. For a"" = 1 when a: = 0. 53-t. The logarithm of the base itself is unity. For a' = a when x=l. 535. The logarithrji of a product is equal to the siim of the logarithms of its factors. For let X- log^^m, y — log^n ; therefore m — a', n = a'^] therefore nin = a'^a^ = a'''^^ ; therefore log„ mn = x + y — log„ 7n + log„?i . 536. The logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. For let X = log„7n, y = log^n ; 332 LOGARITHMS. therefore m = cf, n = a^ ; therefore — - -r, = ^*~^ j n a" therefore log^ - = x—y = log^m — log^n. 71/ 537. :77i€ logarithm of any power, integral or fractional, of a number is equal to the product of tJie logarithm of the number and the index of the power. For let m = a*; therefore m*" = (a*)'' = a**, therefore loga(m'") = xr = r log^m. 538. To find the relation between the logarithms of the same number to different bases. Let rc = = log<,w, 2/ = log6m; therefore m^a" and =W ) therefore a'=b''; therefore a y a^=b, and 6*= a; therefore y~ = log<,6, and ^ = log, a. S7 "FTennA ♦/ = 1 J ^ 'v? Inof. n. n-nri — Hence the logarithm of a number to the base b may be found by multiplying the logarithm of the number to the base a by log.a, or by j-i^. We may notice that log^a x log^S = 1. 539. In practical calculations the only base that is used is 10; logarithms to the base 10 are called common logarithms. We will point out in the next two Articles some peculiarities which constitute the advantage of the base 10. We shall require the fol- lowing definition : the integral part of any logarithm is called the characteristic, and the decimal part the mantissa. EXAMPLES. XXXVIII. 383 040. In the common system of logarithms, if the logarithm vf any number he known we can immediately determine the loga- rithm of tJie product or quotient of that member by any power of 10. For log.„(iVx 10")=.log,„iV^4-log,„10" = log,,iV^+7^, That is, if we know the logarithm of any number we can determine the logarithm of any number wliich has the same figures, but differs merely by the position of the decimal point. 541. In the common system of logarithms the characteristic of the logarithm of any number can be determined by itispection. For suppose the number to be greater than unity and to lie between 10" and lO""^^; then its logarithm must be greater than n and less than n + 1: hence the characteristic of the logarithm is n. Next suppose the number to be less than unity, and to lie 11. between yf^ ^^^ TTv^i' ^^^* ^^' between 10 " and 10 ^""^^^j then its logarithm will be some negative quantity between — n and — {n+\): hence if we agree that the mantissa shall always be positive, the characteristic will be —{n+V). Finisher information on the practical use of logarithms will be found in works on Trigonometry and in the introductions to Tables of Logarithms. EXAMPLES OF LOGARITHMS. . 1. Find the logarithm of 144 to the base 2^3. :?■ ^ 2. Find the characteristic of the logarithm of 7 to the base 2. 3. Find the characteristic of loga 5. ^ 4. Find log^ 3125. 5. Give the characteristic ol logio 1230, and of log,. -0123. 334) EXAMPLES. XXXVIII. 6. Given log 2 = -301030 and log 3 =-477121, find the loga- rithms of -05 and of 5-4:. 7. Given log 2 and log 3 (see Example 6), find the logarithm of -006. 8. Given log 2 and log 3, find the logarithms of 36, 27, and 16. 0. Given log 648 = 2-81157501, log864 = 2-93651374, find log 3 and log 5. 10. Given log 2, find log ^(1-25). 11. Given log 2, find log -0025. 12. Given log 2, find log ^(-0125). 13. Given log 2 and log 3, find log 1080 and log (-0045)^. 14. Given log,„ 2 = -301030 and log,o7 = -845098, find the / 4 \4 logarithm of I kjo ) **^ *1^® ^^^® 1000. 15. Find the number of digits in 2^*, having given log 2. 16. Given log 2, and log 5-743491^-7591760, find the fifth root of -0625. 17. If P be the number of the integers v^hose logarithms have the characteristic p, and Q the number of the integers the logarithms of whose reciprocals have the characteristic — q, she^y that log P - log Q = 2) - q + 1. 1 1 1 18. If 2/ = 10^-1°°^ and z = lO'-'^^^, prove that a;= lO^-^''^*. 19. If a, h, che in g. p., then log^n, log^n, log^oi are in h. p. 20. If the number of persons born in any year be -p^ th of the Avhole population at the commencement of the year, and the number of those who die —j: th of it, find in how many years the population will be doubled; having given log 2 = -301030, log 180 = 2-255272, log 181 = 2-257679. ( 335 ) XXXIX. EXPONENTIAL AND LOGARITHMIC SERIES. 542. To expand a* in a series of ascending powers of is.; that {.^, to expand a number in a series of ascending powers of its logaHthm to a given base. a' = {\ ■{■ {a - \)Y ] and expanding by the Binomial Theorem we have {\ + {a-l)Y=l+x{a-\)^ '^^'^~^\ a-lY x(x-V)(x-'2) 1^3 , x(x-\){x-2){x-^) ,^ , ^ 1727^ ^''~^^ "" 1.2.3.4 ^"""^^ ^ '" = l+x{a-l-i(a-iy+^{a~lf-i(a-lY+ } + terms involving x^, x^, &,c. This shews that a" can be expanded in a series beginning with 1 and proceeding in ascending powers of x ; we may there- fore suppose that ^^ a* = 1 +c^x-\- c.^x" + c^x^ + c^x* + where c^, Cj, Cg, are quantities which do not depend on x, and which therefore remain unchanged however x may be changed ; also c, = a-l-i{a-lY+l(a-iy-i{a-iy+ while (1). ° n {m + n 3 \m + nj 5 \m + n/ ) T. A. 338 EXPONENTIAL AND LOGARITHMIC SERIES. Put n = l, then *=" (7/1+1 3\m + l/ 5\m+lJ ) ^' Again, in (1) put ??i = 7i + l, thus we obtain the value of ?i + 1 log, ; therefore log, (71 + 1) — log, n "^ [2^^^ "^ 3 (271 + 1)' '^5(2?i + l)^'*" I ^^^* 546. The series (2) of the preceding Ai-ticle will enable us to find log, 2 ; put m = 2, then by calculation we shall find log, 2 = -69314718 From the series (3) we can calculate the logarithm, of either of two consecutive numbers when we know that of the other. Put n = 2, and by making use of the known value of log, 2, we shall obtain log, 3 = 1-09861229 Put n=9 in (3) ; then log, n = log, 9 = log, 3^ = 2 log, 3 and is therefore known ; hence we shall find log, 10 = 2-30258509 547. Logarithms to the base e are called Napierian loga- rithms, from Napier the inventor of logarithms ; they are also called natural logarithms, being those which occur first in our investigation of a method of calculating logarithms. We have said that the base 10 is the only base used in the practical appli- cation of logarithms, but logarithms to the base e occur frequently in theoretical investigations. 548. From Art. 538 we see that the logarithm of a number to the base 10 can be found by multiplying the Napierian loga- rithm by j^^, that is, by 2:30^509' "' ^^ -^3429448; this multiplier is called the modulus of the common system. The base e, the modulus of the common system, and the loga- rithms to the base e of 2, 3, and 5 have all been calculated to upwards of 260 places of decimals. See the Proceedings of the Royal Society of London, Vol. xxvii. page 88. EXPONENTIAL AND LOGARITHJVflC SERIES. 339 The series in Art. 545 may be so adjusted as to give common logarithms ; for example, take the series (3), Multiply throughout by the modulus wliich we shall denote by /x ; thus that is, log,. (« + 1) - iog.,« = 2^y-j + gp^iy, -^jj^hrrr^-} ■ 549. By Art. 542 we have (a--l)" = (. + g-.|%|V )" = a;" + terms containing higher powers of a; (1 ). Again, by the Binomial Theorem, (e-'- 1)" ^ e"" - ne<"-'^" + '' ^'' ~ ^ ^ e<-»)- - (2). Expand each of the tenns e"", 6^""'^'', ; thus the coefficient of ic'" in (2) will be ^_ (^-ly n(n-l) (n-2y n{n-l){n-2) {n- Sy Hence from (1), by the same principle as in Art. 542, we see that n(n-l) n(n-l)(n-2) rf -n(n-iy+ \ — -{n- 2y ^ ^ (n-3/ + h =\n \£r = n, and is = if r be less than n. It is easy to see that the tenn on the right-hand side of (1) which involves x"'^^ is k^"^^- Thus we get, by the same principle as before, n"-^^-7i(n-ir'+''^''~^\n-2r'- = in \n+l . 550. We will give another method of arriving at the expo- nential theorem. By the Binomial Theorem 90 o 340 EXPONENTIAL AND LOGARITHMIC SERIES. H ^"^ 1 'nx{nx-l) 1 nx{nx — l){nx- 2) 1^ "''n'^ [2 n' "^ [3 n» nj; (nx - 1 ) (woj - 2) (nx - 3) 1 J ^ —^ -— —J + . Li ^ that is, X \ X — (^-n) K^-»)(''"3 ^+3'=^^*^-Tr^^ — [3 £C I £C ) Ix ) (x nj \ nj \ n, + — ^ + .. Put x = \, then (l + - j n \ n) \ oi) \ n) \ nj \ n, = i + i + _+ + ^ _ + X (x I X I X — \ \ ^--] 1 1 V 01 J \ nJ \ nl ^ hence 1 + aj + , -^ + ,0 + Lf \1 f , , n \ nJ \ 01 =-l 1 + 1+—--. + + Now this being true however large n may be, will be truo when n is made infinite : then - vanishes and we obtain .3 „4 . Ill l+a; + — +,-;+, ^+ = {1 + 1 + ;-?;+, -s+,-T + ^ [3 11 I \1 \1 \i that is, = e'. We have thus obtained the expansion of e' in powers of x ; to find the expansion of a" suppose a = e" so that c = log^ a, thus [2 L? K EXPONENTIAL AND LOGARITHMIC SERIES. 341 551. The student will notice that in the preceding Article wo have used the Binomial Theorem to expand a power of 1 + - , and if - is less than unity, we are certain that the expansion gives an arithmetically true result (Art. 519). In the proof given of the exponential theorem in the first Article of this Chapter, if a — 1 is greater than unity, the expansion by the Binomial Theorem with which the proof commences will not be arithmetically intelligible ; and consequently the proof can only be considered sound pro- vided a is less than 2. With this restriction the proof is sound, and X may have any value. In order to complete that proof we have to shew that the theorem is true for any value of a ; and as e is greater than 2 we ought not to change a into e until we have removed this restriction as to the value of a. This restriction can be easily removed ; for in the theorem a' = \ + (^og,a)x^- ^ ""'^ + ^ '=' ' 4- put a = A^, and by taking y small enough A may be made as gi'eat as we please, while a is less than 2. Then log, a=y log, A ; thus ^^^ = 1 + (log, ^) yx + ^ ^ ' ^ — + --^^-^ — + ; therefore, putting z for yx, ^' = 1 + (iog,^)^+ ^ '^^^ + —-3-^ + ; thus the exponential theorem is proved universally. 552. "We have found in Art. 550, that when 71 increases / \y without limit ( 1 + - J ultimately becomes e* ; in the same way we may shew that when n increases without limit ( 1 4- - ) ■ tdtimately becomes c". 342 EXAJVIPLES. XXXIX. EXAMPLES OF LOGARITHMIC SERIES. 1. Prove that log, (oj + 1) = 2 log, a; - log, (x-l) ~ I'Zx'^-i 6\: Given log^^ 3 = -47712 and ^_ ■,^ = -43429, api% the above 2aj^-l 3\2x'-iJ /• l^gelO series to calculate log^^ 11. 2. Shew that log, (x + 2h) = 2 log, {x + h) - log, x h' 1 h* 1 h' ) Jqc + K)'"" 2{x + hy^ 2>{x + hf^ J* 3. li a, hy c be three consecutive nuinbers, Iog^c = 21og,6-log,a ^f 1 1 1 ) "l2ac + l "^ 3(2ac+ 1/ "^5(2^0+ 1)' "^ j ' 4. If X and /a be the roots of ax^ + hx-k-c = 0, shew that log, {a-hx + cx^) = log, a + (X + fjt.)x ^r-^ x' + 5. Log, {l + l+a; + (l+ xY} = 3 log, (l+x)- log, x 1 1 1 1 ) {i + xf'^'2{i + xy'^3{].+xy'^ /• Ax ^x—1 6. Log, (a: + 1) = 2^-^ log, 0.-^^-^^ log, (a; -1) 2 f 1 2 3 ) 2a;+ll2.3.aj''^3.5.a;' ■^4.7.0;^'^ /' 7. Log,{(,..y-?(i_.y-}.j^+3^ 4 + 5r6 + EXAMPLES. XXXIX. 34-3 501 8. Find the Napierian logarithm of t^. To how many decimal places is your result correct ? 9. Assuming the series for log, {I + x) and e', shew that 1 + '') = 71 ■-(-£) nearly when n is large ; and find the next tenn of the series of which the expression on the second side is the commencement. 10. Find the coefficient of cc' in the development of a + bx + cdt? 1 1. Shew that log, 4 = 1 + j-|^ + ^ + ^J_ + 12. Shew that «"*' - « (re - 1)"+' + "^^ ^^ (re - 2)"' - ( n n{n~ 1)\ , XL. CONYERGENCE AND DIYERGENCE OF SERIES. 553. The expression u^-^-ii^ + u.^ + u^ + in which the successive terms are formed by some regular law, and the number of the terms is unlimited, is called an infinite series. 554. An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quan- tity however great n may be. 555. An infinite series is said to be divergent when the sum of the first n terms can be made numerically greater than any finite quantity, by taking n large enough. 344; CONVERGENCE AND DIVERGENCE OF SERIES. 556. Suppose that by adding more and more terms of an infinite scries we continiiallj approximate to a certain result, so that the sum of a sufficiently large number of terms will differ from that result by less than any assigned quantity, then that result is called the suin of the infinite series. For example, consider the infinite series l + x + x^ + , and suppose x a positive quantity. We know that 1 - a-" \ +x + x^ + +x n-1 \ — x Hence if a; be less than 1, however great n may be, the sum of the first n teiTus of the series is less than ^^ : the series is \ — x therefore convergent. And, as by taking n large enough the sum of the first n terms can be made to dififer from by less than l-x -^ any assigned quantity, ^j is the sum of the infinite series. If ic = 1, the series is divergent ; for the sum of the first n terms is n, and by taking sufiS.cient terms this may be made greater than any finite quantity. If X is greater than 1, the series is divergent; for the sum £C" — 1 of the first n terms is r-, which may be made greater than any finite quantity by taking n lai'ge enough. 557. An infinite series in which all the terms are of the satne sign is divergent if each term is greater than some assigned finite quantity, however small. For if each term is greater than the quantity c, the sum of the first n terms is greater than nc, and this can be made greater than any finite quantity by taking n large enough. CONVERGENCE AND DIVERGENCE OF SERIES. 345 558. An infinite series of terms, the signs of which are alter- nately positive and negative, is convergent if each term is numeri- cally less than the preceding term. Let the series be u^ — u^ + «*3 - w^ + .... ; this may be wiitten (^1 - w,) + {u^ - n^ + {u^-u^A- , and also thus, ^i - K - ^'3) - (^'4 - -^5) - (wg - u:) - From the first mode of writing the series we see that the sum of any number of terms is a positive quantity, and from the second mode of writing tlie series we see that the sum of any number of terms is less than u^ ; hence the series is convergent. It is necessary to shew in this case that the sum of any number of terms is positive ; because if we only know that the sum is less than u , we are not certain that it is not a nesrative quantity of unlimited magnitude. An important distinction should be noticed with respect to the series here considered. If the terms u^, w,, u^, ... diminish without limit the sum of n terms and the sum of ?i + 1 terms will differ by an indefinitely small quantity when n is taken large enough. But if the terms u^, u^, u^, ... do not diminish without limit the sum of ?i terms and the sum of n + 1 terms will always differ by a finite quantity. The series continued to an infinite number of terms will have a sum, according to the definition of Art. 556, in the former case, but not in the latter case. In both cases the series is converojent accordinsc to our definition. But some writers prefer another definition of convergence ; namely, they consider a series convergent only when the sum of an in- definitely large number of terms can be made to differ from one fixed value by less than any assigned quantity : and according to this definition the series is convergent in the first case, but not in the second. 346 CONVERGENCE AND DIVERGENCE OF SERIES. 559. An infinite series is convergent if from and after any fixed term tJie ratio of each term to the preceding term is numerically less than some quatUity which is itself numerically less than unity. Let the series beginning at the fixed term be u^ + u^ + W3 + and let S denote the siiin of the first n of these terms. Then ^ = w, + 2^ + w, + + w 13 3 n f- U^ W, U^ It 71. U^ ) i u^ u^u^ u^u^u^ ) Now first let all the terms be positive, and suppose -^ less than k, — less than k, — less than ^, u. Then S is less than u^{l+k + k'+ +F '} ; that is, less 1 — /fc" ,n u r • Hence if k be less than unity, S is less than ^ I — k — '— : thus the sum of as many teiTQS as we please beginning 1 — ^ with u is less than a certain finite quantity, and therefore the series beginning with u^ is convergent. Secondly, suppose the terms not all positive ; then if they are all negative, the numerical value of the sum of any number of them is the same as if they were all positive ; if some terms are positive and some negative, the sum is numerically less than if the terms were all positive. Hence the infinite series is still con- vergent. Since the infinite series beginning with u^ is convergent, the infinite series which begins with any fixed term before u^ will be also convergent ; for we shall thus only have to add a finite number of finite terms to the series beginning with w, . 560. An infinite series is divergent if from and after any fixed term the ratio of each term to the preceding term is greater than unity J or equal to unity, and the terms are all oftJie same sign. CONVERGENCE AND DIVERGENCE OF SERIES. 347 Let the series beginning at the fixed term be u^+u^ + u^ + , and let S denote the sum of the first n of these terms. Then jS = u+ u,+ u+ + u = ?^ - 1 + ^ + -^ -^ + — ' -? -= + [. 'I u^ u^ u^ u^ u^ u^ ) Now, first suppose nt 7/ 7/ -* orreater than 1, — ^ cjreater than 1, -"* m-eater than 1, u, u^ w, ' 1 1 3 Tlien S is numerically greater than u^ {1 + 1 + + 1 }, that is, numerically greater than nu^. Hence S may be made numerically gi^eater than any finite quantity by taking n large enough, and therefore the series beginning with u^ is divergent. Next, suppose the ratio of each term to the preceding to be unity; then S^oiu^, and this may be made greater than any finite quantity by taking n large enough. And if we begin with any fixed term before u^ the series will obviously still be divergent. 561. The rules in the preceding Articles will determine in ii\any cases whether an infinite series is convergent or divergent. There is one case in which they do Twt apply which it is desirable to notice, namely, when the ratio of each term to the preceding is less than unity, but continually approaching unity, so that we cannot name any finite quantity h which is less than unity, and yet always greater than this ratio. In such a case, as will appear from the example in the following Article, the series may be con- vergent or divergent. 562. Consider the infinite scries 1 1 1 1 + + + + p 'A^ '6' 4'' Here the ratio of the n^^ term to the (n — \y^ term is ( J ; if /? be positive, this is less than unity, but continually 348 CONVERGENCE AND DIVERGENCE OF SERIES. approaclies to unity as n increases. This case tlicn cannot be tested by any of the rules already given ; we shall however prove that the series is convergent if p be greater than unity, and divergent if 7; be unity, or less than unity. I. Suppose J) greater than unity. The first term of the series is 1, the next two terms are toge- 2 ther less than — , the follo^ving four terms are together less A 8 than — • , the followin2: eis^ht terms are tosrether less than 7,,. , and so on. Hence the whole series is less than , 2 4 8 l + rP+4-p-8"^+ that is, less than \+ X +X' + x^ + 2 where x — — . Since p is greater than unity, x is less than unity : hence the series is convergent. II. Suppose^:) equal to unity. ^, . . _ 1 1 1 1 Lhe series is now l+-:r-l ' 1 \- 2 3 4: 5 The first term is 1, the second term is - , the next two tenns 2 1 are together greater than -- oy -, the following four terms are 4 1 together greater than ^ or - , and so on. Hence by taking a sufficient number of terms we can obtain a sum greater than any finite multiple of ^ ; the series is therefore divergent. III. Suppose p less than unity or negative. Each terra is now greater than the corresponding term in II. ; the series is therefore a fortiori divergent. CONVERGENCE AND DIVERGENCE OF SERIES. 349 563. We will now give a general theorem wliich can he proved in the manner exemplified in the preceding Article. If (f> (x) be positive for all positive integral values of x, and con- tinually diminish as X increases, and m he any positive integer, then the two infinite series (^0 + ^^^ {m^) and (wi^"^^), including the last and excluding the first, k being any positive integer ; the number of these terms is m^'^^ — ??i*, and their sum is therefore greater than ni^i^ni— 1) ^(Ayt^"*"^). Thus all the first series beginning with the term <^{iu^+\) will be 11X — 1 greater than — — - times the second series besfinnin^x with the ^ 111 o o term 7?^''"+^ ^ {m^'^^) . Thus if the second series be divergent, so also is the first. Again, the tenns selected from the first series are less than m^{in-\) (^{in^). Thus all the first series beginning with the term <^ (??**' +1) will be less than in - 1 times the second series beginning with m*"^ (in^). Thus if the second series be convergent, so also is the first. As an example of the use of this theorem we may take the following : the series of which the general term is — -, is con- ^ -^ ^ n(logn)P vergent if ''^ he greater than unity, and divergent if ^ he equal to unity or less than unity. By the theorem the proposed series is convergent or divergent according as the series of which tlie general term is ;:; is convergent or divergent ; the latter m" (log m"Y treneral term is -; , so that it beai-s a constant ratio to the *=* (log my n^ ' general term — for all values of n. ITence tlie required result follows by Ai-t. 502. 350 CONVERGENCE AND DIVERGENCE OF SERIES. 5G4. The series obtained by expanding {\+xY by the Bino- mial Theorem is convergent if x is numerically less tJian unity. For the ratio of the (r+1)^ term to the r*^ is — — x. Ifn • Ti 1 • 1 /. n-r + 1 is negative and numerically greater than unity the factor is numerically greater than unity ; but it continually approaches unity, and can be made to diifer from unity by less than any assigned quantity by taking r large enough. Hence if a; is nume- , - , 7Z, — r + 1 , . , ncally less than unity the product ^ cc, when r is large enough, will be numerically less than a quantity which is itself numerically less than unity. Hence the series is convergent. (Art. 559.) If n is positive the factor is numerically less than unity when r is greater than n ; if n is negative and numerically less than unity this factor is always numerically less than unity ; if n = — \ this factor is numerically equal to unity : thus in the first case when r is greater than n, and in the other two cases • • -111 • ^ n — r+\ always, if x is numerically less than unity the product — — ; x is numerically less than a quantity which is itself numerically less than unity. Hence the series is convergent. (Art. 559.) 5^5. The series obtained by expa^iding log (1 + x) in powers of n is convergent if :s. is numerically less than unity. rx For the ratio of the (r + 1)^^ term to the r'^^ is r- . If then V / r4- 1 x be less than unity, this ratio is always numerically less than a quantity which is itself numerically less than unity. Hence the series is convergent. (Axt. 559.) 566. The series obtained by expanding a"^ in powers of x is always convergent. EXAJMPLES. XL. 351 sc locr CL For the ratio of the (r + lY*" terai to the r'*" is 2!_ . What- ever be the value of x, we can take r so large that this ratio shall be less than unity, and the ratio will diminLsh as r increases. Hence the series is always convergent. (Art. 559.) EXAMPLES OF CONVERGENCE AND DIVERGENCE OF SERIES. Examine whether the following ten series are convergent or divergent : I . — — 1 4- \. x(x + a) [x + 2a) [x + 3a) (x + ia) (x + 5a) ^ 3 5x' 7x' 9x* 2n+l „ 2 5 lU 17 n +1 m +2^ m + 2]) m+ 3p O. 4- 3 "r 5 "}" a a a 4. {a + iy + {a+ 2)' X + {a + 3y x' + 5. V + 2'x + 3'x'+ - ^ 1 1 1 2 ^ M./ tAy %Aj 1 1 + • l + x^ 1+a;^ 1+a;^ o 1 1 1 1 8. 1 1 1 h V 3^ 5^ V 9. r + 2"£c+3V+ a; a;' a;' {a + hy "^ (a + 26)" "^ (a + 36)" "^ I I . Suppose that in the series u^-¥u^^u^^u^-¥ each temi is less than the preceding ; then shew that this series and the series u^ + 2u^ + 2Si.^ + 2^u^ + 2\i^^ + are both convergent or both divergent. 12 3 12. Shew that the series 1 + ^ + — + - - + is convergent Is O 4: if n be greater than 2, and divergent if n be less than 2 or equal to 2. 8-)2 l^'TEBEST. XLI. INTEREST. 567. Interest is money paid for the use of money. Tlie sum lent is called the Principal. The Amount is the sum of the Principal and Interest at the end of any time. 568. Interest is of two kinds, simple and cojnpomid. "When interest of the Principal alone is taken it is called simple interest ; but if the interest as soon as it becomes due is added to the })rincipal and interest charged upon the whole, it is called com- pound interest. » 569. The rate of interest is the money paid for the use of a certain sum for a certain time. In practice the sum is usually £100 and the time one year; and when we say that the rate of interest is £-t. Qs. 8d. per cent., we mean that <£4. Qs. Sd., that is, X4^, is due for the use of ,£100 for one year. In theory it is convenient, as Vv^e shall see, to use a symbol to denote the interest of one pound for one yccir. 570. To find the amount of a given sum in any time at simple interest. Let P be the principal in pounds, n t];e number of years for which interest is taken, r the interest of one pound for one year, M the amount. Since r is the interest of one pound for one year, Pr is the interest of P pounds for one year, and therefore nPr the interest of P pounds for n years ; therefore M^P A-Pnr. From this equation if any three of the foiu' quantities J/, P, Tij r are given, the fourth can be found ; thus p = ^^ - ^^--^ _ ^zl 1 + ar ' '^~ Pr ' "* ~ ~P^~ ' INTEREST. 353 571. To find the amount of a given sum in any time at com- pound interest. Let B denote the amount of one pound in one year, so that it - 1 + r, then FB is the amount of P in one year ; the amount of PJi in one year is FliE or FJi^, which is therefore the amount of F in two years at compound interest. Similarly the amount of FF' in one year is FF^, which is therefore the amount of F in three yeai'S. Proceeding thus we find that the amount of F in n years is FF" j therefore denoting this amount by 3f, M=FR\ M logi¥-logP /J/\i Hence P = ^„ n^ ^^^^ , ^^Kf) ' The interest gained in n years is M-F or F (R" — 1). 572. Next suppose interest is due more frequently than once a year; for example, suppose interest to be due every quarter, and let - be the interest of one pound for one quarter. Then, at 4 compound interest, the amount of F in n years is P f 1 + - 1 ; for the amount is obviously the same as if the number of years r • . were in, and 7 the interest of one pound for one year. Simi- larly, at compound interest, if interest be due q times a year, and r the interest of one pound be - for each interval, the amount of 2' / r\^ F in n years is P ( 1 + - ) . At simple interest the amount will be the same in the cases supposed as if the interest were payable yearly, r being the inter- est of one pound for one year. 573. The formula} of the preceding Articles have been ob- tained on the supposition that n is an integer ; we may therefore ask whether they are true when n is not an integer. Suppose T. A. 2o 354 INTEREST. n = 7)1 + , ^vllere 7n is an integer and - a pix)per fraction. At »imple interest the interest of P for m years is Pmr ; and if the borrower has agreed to pay for an^ fraction of a year the same Ft . fraction of the annual interest, then — is the interest of P for /1\*^ . . Pr . ( - ) of a year ; hence the whole interest is Pmr -\ , that is, Pnr, and the formula for the amount holds when n is not an integer. Next consider the case of compound intei'est ; the amount of P in 7?i years will be Pit"* ; if for the fraction of a year interest is due in the same way as before, the interest of PB!^ for /l\th ^ Pi?'V . / r\ ( - ) of a year is , and the whole amount is PR^ ( 1 + - ) . On this supposition then the formula is not true when n is not an integer. To make the formula true the agreement must bo /IV^ that the amount of one pound at the end of ( - ] of a year \ /l\t^ shall be (1 + 7')"^, and therefore the intei'est for ( - ) of a year (l + ?-)'^-l. This supposition though not made in practice is often made in theory, in order that the formulee may hold uni- versally. Similarly' if interest is payable q times a year the amount of P in n years is P ( 1 + - j , by Art. 572, if n be an integer ; and it is assumed in theory that this result holds if n be not an integer. 574. The amount of P in n years when the interest is paid q times a year is P ( 1 + - j , by Art, 572 ; if we suppose q to increase without limit, this becomes Pe" (Art. 552), which will therefore be the amount when the interest is due every moment. 575. The Present value of an amount due at the end of a given time is that sum which with its interest for the given time EXAMPLES. XLI. 355 ■Nvill be equal to the amount. That is, (Art. 567), the Principal is the present value of the amount. 576. Discount is an allowance made for the payment of a siun of money before it is due. From the definition of 2'>'>'^sent value, it follows that a debt due at some future period is equitably discharged by paying the present value at once ; hence the discount will be e<|iial to the amount due diminished by its present value. 577. To find the present value of a sum due at the end of a given time and the discount. Let F be the present value, Jf the amount, D the discount, r the interest of one pound for one year, n the number of years, Ji the amount of one pound in one year. At simple interest : Jf^P{l + nr), (Art. 570); therefore F = -J^ , D = M-F= ^^^"''' 1 + nr^ " \ + nr' At compound interest : M=FR\ (Art. 571); therefore P-^, D =r M - F ^^^^^^^j^ . 578. In practice it is very common to allow the interest of a sum of money paid before it is due, instead of the discount as here ]\[iir defined. Thus at simple interest, instead of -. the payer would be allowed Mnr for immediate payment. EXAMPLES OF INTEREST. 1. Shew that at simple interest the discount is half the har- monic mean between the sum due and the interest on it. 23—2 350 EXA]^IPLES. XLI. 2. At simple interest the interest on a certain sum of money is £180, and the discount on the same sum for the same time and at the same rate is £150 : find the sum. 3. If the interest on £A for a year be equal to the discount on £B for the same time, find the rate of interest. 4. If a sum of money doubles itself in 40 years at simple interest, find the rate of interest. 5. A tradesman marks his goods with two prices, one for ready money, and the other for a credit of 6 months : find what ratio the two piices ought to bear to each other, allowing 5 per cent, simple interest. 6. Find in how many years £100 will become £1050 at 5 per cent, compound interest ; having given log 14 = 1-14613, log 15 = M7609, log 16 = 1-20412. 7. Find how many years will elapse before a sum of money trebles itself at 3 J per cent, compound interest ; having given log 10350 = 4-01494, log 3 = -47712. 8. If a sum of money at a given rate of compound interest accumulate to p times its original value in m years, and to q times its original value in n years, prove that n = m log^ q. XLII. EQUATION OF PAYMENTS. 579. When different sums of money are due from one person to another at different times, we may be required to find the time at which they may all be paid together, so that neither lender nor borrower may lose. The time so found is called the equated time. 580. To find the equated time of payment of two sums due at different times supposing simple interest. Let P|, P^ be the two sums due at the end of t^, t,^ years EQUATION OF PAYMENTS. 357 respectively ; suppose t^ greater than t^ ; let r be the interest of one jx>und for one year, x the number of years in the equated time. The condition of fairness to both parties may be secured by supposing that the discount allowed for the sum paid before it is due is equal to the interest charged on the sum not paid until after it is due. The discount on P„ for t„ — x years is ~-~ , : the interest on P^ for x— f^ years is F^ (x — tj7'; therefore -P.fe-^;) ^ ^ , _ ^ . l + {t^-x)r ' ^ " This will give a quadratic equation in x, namely, P,rx' - {P,r {t, + f,) + P^ + PJ cc + P^rt^t^^ f P/, + Pj^ = ; that root must be taken which lies between t^ and t^. 581. Another method of solving the question of the preced- ing Article is as follows : P The present value of P^ due at the end of t^ years is ^j — ' — ; P the present value of P^ due at the end of t^ years is .j — ^— ; . P +P the present value of P^ + P^ due at the end of x years is y f . Hence we may propose to find the equated time of payment, X, from the equation P P P +P I + tiV I + t^r I +xr ' 582. If such a question did occur in practice however the method would probably be to proceed as in the first solution, witli this exception, that the lender would allow interest instead of dis- ,So8 EQUATION OF PAYMENTS. coind on the sum paid before it was duo ; thus we should find x from therefore {I\ + P,) x = F,t^ + l\t,. In this case the interest on Pj + P^ for x years is equal to the sum of the interests of P^ and P^ for the times t^ and t„ respect- ively ; this follows if we multiply both sides of the last equation by r. This rule is more advantageous to the borrower than that in Art. 580, for the inlerest on a given amount is greater than the discount. See Art. 577. 583. Suppose there are several sums P^, P^, P^, due at the end of t^, t^, t.^, years respectively, and the equated time of payment is required. The first method of solution (Art. 580) becomes very compli- cated in this case, and we shall therefore omit it. The second method (Art. 581) gives for determining x, the number of years in the equated time, P, P, P, P^p+p+ 1 + ^^r 1 + t^r 1 + t^r '" \ + xr P P P P Denote the sum -= — ' \- - — \- - — 1- by '% 1 + tr 1 -f tr 1 + t..r •" I + Ir' \ 2 o and the sum P^ + P^ + P^+ by SP j then we may write the above result thus, \1 + trj 1 + xr The third method (Art. 582) gives x(p, + p,+P3 + ) = i'A+^/. + ^A+ ; which may be written x'^P = ^Pt. 584. Equation of payments is a subject of no practical im- portance, and seems retained in books chiefly on account of the apparent paradox of different methods occurring which may EQUATION OF PAYMENTS. 359 appear equally fair, but which lead to different results. "VVe refer the student for more information on the question to the article Discount in the English Cyclopcedia, We may observe, however, that the difficulty, if such it be, arises from the fact that simple interest is almost a fiction ; the moment any sum of money is due, it matters not whether it is called principal or interest, it is of equal value to the owner ; and thus if the interest on bor- rowed money is retained by the borrower, it ought in justice to the lender, to be united to the principal, and charged with in- terest afterwards. 585. If compound interest be allowed, the solutions in Ai-ts. 580 and 581 will give the same result. For the solution according to Art. 580 will be as follows : the discount on P^ for t^-x years is P^(l— yr^. ) , the interest on P^ for x-t^ years is P^ [E^h — V) ; therefore P,(l- -^) = P, {R^U _ 1 ). From this equation x must be found ; by transposition we shall see that this is the same equation as would be obtained by the method of Art. 581 ; for we obtain P P A. P ^ ?- + P Rxr-i ■ , ^ ^, + ^, P. P. therefore -^W ^ W^ "^ ' R^'^ ^ which shews that x is such that the present value of P + P, due at the end of x years is equal to the sum of the present values of P^ and P^ due at the end of t^ and ^^ years respectively. 586. If there are different sums Pj, P^, P^, due at the end of {m - ly 2) cji (n - m + 1, 3) + 10. With the same notation shew that a-(a + P)(n, 1) + (a + 2^) {n, n) = 0. 11. If s be the sum of n terms of a geometrical progression whose first term is a and common ratio 1 + rz;, where x is very small, shew that n = -\l — - — ' i approximately. 12. If a quantity change continuously in value from a to 6 in a given time t^, the increase at any instant bearing a constant ratio to its value at that instant, shew that its value at any time t will be a (^Y\ (Art. 574.) 302 ANNUITIES. XLIII. ANNUITIES. 588. To find the amount of an annuity left unijaid for any Clumber of years, allowing simple interest upon each sum from the li)7ie it becomes due. Let A be the annuity, n the number of years, r the interest of one pound for one year, M the amount. At the end of the first year A becomes due, and at the end of the second year the interest of the first annuity is rA ; at the end of this year the principal becomes 2A, therefore the interest due at the end of the third year is 2rA ; in the same way the interest due at the end of the fourth year is 3rA j and so on ; hence the whole interest is rA + 2rA + SrA + + (7i — l)rA; that is, flj /^o 2 ) rA — ^^ — ^r-^ , by Art. 450 ; and the sum of the annuities is 7iA : ir- 4 n(n — l) . therefore M = nA H rA, 'A 589. To find the present value of an annuity, to continue for a certain number of years, allowing simple interest. Let P denote the present value ; then P with its interest for n years should be equal to the amount of the annuity in the same time ; that is, _, „ . n{n-\) . P + Pnr ^nA+ -^—~ — - rA ; ,, » „ nA + hn {71 - 1) rA therefore P = v^ • 1 + 7ir 590. Another method has been proposed for solving the question in the preceding Article. A The present value of A due at the end of 1 year is , (Art. 577) ; the present value of A due at the end of 2 years is A A - — -r— : the present value of A due at the end of 3 years is -:; — , and so on ; the present value of the annuity for n years should ANNUITIES. 363 be equal to the sum of the present viihies of the different pay- ments : hence f 1 1 1 1 ) [l + r'^ l + '2r'^ l + 3r'^ "^l + nrj* 591. Some writers on Algebra have ado|Dted the solution given in Art. 589, and others that in Art. 590 ; we liave already intimated in a similar case (Art. 584), that the solution of such questions by simjyle interest must be unsatisfactory. The student may consult on this point Wood's Algebra, the Treatise on Arith- metic and Algebra in the Library of Useful Knowledge, p. 102 ; Jones on the Value of Annuities and Reversionary Payments j Vol. I. p. 9 ; and the article Discount in the English Cyclopoedia. 592. The fonnulse in Arts. 589 and 590 make the value of a perpetual annuity infinite. For the value of P in Art. 589 may be written A ->r ^{n-V)rA 1 ' . n '^ when n is infinite the denominator of this expression becomes r, and the numerator becomes infinite ; thus P is infinite. The series given for P in Art. 590 also becomes infinite when n is infinite. This result is another indication that the value of annuities should be estimated in a difierent way. We proceed to the sup- position of compound interest. 593. To find the amount of an annuity left unpaid for any number of years, allowing comipound interest. Let A be the annuity, n the number of years, R the amount of one pound in one year, M the requii-ed amount. At the end of the first year A is due ; at the end of the second year RA is the amount of the first annuity, hence the whole sum due at the end of the second year is RA + A, that is, (7? + l) yl ; similarly at the end of the third year the whole sum due is RiJi-^X) A + A, that is, ifl^ ->r R a-V) A \ and so on ; hence the SG4 ANNUITIES. whole sum due at the end of n years is (jR""' + i?"~' + + 1) ^ ; tlnis Al = -^ — - A, K— 1 594. To find the i^resent value of an annuity, to continue for a certain number of years, allowing compound interest. Let P denote the present value; then the amount of P in n years should be equal to the amount of the annuity in the same time ; that is, therefore P = — - — — A = ^ — A. It — 1 r 595. We may also solve the; question of the preceding Article by supposing P equal to the sum of the present values of the different payments. A The present value of A due at the end of 1 year is — , the present value of A due at the end of 2 years is — ; A the present value of A due at the end of 3 years is ^3 ; and so on ; p_^A A^ A A^ R\ PV A{l-R-") 1 _ 1 ~ ^-1 * R If the present value of an annuity A for any number of years be mA^ the annuity is said to be worth m yeari purchase. 596. To find the present value of a perpetual annuity. Suppose n to be infinite in the formula P = — ^ — z — - , ■thus P=. ^ =J. R-l r therefore ANNUITIES. 365 597. To find the preserd value of an annuity, to commence at the end of]y years, and then to continue q years. The present value of an annuity to commence at the end of p years, and then to continue q years, is found by subtracting the present value of the annuity for p years from the present value of the annuity for p + q years ; thus we obtain A -^-^- - A -^-j- , that iB, ^^ {R-' - if-'-.). If the annuity is to commence at the end ofp years, and then to continue for ever, we must suppose q infinite, and the present AR~P value becomes -^ — =-. This may be obtained directly; for the present value is the sum of the following infinite series, AAA 598. The preceding Article may be applied to calculate the fine which must be paid for the renewal of a lease. Suppose an estate to be worth £A per annum, and that a lease of the estate is granted for p + q years for a certain sum of money paid down ; and suppose that when q years have elapsed, the lessee wishes to obtain a new lease for p + q years ; he must therefore pay a sum equivalent to the value of an annuity of £A to begin at tlie end of p years, and to continue for q years. This sum is called the fine to be paid for renewing q years of the lease. 599. We have hitheiix) in the present Chapter confined ourselves to the case in which the interest and the annuity are due only once a year. We will now give a more general pro- position. To find tlie amount of an annuity left unpaid for n years, at compound interest, supposing interest due q times a year, and the annuity payable m times a year. r /1\^ Let - be the interest of one pound for f - j of a year ; then by Art. 573, the amount of one pound in 8 years is SGG EXAMPLES. XLIII. 1 + - ) wlietlier s he an integer or not : thus the amount of one pound for f — j of a year is f 1 + j ; we shall denote this by p. Let a be the instalment of the annuity that should be paid each time ; then the amount of the annuity at the end of n years is the sum of the following mn terms : „ (mn—l , mH—2 , mn— 3 . i _ i 1 ) a{p + p + p + + p + 1 i , that IS, a —- , that is, a 1 ' 9 (i + '-Y-i ^/ EXAMPLES OF ANNUITIES. In the examples the interest is supposed compound unless otherwise stated. . 1. A person borrows £600. 5^. : find how much he must pay annually that the whole debt may be discharged in 35 years, allowing simple interest at 4 per cent. 2. DeteiTuine what the rate of interest must be in order that the present value of an annuity for a given number of years, at simple interest, may be equal to half the sum of the annuities. 3. A freehold estate of £100 a year is sold for £2500 : find at what rate the interest is calculated. . 4. The reversion, after 2 years, of a freehold worth £168. 2.9. a year is to be sold : find its present value, supposing interest at 2^ per cent. 5. If 20 years' purchase must be paid for an annuity to con- tinue a cei-tain number of years, and 26 years' purchase for an annuity to continue twice as long : find the rate per cent. 6. When 3-1 per cent, is the rate of interest, find what sum must be paid now to receive a freehold estate of £320 a year 1 years hence ; having given log 1 032 = -0136797, log 7-29798 = -8632030. EXAMPLES. XLIII. 8G7 7. Supposing an annuity to continue for ever to be wortli 25 years' purchase, find the annuity to continue for 3 years whicli can be pui'chased for £G'2o. 8. A sum of cfilOOO is lent to be repaid with interest at 4 per cent, by annual instalments, beginning with X40 at the end of the first year, and increasing 30 per cent, each year on the last preceding instalment. Find when the debt will be paid off; having given log 2 = -30103, log 3 = -47712. 9. Find the present value of an annuity which is to com- mence at the end of p years, and to continue for ever, each pay- ment being m times the preceding. What limitation is there as to 711 1 10. Find what sum will amount to ^1 in 20 yeai-s, at 5 per cent., the interest being supposed to be payable eveiy instant. 11. If interest be payable every instant, and the interest for / 1 \'^ one year be { — ) of the principal, find the amount m n yeai-s. 12. A person borrows a sum of money, and pays off at the end of each year as much of the principal as he pays interest for that year : find how much he owes at the end of n years. 13. An estate, the clear annual value of which is £A, is let on a lease of 20 years, renewable every 7 years on payment of a fine : calculate the fine to be paid on rene%\ing, interest being allowed at six per cent. ; having given log 106 = 2-0253059, log 4-688385 = -6710233, log 3-118042 = -4938820. 14. A person with a capital of £a, for which he receives interest at r per cent., spends every year £b, which is more than his original income. Find in how many years he will be ruined. Km. If a =1000, r = 5, 6 = 90, shew that he will be niiiu-d before the end of the 17th year; having given log 2 = -3010300, log 3 = -4771213, log 7 = -8450980. 3()8 CONTINUED FRACTIONS. XLIY. CONTINUED FRACTIONS. h 600. Every expression of the form a ± -j is called c=fc a continued fraction. ^ "^ ^^' We shall confine our attention to continued fractions of the form a + , where a, b, Cy are all positive integers. 6 + — V For the sake of abbreviation the continued fraction is some- 1 1 times written thus : a + -, — , 6 + c + (fee. When the number of the terms a, h, c, infinite, the con- tinued fraction is said to be terminating ; such a continued frac- tion may be reduced to an ordinary fraction by effecting the operations indicated. 601. To convert any given fraction into a continued fraction. m Let — be the given fraction ; divide m by n^ let a be the quotient and » the remainder ; thus — = « + - = « + -. Next di- *^ ' n n n P vide n by p, let h be the quotient and q the remainder ; thus n^'ljt? r 1 z=6 + - =6 + -. bimilarly, -'- = c + -=c+-, and so on. V P P ^ ^ g q r m m \ Ihus —=a + b + c+ &c. If m be less than n, the first quotient a is zero. We see then that to convert a given fraction into a continued fraction, we have to proceed as if we were finding the greatest common measure of the numerator and denominator ; and we CONTINUED FRACTIONS. 869 must therefore at last arrive at a point where the remaiilder is zero and the operation teiTninates : hence every fraction can be converted into a terminating continued fraction. 602. The fractions formed by taking one, two, three, ... of the quotients of the continued fraction a + ; ^ — are called h + C + &LC. converging fractions or convergents. Thus the first convergent is a ; the second convergent is formed from cl + r, it is therefore — 7 — ; the third convergent is formed from a -f ■ r , that is, from c c .,..■, „ ohc + a + c , « + ; =-, it IS therefore — , ^ — : and so on. bc+l bc + 1 ' 603. The convergents tahen in order are alternately less and greater than tlie continued fraction. The first convero^ent a is too small because the part 1 ; — is omitted ; a + - is too great because the denominator h is too small ; a H is too small because 6 + - is too great ; and 6 + - c so on. 604. To ijrove the law of formation of the successive con- vergents. _,--,,, a ah-\-\ ahc + a + c ,. Ihe nrst three convergents are j, — 7 — , — -, ^ — ; tne numerator of the third is c(ah + \) + a, that is, it may be formed by multiplying the numerator of the second by the third quotient, and adding the numerator of the first ; the denominator of the thii'd convergent may be foi-med in a similar manner by multi- plying the denominator of the second by the third quotient, and adding the denominator of the first. We shall now shew by in- duction that such a law holds universal! v. T. A. * 2i m>"+/ 370 CONTINUED FRACTIONS. Let - , -^ , 7, , be three consecutive convergents ; m, m', m", q q 9. the corresponding quotients ; and suppose that f=m"p'A-j), q"=m"q'-¥q. Let m'" be the next quotient; then the next convergent diffei*s from -S/ only in taking in the additional quotient m"^ * „ 1 . so that we have to write m" ^ ^, instead of m"\ thus the next m convergent y''" "^ ^7 ^' "^^ m" {my + jj) +/ ~ / „ 1 \ , ~ m!" (m"q +q) + q'~ m''Y + q ' If therefore we suppose p = m p' + p and q =m q +qy P • P the next convergent to ^ will be equal to ^, , thus the converg- /// p ent —^, may be formed by the same law that was supposed to p'' hold for -y. ; but the law has been proved to be applicable for the third convergent, and therefore it is applicable for every subsequent convergent. "We have thus shewn that the successive convergents 7na7/ be formed according to a certain law ; as yet we have not proved that when they are so formed each convergent is in its lowest terms, but this will be proved in Art. 606. 605. The difference between any two consecutive convergents is a fraction whose numerator is unity, and whose denominator is tlie j^i'oduct of the denominators of the convergents. This is obvious with respect to the first and second converg- _ a6 + 1 a 1 ents, for -^ - J = ^ . CONTINUED FRACTIONS. 371 Suppose the law to hold for any two consecutive convergents - , -, ; that IS, suppose p'q -i^q = ± 1, so that then, 2^"9' ~p' and greater than qij^q+q) q(q+^) ^^ — T—, , . Since r/ is sreater than q, the error a fortiori is less qiq'-^q) ^ ^ ^ 1 1 . . . than — 2 and greater than ^^-7^ ; these limits are simpler than those first given, though of course not so close. 609. In order that the error made may be less than a given quantity y , we have therefore only to form the consecutive con- vergents until we arrive at one - , such that (f is not less than Ic. 610. Any convergent is nearer to the continued fraction than any other fraction which has a smaller denominator than the convergent has. r Let — , be the convergent, and - a fraction, such that s is q ° S P less than (/. Let x be the continued fraction, and - the con- ^ q . . . p pp., vergent immediately preceding ^ . Then - , x, —, are either ui r ascending or descending order of magnitude by Art. 603. Now - o p p . r p cannot lie between - and —. : for then the difference of - and - q q' s q CONTINUED FKACTIONS. 373 would be less than the difference of - and — , that is, less than q q — — , and therefore the difference of ps and qr would be less than qq s ,, that is, an integer less than a proper fraction, which is ini- possible. Thus either ^ , x,-,, - or -,■'-, x,— must be in ^ q q s s q ' q' T order of mamitude. In the former case - differs more fi'ora x s r than — , does : in the latter case - differs more from x than - does, q s gr ' and therefore a fortiori more than —, does. q 611. Suppose - , -, t\yo consecutive convergents to a con- tinned fraction x. then -^ is greater or less than oj^ according qci ^ ^ P . 7) as - IS greater or less than — , . For, as in Art. 607, we have x = ^^^; therefore ^ -^ =Pp^-i^j^^. f^q +q qx p q {fxj) +p) p {y.q + q) Reduce the fractions on the right-hand side to a common de- nominator ; then the numerator is pp {y^q -^^ qf- qq (/>i/->' + pY, tliat is, l^'{pp'q'^-qq'iy') +VP'^''-m'P^ t^^t is, {j^^p'q'-pq) (pq'-p'q). The factor \}^pq —pq is necessarily positive ; the isioXor pci — p q P 7)' IS positive or negative, according as - is greater or less than -, ; 9. q hence is greater or less than — ^ , that is, —, is greater or less qx p qq than x', according as - is greater or less than -7 . 374 EXAMPLES. XLIV. EXAMPLES OF CONTINUED FRACTIONS. Convert the following four fractions into continued fiuctions : 1380 9 445 19763 743 • 1051' Gi2* 44126* ' 611* 5. Find three fractions converging to 3 '14 16. 6. Find a series of fractions converging to the I'utio of 5 houi-s 48 minutes 51 seconds to 24 hours. V V V 7. If — , — , — be three consecutive converojents, shew 9l ' + p'> - Nq\ a"q' + p"q = p' ; therefore a" (pq' -p'q) = pp - qq^^t p"(pq'-pq)'(^ —p'q^, pp —qqN^ and q^N —p'^ have the same sign ; hence a' and p' are positive integers. This investigation may be applied to any corresponding pair of quantities in (1) and (2) except the first two pairs; it cannot be v v' applied to these because two eonvercjents - and S- are assumed to 2 2 p" precede the convergent ^, . But the first two pairs of quantities in (1) and (2), na,mely and 1, and a and r, are known to be positive integers. Thus all the quantities in (1) and (2) are {>ositive integers. 615. The greatest term in (1) is a. For by the mode of fonnation of the series, pp = N— a'; since p and p' are positive, a'' is less than iV, and therefore a' is not greater than a. 616. No term in (2) or (3) can be greater than 2a. For by the mode of formation of the series, a + a" = p'(S^; and since neither a' nor a" can be greater than a, neither p nor ^' can be greater than 2a. 617. If p"= 1, then a" — a. For, by Art. 614, a" + p"^, = K, therefore if p'=l a" + a fraction = —, . Now — , is a nearer approximation to ^N than a is, and a is less than ^/-^j therefore ^ is greater than a j hence a" =: a. TO A CONTINUED FRACTION. 379 618. If any term in (1), excluding the fii-st, be subtracted from a, the remainder is less than the corresponding term in (2). For, by Art. G14, d'q + p"q^ 2^' > therefore -?', = i/^'- q"V 9 P \Q J theiefore — - a'' is less than p'' ; therefore, a fortioriy a - a" is less than p'\ This demonstration will only apply to the third or any fol- lowing temi, because in Art. 614 it is supposed that two terms a, a' precede a". The theorem, however, holds for the second term, as is obvious by inspection, for a — a, or zero, is leas than r. ' G19. It is shewn in Arts. 615 and 61 G that the values of the terms in (1) and (2) oaimot exceed a and 2a respectively ; hence the same values must recur in the two series simultaneously, and there cannot be more than 2a^ terms in each series before this takes place. 620. Let the series (1) be denoted by tti, a^, a^, ^m-l) ^m) ^OT+l> *«-l> ^H) ^n+l) and let a similar notation be used for (2) and (3). We have proved that a recurrence must take place, suppose then that tlifi terms from the m*^ to the {n — 1)*^ inclusive recur, so that ^n = ^m> ^n + l = ^»n+l) ^n+2 = ^m + 25 ' ' ' ' ' ' "n — t»^j ^n + \ — ^m + \^ ^n42~^'n + 2> 'n^^'mi 'n + l^^'m+1? '^n + 2~'w» + 2) We shall shew that We have r„_i r,^ = N- a J, t^^xT^ = N- a,.', but r^ = r^j and a„ = a^; thereforo r„_, = r,„_ , . Again, a^_^ + a„, = r,^_^b^_^ , a„_^ + «» = r^_i K.i ; therefor© a„_^ - «,„_ ^ = (5„_i - 5„_i) 7-„_ , ; therefore -^^ ^^^ = &n-i - ^m_i = zero or an integer. 380 REDUCTION OF A QUADRATIC SURD But, by Art. 618, a-a„,_, is less than r„_i, and a-a„_i is less than ?Vi, so that a-a„_i is less than r„,_i ; therefore Oh_i - «m-i is less than r^.^ ; therefore -^^ ^^ is less than 1. Comparing this with the former result, we see that "~^^ — ^^ ' m— 1 must be zero ; therefore a„_i = a,„_i, and hn_i = 6„,_i. Hence, knowing that the m^^ term recurs, we can infer that the (m-iy^^ term also recurs. This demonstration holds as long as 7n is not less than 3 ; for it depends on the theorem established in Art. 618. Hence the terms recur beginning with the complete quotient ; . 621. The last quotient will always be 2a. . /(iV) + a„ , For let the last complete quotient be -^^ ^ ■ , then the n next is ^~^^ ; hence a„ + a = r„6„, rj- = N- a^ ; but r = N- a'; r therefore r„= 1 ; therefore, by Art. 617, <^„ = a; therefore 6„ = 2a. 622. Every peHodic continued fraction is equal to one of the roots of a quadratic equation with rational coefficients. 1 111 Let x — a-\- where y -r + b + h+ k + y' 1 111 s + u + V + y so that a, b, h, k are the quotients which do not recur, and r, s, u, V are those which recur perpetually. Let — be the converj^ent formed from the quotients a, h, ... down to k inclusive ; and let ^ be the convergent immediately p' preceding — ; then, as in Art. 607, «=-?^ ..(1). 91/ + t^^<^i^ r^j + P '■^-QYTQ (2)- From (1) and (2) by eliminating y we obtain a quadratic equation in x with rational coefficients. To obtain x we must solve this equation : or we may take the positive value of y found from (2), that is, from Q'l/ + (^Q - P') y - p = Q, and substitute it in (1). 623. The following theorem in continued fractions may be noticed. Let J -, J-, be the development of a proper 0+ c + m + m + m ^ ^ ^ . P fraction -^ ; and let the corresponding series of convergents be 1 C p J) 'p" P h' 'cb + l' q' q" Y" Q' a" . then the development of -^ will be 111 11 that is, the same quotients will occur but in the reverse order. For Q = ml'q' + 5^, therefore « «"+« / i q" = m'q + q, therefore 7, = — — ; and so on. Q ~ m" ■¥ m' + m -^ c + 6 * 5 / , 5' Hence 882 REDUCTION OF A QUADRATIC SURD 624. The preceding theorem will furnish an addition to the results obtained in the present Chapter. Let - and — , be two successive convergents to JNj such 2 9. r that ^ is tlie last convergent formed before the quotients recur ; therefore by Ai-ts. 614 and 621, p' = aq' + q. Kow the development of ■^- — ^-^, that is of^-a, will be 2 9. with the notation of Art. 620 111 111 ^, + ^3 + ^ + K-z + K-, + K-x' and the last convergent will be . But we have just seen that q =p —aq'. Hence by Art. 623 62o. There is also a recurrence of the same terms in the reverse order with respect to the second and the third series of Arts. 614 and 620, like that which has just been demonstrated with respect to the first series. We have universally r ,r =N-a^ (1), a., + a^=^r,h^_^ (2). Put in (1) for m successively the values 2 and n\ thus r,r^ = N- a/, r„_//^ - ^- «.' \ we know that a, — a for each = or, and that r. =r for each = 1 : therefore r„ = r , . Put in (2) for m successively the values 3 and n ; thus wo know that a„ = a , that r„-r , , and that 6„ = 6 , : therefore au' 2 n—\' 2 «— 1 8 « - 1 Again, put in (1) for m successively the values 3 and n — 1 : hence we obtain ii\-ii\_^. Put in (2) for m successively the values 4 and n—\\ hence we obtain a. = a „ . And so on. 4 H — 2 TO A CONTINUED FRACTION. 383 62G. Tlie following theorem relating to continued fractions was communicated to the present writer by Mr Rickard of Bir- mingham. The theorem will fiiniish high convergents to the square root of a number with little labour. Let ^ be a positive integer which is not an exact square, and let the convergents to pj^ be sup2X)sed formed in the usual way; let c be the number of recurring quotients in one com- plete cycle, or any multiple of that number ; let — be the c* 1%) convergent, and — " the (2cy^ convergent ; then will S'2c '^ \qc Pc J Let a be the gi'eatest integer in JN'y and let the quotients obtained by converting JI^ into a continued fraction in the usual way, be denoted by ^1, K K '" h, K+i, 5c+2j ••• Ki ••• Then from Arts 620, 621 we have h = K+2y h = h+z, h = ^c+i (1); also bi = a, h,^.i=2a (2). Let -^ and -^^ be the convergents immediately pi'eceding and foUowing ^ : then ^^ = KxPc-^Pc-x ^ 9c ^c+l Oc + l^c + ^c-l Now ^y differs from -^ in this respect ; instead of using 5'c+i the quotient bc^.i we must use the cori-esponding comjjlete quotieni, which is a+ JN, by Art. 621. Therefore ^i^ . (^±^^-^- ; multiply up, and equate tlie rational and the irrational parts ; thus «i>«+i^-i = ^%, «!7c + 7c-i=;'c (3). 384! REDUCTION OF A QUADRATIC SURD Airain, — differs from ^^^ in this respect: instead of usin>-('^)' ^KS^f) We can give an interesting geometrical illustration of the, Pc theorem. If iV denote the area of a rectanojle and — be taken for one side, the other side is — — . Thus ^^ is equal to half the sum Pc ^2. of the sides of this rectangle. Lst h and k denote the sides of one rectangle ; then if |^ (A + k) denote a side of another rectangle of 2hk the same area the other side will be - — ^ ; the difference of these h + k 111 - kY two sides will be -J—- — '—, which is less than h-k. Now in 2 {h + k) seeking JX we in fact desire the side of a square of which the area is N ; and the present theorem may be considered to supply a series of rectangles, in which a side of each rectangle is half the sum of the sides of the preceding rectangle ; so that each rectangle is more nearly equilateral than the preceding rectangle : and the rectangles tend to the form of a square. This illustration has lieen suggested by a paper entitled The Rectangular Theorem by Henry Brook. Suppose for an example that A"^ = a' + 1 ; then the quotients ars' a, 2a, 2a, 2a, ... ; that is, the cycle of recurring quotients re- TO A CONTINUED FRACTION. 385 duces to the single quotient 2a. In this case then c may be any whole number whatever. Suppose for another example that iV = a' — 1 ; then the quo- tients are a-1, 1, 2(a-l), 1, 2(a-l), ...; thus the cycle of recurring quotients consists of the two quotients 1 and 2{a—\). Thus in the above theorem c may be any even whole number. In this case however the theorem will also be true if c be any odd whole number, as we will now shew. Suppose c any odd whole number. Since the (c + 1)'^ quotient is unity we have Pc:,i^Pc + Pc-u 9!c+i = qc + qc^i (4). And, in the same mamier as equations (3) wore proved, we have {^-'^)Pc+i + Pc=^%+i, (a-l)'7c+i + 5'c=/'c+i (5). Now ^' differs from ^^ in this respect; instead of using the quotient unity we must use the continued fraction 1 + n <• _ i\ j_ T ^ ^^^^^ *^^^^ continued fraction is equal to 1 . q ^— — ^ , that is^ to -^^ by the second of equations (-3). Yc+l p,^+p,_, P,^+p,,,-p, Thus ^=—2s -— %- , by (4). q^c qc^i + qc-i Hc^x-qo From equations (5) since N =a^ —1, it may be deduced that _ {a-\)p, + Nq, _ (a-\)q^ + p^ ^'^'~ 2{a-l) ' ^'^'~ '2{a-l) ' Substitute these values in the last expression for - and q-ic Nq^ + P^-1 we obtain ^ -^ = i- . '?2o -P.- T. A. 25 3SG EXAMPLES. XLV. EXAMPLES OF CONTINUED FRACTIONS FROM QUADRATIC SURDS. Express the following fourteen snrds as continued fractions, and find the first four convergents to each : 1. ^/8. 2. ^/(lO). 3. J{U). 4. ^(17). 5. JilOy 6. J{26). 7. V(27). 8. ^/(46). 9. ^/(53). 10. JilOl). 11. J(a«4-1). 12. J{a'-1). 13. V(»' + 4 14- n/K-4 15. "Find the S^^ convergent to V(13). 16. Find the 8*^ convergent to J(^l). 17. Shew that the 9^^ convergent to ^(33) will give the true value to at least 6 places of decimals. 211 18. Find limits of the error when — - is taken for ^^(23). 916 19. Shew that y^ differs from J (23) by a quantity less X «/ JL 1 , 1 than rrwTXo and greater than (1917^''"^ '^"'^^^ 2(240)^' • 1151 . 20. Find limits of the en-or when -^tq- is taken for ^/(23). 21. Find limits of the error when the 8*^ convergent is taken for ^/(31). . . 1111 //5\ 22. Shewthatl4-— 2T3T2^ = V U/ ' 23. Shew that / _l__l__l._i_ W±J_JLJL ] = '! \ b+a+b+a+ J\b + a + b + a+ / b' 24. Shew that 2a + ^ ^ _L J- =2^f(l + a')', a + 4a + a + 4« + shew that the second convergent difiers fi'om the true value by a quantity less than I -^ a (4a^ + 1) j and thence by making a = 7, 99 . 1 shew that ^ differs from ^2 by a quantity less than ..^..^.q . EXAMPLES. XLV. 38? 25. Shew that the 3^^ convergent to J(a" + a + 1) is J (2a + 1), v/3 13 26. Find convergents to -— ; shew that — exceeds the ti-ue 4 ok) vahie by a quantity less than . 27. Find the G*** convergent to /( ^ 28. Find the G''' convergent to the positive root of 2a'-3a;-G = 0. 29. Find the G^^ convergent to each root of x'-5x + 3 = 0. 30. Find the 7*^ convergent to the greater root of 2x'-7^: + 4 = 0. 31. Find the 5*** convergent to -r-r^, . V(45) 32. Find the value of 1 + ?r- tt- 2 + 2 + 33. Find the value of - — - — ^j — pr — 1+2+1+2+ 34. Find the value of 1 + -—- ^r— = — 7; — x — -r — ■ .. 2+3+1+2+3+1+ 35. Find the value of ^ — - — z — 7—- ^r— :;; — 3+2+1+3+2+1+ 36. Find the vahie of 2 + ~ ^ J^ -^- -^ ^ ... 1+3+5+1+5+1+ XLYI. INDETERMINATE EQUATIONS OF THE FIRST DEGREE. C27. \Vlien only one equation is given involving more than one variable, we can generally solve the equation in an infinite number of ways ; for example, if ax + 1)7/ = c, we may ascribe any value we please to x, and then determine the corresponding value of y. 25—2 388 INDETERMINATE EQUATIONS OF THE FIRST DEGREE. Similarly, if tLere be any number of equations involving more than the same number of variables, there will be an infinite number of systems of solutions. Such equations are called in- determinate equations. 628. In some cases, however, the nature of the problem may be such, that we only want those solutions in which the variables have positive integral values. In this case the number of solutions 7)uiy be limited, as we shall see. We shall proceed then to some jiropositions respecting the solution of indeterminate equations in 2')ositive integers. The coefiicients and constant terms in these equations will be assumed to be integers. Before w^e give the general theory we will shew by an example how such equations are often solved in practice. Required to find corresponding integral values of x and y in the equation 5x + 8?/ = 37. Divide the given equation by 5, the least coefficient : thus % ^ 2 ^ 2-3v , , , , . ^ + y + '^='+F> ov x + y-i = — — ^ . As X and y are to be m- tegers — p — must be ah integer; denote it by j9S0 that 2 — 3y=5p. Divide by 3 : thus ■^ — ^ =p + ^, ov p + y= '^ . Hence 2-2p — ^^ must be an integer ; denote it by q, so that 2 —2p = 3q. o Divide by 2 : thus 1 — /> = 2' + ^ • Hence ^ must be an integer ; denote it by s, so that q = 2s. Then l — p = 2s + s, so that p = l—3s. Then 2-3y = 5p=-5-15s, so that y = 5s-l. Then 5x = 37 - Sy = 45 - 405, so that x=d-8s. We have then y = 5s—l and x = — Ss ; and if we ascribe aiiy integral value to s we shall obtain corresponding integral values of x and y : but the only positive integral values of x and y are obtained by putting s = 1 ; then y = 4, and x=l. INDETERMINATE EQUATIONS OF THE FIRST DEGREE. 389 629. Xeitherofthe equations ax+by = c, ax — by = c can he solved in integers \f a a^id b /(at;e a divisor which does noi divide c. For, if possible, suppose that either of the equations has such a solution ; then divide both sides of the equation by the common divisor ; thus the left-hand member is integral and the right hand member fractional, v^^hich isi impossible. If a, h, c have any common divisor, it may be removed by division, so that we shall in future suppose that a aixd b have no common divisor. 630. Given one solution qfa.x — hj = c i^ ^:>05i7ii:e integers, ta Jind the general solution. Suppose x = a, 2/ = /? is one solution of ax — hy = c, so that aa — b[i = c. By subtraction a{x-a)-b(7/-l3)=^0; therefore ^ = ^^ . Since y is in its lowest temis, and x and y are to have integral values., we must have (as will be shewn in the Chapter or\ the. Theory of Numbers], X - a = bt, 1/ — /3 - aty where t is an integer ; therefore x = a + bt, y = p + af. Hence if one solution is known, we may by ascribing to t dif- ferent positive integral values, obtain as many solutions as we please. We niay also give to t such negative integral values as make bt and at numerically less than a and /5 respectively. We shall now shevr that one solution can always be found. G31. A solution of the equation ax - by = c in positive integers can always be found. Let -J- be converted into a continued fraction, aRcl the succes- b 390 INDETERMINATE EQUATIONS OF THE FIRST DEGREE. sive convergents formed; let - be the convergent immediately preceding ^ ; then aq-hj) = ^ 1. Fii'st suppose aq-hjj = 1, therefore aqc-bpc = c. Hence x = qc, y —2)c is a solution of ax — hy = c. Next suppose aq - 5;j = - 1, then a{h — q)-h{a -p) = 1 ; there- fore a{b — q) c - h (a - p) c = c. Hence x = (b — q)c, y ^ia-p) c is a solution of ax — by = c. If a=l, the preceding method is inapplicable; in this case the equation becomes x — by = c ; we can obtain solutions ob- viously by giving to y any positive integral value, and then making x = c + by. Similarly if 6 = 1. G32. Given one solution of the equation ax + by = c in p)ositive integers, to find the general solution. Suppose that x = a, 2/ = /? is one solution of ax -^by = c, so that aa + b/3 = c. By subtraction, a{x-a)+b(y-B) = 0: therefore J^^Zl^. ^ ^ w ' / b x-a a Since y is in its lowest terms and x and y are to have inte- gral values, we must have x — a = bt, ^ — y^atj where t is an integer; therefore X -a + bt, y = (3 — at. 633. It may happen that there is no such solution of the equation ax + by — c. For example, if c is less than a + b, it is impossible that c = ax + by for positive integral values of x and ?/, excluding zero values. By the following method we can find a solution when one exists. Let ,- be converted into a continued fraction, and let - b q be the convergent immediately preceding - ; then aq — bp-^1. INDETERMINATE EQUATIONS OF THE FIEST DEGHEE. 391 First suppose aq-bp = l^ then aqc -bpG = c; combine this with ax + bj/ = c; therefore a{qG~x) -b (pc + y) = ; therefore qc — x=btf pc + y-- aty where t is some integer. Hence x—qc—bt, y = at—pc. Solutions will be found by giving to t, if possible, positive integi'al values greater than — and less than -- . a b Next suppose aq - bp = - 1, then aqc - bpc =-c ; combine this with ax + by=c, therefore a{x+ qc) -b (jjc -y) = 0. Hence ♦ x = bt-qc, y = pc- at. Solutions will be found by giving to t, if possible, positive integral values greater than - and less than - . 634. To find tJie number of solutions in positive i7itegers of the equation ax + by = c. Let - be converted into a continued fraction, and let ^ a b be the convergent immediately preceding - ; then aq -bp -^1. Suppose aq — bp)= 1 . Then by the preceding Article, x = qc — btj y = at - pc. c c I. Suppose - and - not to be integers. -Let — = m+ f. ^j- = n + a, a *" b ^ a b pc qc where m and n are integers, and y and g are proper fractions. Then the least admissible value of t is in + \, and the gi-eatest is n; thus the number of solutions is n- in, that is, f- — — -^f— g-, a /» that is, — T +/— g. And as this result must be an integer it must c be the nearest integer to —y , superior or inferior according as forg is the greater. 392 IXDETEKMINATE EQUATIONS OF THE FIRST DEGREE. c IT. Suj^pose - an integer. Then f= ; thus wlien t = m tlie value of y is zero. If wo include this solution the number of solutions is equal to the gi-eatest integer in — + 1 ; if we exclude this solution the number of solutions is equal to the greatest integer in — r . III. Suppose - an integer. Then ^ — ; tlius when t = n the value of x is zero. If we include this solvition the number of solutions is equal to the c . Cfreatest intec^er m -r + 1 ; if we exclude this solution the number t' » ah ' of solutions is equal to the greatest integer in ~j . G C IV. Suppose - and y to be integers. Then /= 0, and g = ; tlius when t = m the value of y is zero, and Avhen 1 = 91 the value of x is zero. If we include these solu- tions the number of solutions is equal to -^ + 1 : if we exclude ao these solutions the number of solutions is -^ — 1 . ao Thus the number of solutions is determined in every case. Similar results will be obtained on the supposition that aq — lp = —l. 635. To solve the equation ax-^hy + cz = d in positive inte- gers we may proceed thus : w^rite it in the form ax + by = d- cz, then ascribe to z in succession the values 1, 2, 3, and de- termine in each case the values of x and y by the preceding Articles. G36. Suppose we have the simultaneous equations ax + hy + cz = d, a'x + h'y + c'z = d' ; eliminate one of the variables, z for example, we thus obtain an EXAMPLES. XLVI. 393 equation connecting the other two variables, Ax + By - C, sup- pose. Now if A and B contain no common factors except such as are also contained in (7, bj proceeding as in the previous Articles, "we may obtain x = a + Btj 1/^(3 -At. Substitute these values in one of the given equations, we thus obtain an equation connecting t and z, which we may write A't + B'z = C\ From this, if A' and B' contain no common factors except such as are also contained in C\ we may obtain t = a' + B't', z = ^'-A't\ Substitute the value of t in the expressions found for x and y ; thus a: = a + (tt' + B't') B, y = p-{a' + B'f) A, or x = a + Ba'-¥BBr, y = l3-aA-ABr. Hence we obtain for each of the variables x, y, an expression of the same form as that already pbtained for z. EXAMPLES OF INDETERMINATE EQUATIONS. Solve the following six equations in positive integers : 1. 8x+65y = SI. 2. 17aj+ 23y = 183. 3. 19a:+ 5?/ = 119. 4. 7a: + 10y = 297. 5. Zx+ 7y = 2D0. 6. I3x+19y=m0. Find the general integral values in each of the following four equations, and the least values of x and y which satisfy each : 7. 7x-9y = 29. 8. 9x-Uy = 8. 9. 19x-5y = n9. 10. 17a;- 49y + 8 = 0. 11. Find in how many ways £500 can be paid in guineas and five-pound notes. 12. Find in how many ways £100 can be paid in guineas and crowns, 13. Find in how many ways £100 can be paid in half-guineas and sovereiims. 3.94 EXAMPLES. XLVI. 14. Find in how many ways 195. 6d. can be paid in florins and half-crowns. 15. Find in how many ways £22. 35. Gel. can be paid with French five-franc pieces, value 45. each, and Turkish dollars, value 35. 6d. each. 16. If there were coins of 7 shillings and of 17 shillings, find in how many ways <£30 could be paid by means of them. 17. Find the simplest way for a person who has only guineas to pay IO5. 6cL to another who has only half-crowns. 18. Supposing a sovereign equal to 25 francs, find how a debt of 44 shillings can be most simply paid by giving sovereigns and receiving francs. 19. Divide 200 into two parts, such that if one of them be divided by 6 and the other by 11, the respective remainders may be 5 and 4. 20. Find how many crowns and half-crowns, whose diameters are respectively '81 and "666 of an inch, may be placed in a row together, so as to make a yard in length. 21. Find 71 positive integers in arithmetical progression whose sum shall be 71^ : shew that there are two solutions when n is odd. 22. Find the least number which divided by 28 leaves a remainder 21, and divided by 19 leaves a remainder 17. 23. Find the general form of the numbers which divided by 3, 5, 7, have remainders 2, 4, 6, respectively. 24. Find the least number which being divided by 28, 19, and 15, leaves remainders 13, 2, and 7. 25. Solve in positive integers 17ic + 23y + 3^ = 200. 26. Find all the positive integral solutions of the simul- taneous equations 5x + 4:y +z — 272, 8x + 97/ + dz—656. 27. Find in how many ways a person can pay a sum of £15 in half-crowns, shillings, and sixpences, so that the nimiber of shillings and sixpences together shall equal the number of half- crowns. EXAilPLES. XLVI. 395 28. Find in Low many different ways the sum of £4. I6s. can be paid in guineas, crowns, and shillings, so that the number of coins used shall be exactly 16. 29. Find how £2. is. can be paid in crowns, half-crowns, and florins, if there be as many crowns used as half-crowns and florins together. 30. The difference between a certain multiple of ten and the sum of its digits is 99 : find it. 31. The same number is represented in the undenary and septenary scales by the same three digits, the order in the scales being reversed and the middle digit being zero : find the number. 32. A number consists of three digits which together make up 20; if 16 be taken from it and the remainder divided by 2 the digits will be reversed : find the number. 33. Find a number of four digits in the denary scale, such that if the first and last digits be interchanged, the result is the same number expressed in the nonary scale. Shew that there is only one solution. 34. A farmer buys oxen, sheep, and ducks. The whole number bought is 100, and the whole sum paid = .£100. Sup- posiug the oxen to cost £5, the sheep £1, and the ducks Is. per head ; find what number he bought of each. Of how many solu- tions does the problem admit? 3-5. Find three proper fractions in Arithmetical Progression whose denominators shall be 6, 9, IS, and whose sum shall be 2|. 36. Three bells commenced tolling simultaneously, and tolled at intervals of 25, 29, 33 seconds respectively. In less than half an hour the first ceased, and the second and third tolled 18 seconds and 21 seconds respectively after the cessation of the first and then ceased ; how many times did each bell toll ] 37. Two rods each c inches long, and divided into m, n equal parts respectively, where m and n have no common measure greater than unity, are placed in longitudinal contact w^th their 39G EXAMPLES. XLVI. ends coincident. Prove that no two divisions are at a less distance (^ than — inches, and that two pairs of divisions are at this distance. mn ' ^ If m - 250 and n = 243, find those divisions which are at the least distance. 38. Tliere are three bookshelves each of which vriW carry 20 books ; when books are composed of 3 sets of 5 volumes each, 6 of 4, and 7 of 3, find how they must be distributed, so that no set 'is divided. 39. Determine the greatest sum of money that can be paid in 10 different ways and no more, in half-crowns and shillings; allowing a zero number of half-crowns or of shillings. 40. Detemiine the greatest sum of money that can be paid in 10 different ways and no more, in half-crowns and shillings; excluding a zero number of half-crowns or of shillings. XLYII. INDETERMINATE EQUATIONS OF A DEGREE HIGHER THAN THE FIRST. G37. The solution in positive integers of indeterminate equa- tions of a degi'ee higher than the first is a subject of some com- plexity and of little practical importance j we shall therefore only give a few miscellaneous propositions. 638. To solve in positive integers the e^ucUiovi nixy, ■\- 'nx^ + px + qy = r. This equation contains only one of the squares of the variables, and it can always be solved in the manner indicated in the following example. Required to solve in positive integers the equation 2)xy + 2a;* = 5?/ + 4aj + 5. Here y (Zx - 5) = ■- 2x^ + 4a; + 5 ; therefore y = ^ ^ — j , . « , .. n -2z' + l2z + 4:5 « „ 55 let ox = z: therefore vy = = = — 2« 4- 2 H : ' z-5 z-5 55 therefore dy = - Qx + 2 + ^ — ^ . INDETERMINATE EQUATIONS. 397 Since X and y are to have integral values 3x— 5 must be a divisor of 55, and from this condition we can find by trial the values of x, and then deduce those of y. The oilly cases for examination are the following : 3a:-5-±55, 3a;-5=±ll, 3x- 5 = ^5, 3a; - 5 = * 1. Out of these cases only the following give a positive integral value to X : Sx— 5 = 55, therefore cc = 20 ; 3aj — 5 = 1, therefore x = 2. When a; = 20 we do not obtain a positive integral value for y ; when x = 2 we have y = 5; this is therefore the only solution of the proposed equation in positive integers. 639. The equatioli x' — JTy* = 1 can always be solved in integers when i\^ is a whole number and not a perfect square. For in the process of converting JH into a continued fraction we arrive at the following equation (see Ai-t. 614), p'{p7'-p'q)-i"^-p''; and at the end of any complete period of quotients p'' = 1 (Art. 621) ; thus Suppose now that the number of the recuiring quotients is everij then — is always an even convergent, and is therefore greater than ^JN, and so greater than -. Hence 2^'q~ 9.'P — ^> ^^^ "^'® have ~\ = q"'X—p'^ ; SO that p"' — Nq''=l. Hence we obtain solu- tions of the proposed equation by putting x = 'p' and y — q', where P' -, is any convergent just preceding that formed with the quo- tient 2a, Next suppose that the number of the recurring quotients is odd; P' then when first p" = 1 the convergent — is an odd convergent, 3;)8 INDETERMINATE EQUATIONS. v' wben next />''= 1 the convergeut — is an even convergent, and so on. Hence solutions can Le obtained by restricting ourselves to even convergents occurring just before those formed %\'ith the quotient la. 640. If the number of recurring quotients obtained from jy be odd) then, as appears in the preceding Article, if ^ be any odd convergent immediately preceding that formed with the quotient 2a, we have pq' - pq= q'^'I^ - 2^'^, and pq-pq=l', thus we obtain in this case solutions in integers of the equation 641. The equation x} -Nif — ^a^ by putting x — aoc and y = ay' becomes x'^ - Ny'- = ^\, w^hich we have considered in the prec3ding Articles. 642. The relation 9" {vq-v'ci) = q'^-v'\ that is, ^^'=<^^K-p\ will give solutions of the equation 0(^ — Ny^ = ± c in some cases in w^hich c is different from unity. The method ^vill be similar to that given in Arts. 639 and 640. 643. If one solution in integers of the equation x^ — Ny^ = 1 be known, w^e may obtain an unlimited number of such solutions. For suppose x = p and y — q to be such a solution, so that 2^'- _ Kq- = 1 ; *1^6^ (P - 2' J^) (^ + 5' sl^l = 1 > therefore {p-q s^^y {P + <1 sl^y =\-{x-y JX) (x + yJA-), by supposition. Put then thus x = ^[{p + q J-^y + {p-q v/^^0"} . 2/ = ^. {{p + q xMO" -{p-^ v/^^0"} ; it is obvious that if ?2. be any positive integer, these values of X and y will be positive integers. EXAMPLES. XLVII. 399 644. Similarly, if one solution in integers of the equation 9? — Ni/ = — 1 be known, we may obtain an unlimited number of such solutions. For suppose x = j) and y = q to be such a solution, then (p - q J^) (p + q J^) = — I- Now take n any odd integer ; then = {^-y J-^) (^ + 2/ s/^^), by supposition. Then we proceed as in Art. G43. 645. If one solution in integers of the equation x^ — Nif = a be known, we may obtain an unlimited number of such solutions. For suppose x=p and y = qio be such a solution, and let x = 7)i and y = n be a solution of x^ - Ny"^ = 1 ; then the equation V? — Ny^ = a may be written x' - Ny' ^ {f - Nq') (^' - ^V) =^/m^+ N'q'n^ — N{p'n^ + q~m^) = (jym ± Nqn)' — N { pn ± qvif j we may therefore take x = pm ± JS'qn, y =pn ± qm. EXAMPLES OF IXDETERAIIXATE EQUATIONS. 1. Solve in positive integers 3xy — 4:y + 3x =14. 2. Solve in positive integers xy + x^ = 2x + 3y + 29. 3. Find a solution in positive integers of rc" - 1 3y^ = — 1 , 4. Find a solution in positive integers of x^ —lOly' = —1. 5. Shew how to find series of numbers wiiich shall be at the same time of the two forms n^-1 and lOwi^, and find the value of the smallest. G. A gentleman being asked the size of his paddock an- swered, " between one and two roods ; also were it smaller by 3 square yards, it would be a square number of square yards, and if my brother's paddock, which is a square number of square yards, were larger by one square yard, it would be exactly half as large as mine." Find the size of his paddock. 400 EXAMPLES. XLVII. 7. Find a whole number which is gi-eater than three times the integral part of its square root hy unity : shew that there are two solutions of the problem and no more. 8. Shew that the number of solutions in positive integers of y* + ax' = 6 is limited when a is positive. 9. Find all the solutions in positive integers of 10. Find all the solutions in positive integers of 2x'-dx7/ + 77f=3B. 11. Find a general form for solutions in positive integera of x'— 23y^ = 1, having given the solution a? = 24 and y - 5. 12. Find a general form for solutions in positive integers of X* — 2y' — 7, having given the solution x= 3 and y—l. XLVIII. PARTIAL FEACTIONS AND IXDETER^MI- NATE COEFFICIENTS. C46. An algebraical fraction may be sometimes decomposed into the sum of two or more simpler fractions ; for example, 2a;- 3 1 1 + X' — 3x + '2 X — I X- 2' The general theory of the decomposition of a fraction into simpler fractions, called partial fractions, is given in treatises on the Theory of Equations and on the Integral Calculus. (See Theory of Equations^ Chap, xxiv., Integral Gakulus, Chap, ii.) We shall here only consider a simple case. aou -\- ox ■\- c 647. Let ;; ~ _, , > be a fraction, the denominator {x-a){x-P){x-y) of wliich is composed of three different factore of the first degree with respect to x, and the numerator is of a degree not higher than tlie second with respect to x ; this fraction can be decom- posed into three simple fractions, which have for their denomina- tors respectively the factors of the denominator of the proposed PARTIAL FRACTIONS. 401 fraction, and for their numerators certain quantities independent of ic. To prove this, assume adi? -^hx^-G A B C + ,; + {x-a){x — ^){x — y) X — a x - /3 x-y where A, B, G are at present undetermined; we have then to sliew that such constant values can be found for A, B and C, as will make the above equation an identity, that is, true whatever may be the value of x. Multiply by {x — a) (x — fS) (x— y); then all that we require is that the following shall be an identity, ax' +bx + c^A{x- (3)(x-y) + B(x- a){x — y) + C {x — a)(x- /3); this will be secured if we arrange the terms on the right hand according to powers of x, and equate the coefficient of each power to the corresponding coefficient on the left hand; we shall thus obtain three simple equations for determining A, B and 0. 6i8. The metliod of the preceding Article may be applied to any fraction, the denominator of v/liich is the product of different simple factors, and the numerator of lower dimensions than the denominator. The preceding Article however is not quite satisfactory, because "sve do not shew that the final equations which we obtain are in- dependent and consistent. But as we shall only have to apply the metliod to simple examples, Avhere the results may be easily verified, we shall not devote any more space to the subject, but refer the student to the Theory of Equations and the Integral Calculus. 2a;— 3 G49. Suppose we have to develop -rpy, h '^'^ ^ series l>roceeding according to ascending powers of x ; tliere are various methods which may be adopted. We may proceed by ordinary algebraical division, writing the divisor in the order 2 — ox + x^ and the dividend in the order — 3 + 2x. Or we may develop -- — . ~ by writing it in the form (x^ — 3x + 2y\ and finding the coefficients of the successive powers of a; by the multinomial T. A. 26 402 PARTIAL FEACTIONS AND tlicorem j avc must then multiply the result by 2x - 3. It is however more convenient to decompose the fraction into partial fractions and then to develop each of these. Thus 2a;-3 1111 + - X 3a; + 2 x— \. x-'l 1 - x 2 — x^ - , =-i\-oc) ^ =-[l + x^x^ + x' + ...+x" + ...\, 1 — a; ^ ^ ( j 1 1 /, x\~^ If, X x" x^ x" ) I^ = -~2V-2) =-2V'-i^Wr^ + 2» + /• 2x - 3 Hence the required series for , ^ has for its general X' — ox + ^ term (1^,4)-'. 650. AVithout actually developing such an expression as the above, we may shew that the succeBsi^'e coefficients will be con- nected by a certain relation ; before we can shew this it will be necessary to establish a general property of series. G51. If the series a^ + a^x + a^x' + a^ + is always equal to zero whatever may be the value of x, the coefficients ^0' ^]' '^'i' ^^''' must each separately be equal to zero. For since the series is to be zero vjhatever maij he the value of x, we may put x =^ ; thus the series reduces to a^, which must therefore itself be zero. Hence removing this term we have a,£c + «2^^+a3.'c^+ ... always zero; divide hj x, then ai + a^x + a^x^+ ... is always zero. Hence, as before, we infer that ai -- 0. Proceeding in this way, the theorem is established. If the series a^ + a^x + a„x^ + a^x^' + and A^+A^x+A^x^+J^x'^+ are always equal whatever may be the value of x, then is always zero whatever may be the value of x j hence we infer that ^o-A = ^, «,->4,=0, a^-A^_ = 0, ; INDETERMINATE COEFFICIENTS. 403 that is, the coefficients of like powei"S of a; in the two series are equal. The theorem here given is sometimes quoted as the Principle of Indeterminate Coejicients ; we assumed its truth in Aii;s. 526^ 542, and 549. 652. The demonstration of the preceding Article is that which has been usually given in elementary works on Algebra; there is however a difficulty in it which requires examination. We confine ourselves to the theorem that if the series a^ + a^x + a^x^ + ... is always equal to zero, each coefficient must be equal to zero; the theorem in the latter part of the Article follows from this. When we say that the series is always equal to zero we meiin that it is equal to zero for all such values of x as make the series convergent ; for of course a divergent seiies cannot be said to vanish. In the demonstration we shew that a^x + a^x^ + a^x^ + . . . is always zero; that is xS^ is always zero, where S^ stands for CTj + a^x + a^x^ + ... Hence if a; is not zero S^ must be zero ; but if X is zero xS^ vanishes whatever finite value S^ may ha-\-e : thus in fact we ought not to assume that S^ is zero wlien x is zero, and so the result a^ = is not strictly demonstrated. This is the difficulty we have to examine. We have S^ = a^ + xS^ where S^ stands for a„ + a.jx, + a^x" + . . . ; and although we are not justified in saying that S^ is zero when x is zero, yet we may say that S^ is zero however small x may be. Since the original series is supposed to be convergent *S'^ is also a convergent series, and therefore it will not increase beyond some fixed value when x is made small enough ; and therefore by making x small enough a;AS'„ may be made as small as we please : hence a^ must be zero, for if a, were not zero we could not have S^ zero however small x might be, 26—2 404 EXAMPLES. XLVIII. Thus the result «j = follows strictly if S^ is convergent when X is made as small as we please. In like manner the result a„ = follows strictly if /Sg is convergent when x is made as small as we please, where S^ stands for a^ + a^x + a^x^ + ... And so on. Since the original series i? supposed to be convergent the series aS2, ^S'^, ... are convergent, when x is made as small as we please; and so the theorem of the preceding Article holds. G53. Suppose that the series lo^ + u^x + u^x^ + ujc^ + a "f- ux rein-esents the development of , , : then ■^1 -'px — qx a ^hx ~ {^ — 2?x — qx^) (u^ + u^x + u^x^ + ii^pi^ + ). If n be greater than 1, the coefficient of cc" on the right-hand side is u^^— 2)u^_^ — qu^_^', hence since there is no power of x higher than the first on the left-hand side, we must have by Art. 651, for every value of n greater than 1, And by comparing the first and second terms on each side, we have the last two equations determine u^ and u^^ and then the previous equation will determine U2, ^s, u^, by making successively n = 2, 3, 4, EXAMPLES or PARTIAL FRACTIONS AND INDETERMINATE COEFFICIENTS. Expand each of the following seven expressions in ascending powers of x, and give the general term : 3x-2 1. 7. 1 ^ 5 - 10a; 3 3-2x' " 2-a;-3aj-" "^ 5 1 {l-x){l-px)' ' 1- -2x + x^ l +4:X + X'^ {x -l)(x- 2) {x-3)' 5 + 6x i. ,-, ^, ■,. 5. ^^ -.. 6. ^-j-_3^^,. (l-.# • EXAMPLES. XLVIII. 405 Expand each of the following five expressions in ascending powers of cc as far as five terms, and write down the relation which connects the coefiicients of consecutive terms : 8 1 9 ^ - 10 ^-^ l-x + of ' l-2x + 3x'' ' 2-2x-x'' ,11. , . V , . 12. ^ 1 — px + px* — x" ' 13. Sum the following series to n terms bj separating each term into partial fractions : X ax a^x + 71— zr. . . X +T1 ^T-r. ^+ (1 + a:) (1 + aa;) (1 + aa;) (1 + a^x) (1 + a^x) (1 + cC'x) 14. Sum in a similar manner the following series to n terms : x{\ — ax) ax (1 - a^x) ( 1 + ic) (1 + ax) (1 + d'x) (1 + ax) (1 + a^x) ( 1 + a^x) 15. Determine a, h, c, d, e, so that the n^^ term in the „ a + hx-\-cx^ + clx^ + ex* . . __, expansion oi -p. r^ may be n x . x^ 1 6. Shew how to decompose — - — y^. ^ — into par- {x— a){x — o) {x — c) ... tial fractions, supposing that n is the number of factors in the denominator, and that p is an integer less than n. If 2^ he less than n, shew that a^~' h"'' c^~^ + -r. :-T-, ^ + -. — 7 r^ — + . . . ^ 0. [a -b){a-c)... {b - a) [b -c) ... {c-^ a){c-b) ... XLTX. RECUREIlSrG SERIES. 654. A series is called a recurring series, when from and after some fixed term each tenn is equal to the sum of a fixed number of the preceding terms multiplied respectively by certain constants. By constants here we mean quantities which remain unchanged whatever term, of the series we consider. G55. A geometrical progression is a simple exam2)le of a recun-ing series; for in the series a + ar + ar^ + ar^ + each 40G RFX'URRIXG SERIES. term after the first is r times the preceding term. If u,^_^ and ?/„ denote respectively the {n — \Y^ term and the rt**" term, then w^-rw„_i = 0; the sum of the coefficients of u^ and u^_^ with their 2>i"oper signs, that is, 1 -r, is called the scale of relation. Again, in the series 2 + 4a; + 1 ix^ + 46a;^ + 152a;* + the law connecting consecutive terms is w^ — ^xu^^_-^ — x^u^_^ = ; this law holds for values of ?i greater than 1, so that every term after the second can he obtained from the two terms immediately pre- ceding. The scale of relation is 1 — 3x — a;^ 656. To find the sum ofii terms of a recurring series. Let the series be u^ + u^x + u^x' + u^ + , and let the scale of relation be X—fx-qx^, so that for every value of n greater than unity u^-'pii^_-^-qu^_^ = ^. Denote the first n terms of the series by /S', then S = u^ + u^x + u_^ + u^^ -V ...... + u^^_^x"~\ 2')xS = u^jyx + u^i^x^ + ujyx^ + + w^^_gp.r"~' + u^_^px'', qx^S = u^qx^ + u^qx"" + + ic^_^qx''~'^ + U',^_/IX^+ u^^_^qx''*^'y hence S-pxS-qx'S = u^ + u^x - ti^px - Un_-^px'' - n^_iqx^ - w„_i|?a;"'^', for all the other terms on the right-hand side disappear by vii'tue of the relation which holds between any three consecutive terms of the given series • therefore u^ + x{u^- pu^) - x^ \ pu^_^-\-'qif^_,+ qxu^^,\ 1 —px— qx If the term x^ {pu^^ + 5'^n-2 + 2'^*^n-i} decreases without limit as n increases without limit, we m^y say that the sum of an in- finite number of terms of the recurring series is u^^x(u^-pu;) ^ \ ~px — qx' It is obvious, that if this expression be developed in a series according to powers of x, Ave shall I'ecover the given recurring series. (See Art. 653.) EXAMPLES. XLTX. 407 657. If tlie recurring series be u^ + ic^+u^ + u^-h , and the scale of relation 1 —p — q, we liave only to make x—\ in the results of the preceding Article, in order to find the sum of n terms, or of an infinite number of terms. 658. "When 1 —px — qx^ can be resolved into two real factors of the first degree m x, the expression —°_ ^— ' — -^ may be de- -L poo ~" UtXj composed into j^artial fractions, each having for its denominator an expression containing only the first power of x : see Arts. 337 and 647. In this case, siuce each partial fraction can be developed into a geometrical progression, we can obtain an expression for the general term of the recurring series. We have thus also another method of obtaining the sum of n terms, since the sum of n terms of each of the geometrical progressions is known. EXAMPLES OF RECURRIXG SERIES. Find the expressions from which the following three series are derivable ; resolve the expressions into partial fi'actions, and give the general term of each series : 1. A: + ^x + 2\x'+5W+ 2. l + lla; + 89a.--+G59a;'+ 3. l+3a; + llx-' + 43:c^+...... 4. Find how small x must be in order that the series in Example 3 may be convergent. 5. Find the general term of the series 3 + 11+32 + 84 + 6. Sum the following series to n terms 1 + 5 + 17 + 53 + lGl +48j + 7. Find the general term of the series 10+1 1 + 10 + 6 + ... and the sum to infinity. 8. Find the expression from which the following series is derivable, and obtain the general tei-m 2-x^1ic'-5x''+\0x*-\1x'' f 408 SUMMATION OF SERIES. L. SUMMATION OF SERIES. C59. Series of particular kinds liave been summed in the Chapters on Arithmetical Progression, Geometrical Progi-ession, and Recurring Series; we shall here give some miscellaneous examples which do not fall under the preceding Chapters. 6 GO. To find the sum of the series 1^ + 2^ + 3^ + + 7^^ "We have already found this sum in Arts. 460, 482 ; the fol- lowing method is however usually given. Assume 1^+2'+ 3'+ +n'' = A + Bn+C)i^ + Dn^ + E7i'-\- , where A, B, C, D, E, are constants at present undetermined. Change n into n + \] tlnis p + 22+324. ^-n- ^ {ii + 1)" = A + B {n + \) + G {n + \f + D {n + \Y + E {n+ ly + By subtraction, n^+2n^l=B+C {2n + 1) + i> (3?^" + 3n+ 1) + ^(4?z' + 6?z^4-4?z + l)4- Equate the coefficients of the respective powers of n ; thus E = 0, and so any other term after E would ^ ; W = l; 3D + 2C = 2; D + C + B=l; hence "^"^S' ^'~'>' ^^G' 2 3 07/ 07 07/ Thus 1^ + 2' + 3V ......-^-oi' = A + -^+-^+-^. To determine A we observe that since this equation is to hold for all positive integral values of w, Ave may put oi=l ; thus ^ = 0. Hence the required sum is ^n{oi+l){2n + l). The same method may be applied to find the sum of the cubes of the first n natural numbers, or the sum of their fourth powers, and so on. See also Art. 671. SU:\[MATION OF SERIES. 409 6G1. Suppose tlie n^^ term of a series to be {an + b} {a (n +l) + b}{a{7i + 2) + h] {a (n + 7n-l) + b], where m is a fixed positive integer, and a and b known constants ; then the sum of the first n terms of this series will be {a)i + b}{a(n + I) + b] {a{n + m-l) + b] {a (n + m)+b} ^ {m+l)a where C is some constant. Let u^ denote the ?i*^ term of the proposed series, 8,^ the sum of ?i terms ; then we have to prove that ^ an + b ^ Assume that the formula is true for an assigned value of n ; add the (n + 1)*** tei-m of the series to both sides ; then ~ an -{-b ^, ,,. „ a(n + 7)1 + V) + b ^ a(n+\) + b „ that IS, S^.^ = — ^- — -^ Ur^^i + C = -} ~ ?f„ , o + C ; thus the same formula will hold for the sum of n -vX terms, which was assumed to hold for the sum of n terms. Hence if the formula be true for any number of terms it is true for the next greater number ; and so on. But the formula will be true when n^\ if we take C sucli that CY ^' + ^ /^ XI . • (^ '- ^^ n S = ; n + C , that is, u, = — ?/ -f C7 : ' {m+l)a ■' ' ' ^ (m-fl)a ^ * thus C is determined and the truth of the theorem established. r^. a (m + 1) + 5 , Since u„ = J w, , we have « a + b '' ^ a (m + \) -{-b C = u, ^— ^— , — v, = — bu. a (7)1+ 1) ' a{Tti + \)' TT f^ an+ b bu. Hence \ = r^- w„ , , - ,^ . " {7)1+ I) a "+^ {})i+l)a Thus the sum of the first 7i terms of the proposed series is ob- u n + i'y ^^'® ™^^3^ ^^^ P^^^ ^^^^^ expression into the equi- 410 SUMMATION OF SERIES. tained ))V subtracting the constant quantity 7 !t— from a •^ ° A ^ (m + l)a certain expression which depends on n. This expression is an-¥h (m 4- 1) a valent form — ) r-^ u„, and to aSSi^t the memory we may (m + l)a -^ -^ observe that it can be formed by introducing an additional /actor at the end of u„, and dividing hy the product of the number of factors thus increased and the coefficient of n. 662. We may obtain the result of the preceding Article in another way. As before, let u^, denote [an + h] [a (n + \) + h}{a {n + 2) + h] {a{n + m-l) + h], and let S^ denote the sum of the first n terms of the series of which w„ is the n^^ term. VVe have a (n + 7?i) + 5 amu u ^, = — ^^ ^ u =u -¥ y ; " + ' an + b " " an + b let an + b =1) '} thus P (u ^, - u ) = amu : change n into ?i — 1 , thus similarly, {;? - 3a} (^*„_2 - w„_3) = a»m„_3, {p - (ti - 1) «} {ii^ - Hi) = amu^. Hence, by addition, V K+i - -w,) - a {^^,. + ^^,_, + w_2 + . . . + ^^3 - (?^ - 1) zt,} = amS^^ ; therefore jf' (^n+i - "^i) + '^<^^*i - a»i'S'„ + aS^^ ; , , ^ cr an^b bu^ thcreiore o = -, rr— ^f„^, — " (//i + l)a "-"^ (7?n-l)a* SUMMATION OF SERIES. 411 663. Suppose the n^^ temi of a series to be — , where m, is the same as in the preceding Article ; then the sum of the first n terms of this series will be — -. ^r + C. [m — l)«w„ Assume, as before, S^ = — — + (7, im — 1) au^ 1 add to both sides, then H + l „ 1 an + b ™ u^^^ {7n-l)au, 1 a (m 4- n) + 6 ^ a (n + 1 ) 4 w ^, (>ti — i)au., (m—l)au /I + 1 \ / II + 1 \ I II Hence, as before, the truth of tlie theorem is established, pro- vided C be such that - = ---^Lt^ + C. Thus C = /-^.^-^ , I'l ("^ - 1) «^^i {"I - 1) «Wi ^ am 4-6 an + b and a5„ = 7 r- 7 rr . [771— i) (li^i [m — 1) «it„ This result may also be obtained in the manner of Art. 6G2. G64. A series may occur which is not directly included in the general form of the preceding Article, but may be decomposed into two or more which are. For example, required the sum of Qc terms of the series 4- 1.2.4.5 2.3.5.6 3.4.G.7 Here the n*^ term 71 + 2 (71 + 2y n in + 1) (71 4- 3) (n + 4) 7x (n + 1) (?i 4- 2) (w 4- 3) (u + 4) Now {ii + 2y = 71 {71 + 1) 4- 3;i 4- 4 ; thus the n^^ term n(n + l) + Z71 + 4: 1 71 (n 4- 1 ) (n + 2) (n 4- 3) (n + 4) {n + 2) (n 4- 3) (n 4- 4) 3 4 4- (n 4- 1) (71 4- 2) {n + 3) (7i + 4) ri {71 + I) {71 + 2) (71 4- 3) (ti 4- 4) * 412 SUMMATION OF SERIES. If each term of the proposed series be decomposed in this manner we obtain three series, each of which may be summed by the method of the preceding Article; thus the proposed series can be summed. In the present case the required sum is 1 1 ^ 1 3 J^ 4 '*' 24 4 (n +l){n + 2) (?z + 3) {n + 4) ' 665. Polygonal Numbers. The expression n + ^n[n-\)h is the sum of n terms of an arithmetical progression, of which the first term is unity and the common difference is h. If we make 6 = 0, 1, 2, 3, ... we obtain expressions which are called the gene- ral terms of the 2nd, 3rd, 4th, order of polygonal numhers respectively. The^?*s^ order is that in which every term is unity. Thus we have 1st order, n^ term 1 ; series 1, 1, 1, 2nd order, n^^ term n ; series 1, 2, 3, 4, 5, 3rd order, n*^ teim |-?^ (?i-i- 1) ; series 1, 3, 6, 10, 4th order, ril"^ term n^ ; series 1, 4, 9, 16, 5th order, oi^^ term ^n(^^n— 1); series 1, 5, 12, 22, and so on. The numbers in the 2nd, 3rd, 4th, 5th, series have been called respectively linear, triangular^ square, irentagonal, ^^^. The i-i-^ term of the ?'*^ order of polygonal numbers is n + ^ 71 (?i - 1 ) (r - 2) ; the sum of n temis of this series is, by Art. 661, n(?i+l) r-2 {n—\)n{n-\-V) 2 '^~^r ' 3 ' or l?i(?z+]){(r-2)(?i-l)4-3}. Hence for triangular numbers S^^ = \n{^i+ 1) (?i + 2), for square numbers S^ = \n(n^ 1) (2?^ + 1), and so on. 667. To find the numher of cannon-halls in a pyramidal pile. (1) Suppose the base of the pyramid an equilateral triangle, let there be n balls in a side of the base ; then the number of jp SUMMATION OF SERIES. • 413 balls in the lowest layer is n + {n - 1) + {n-2) + -i- 1, that is, the triangular number ^n(n+l) ; the number in the next layer will be found by changing n into n - 1 ; and so on. Hence, by Art. 665, the number of all the balls is ^ ?i (n + 1) (;i + 2). (2) Suppose the base of the pyramid a square ; let there be n balls in a side of the base ; then the number of balls in the lowest layer is ?^^, in the next layer [n — ly, and so on. The number of all the balls is ^ 7i{n + 1) (2/7, + 1). Similarly we may proceed for any other foi'm of pyramid. We may see from this proposition a reason for the terms triangular 7mmher, square number, If the pile of cannon-balls be incomplete, we must first find the number in the pile supposed complete, then the number in the lesser lAle which is deficient, and the difierence will be the number in the incomplete pile. 668. A question analogous to that in Ai-t. 667 arises when v/e have to sum the balls in a pile of which the base is rectangular but not square. In this case the i:>ile will terminate in a single row at the top ; suppose j) the number of balls in this row ; then the n)-^ layer reckoned from the top has p + n—1 balls in its length and n in its breadth, and therefore contains n{p-\-n—\) balls. Hence the number of balls in n layers is n(n + \) {n-\)n(n+l) , / , i\ /o , o o\ -^^ — Lp ~ -^ L^ -' , or i n {n + 1) {3p + In - 2). If m be the number in the length of the lowest row, 7/i = p-\-n-\, and the sum may be written \n{n+ \){?)m-n-\- l)j as n is the number in the breadth of the lov/est row, the sum is thus expressed in terms of the numbers in the length and breadth of the base. 669. Figurate Numbers. The following series forui \\luit are called the difierent orders oi Jigurate Qiumbers: 1st order, 1, 1, 1, 1, 1, 2nd order, 1, 2, 3, 4, 5, 3rd order, 1, 3, 6, 10, 15, 414 . SUMMATION OF SERIES. the general law is, that the n**" term of any order is the sum of n teniis of the preceding order. Thus the n^^ tenn of the second ?2. (n + 1^ order is n, of the 3rd order is li — ^r— ^, of the fourth order is i • 'U — - — :j — ~-^ , and generally the n^^ term of the r^^ order is 7i(n+l) ... (n + r-2) „,. . • i 4.- t? — ^ ^- ^^ \ Thi? we m^y prove by induction. ±or, I r — 1 assuming this expression for the 7i^^ term of the r^^ order, we may find the sum of the fii.st n terms of the r^^ order by the formula of Art. 661. We have only to put 1 for a, for 6, and ?' — 1 for m. Hence we obtain for the sum n(n+ l){7i + 2) (n + r-l) and then, by definition, this is the expression for the n*^ term of the (r + l)'*" order. 670. We have already shewn that the Binomial Theorem may be sometimes applied to find the sum of a series (see Art. 526); we give another example. Pind the sum of the series where Q^= r{r+l) (r + 2) ...... (r+ q-1), and F^ ^ {n -r) (n-r + l){n -r + 2) (n-r+p-1). We can sec that ^^=^|^xthe coefficient of ic'"' in th3 series for (1 — a:)"^'"^'', and F^=[p X. the coefficient of a""'"' in the series for (1 - x)~^^'^^\ Hence we have so far as terms not higher than o;""^, (l-:.)-»-l= M$, + ^^^+ 5^^'+ ,2^.^.3+ J, Therefore the series which we have to sum is equal to the product of [p [^ into the coefficient of x"~^ in the expansion of the product of (1 -a?)'''"^^^ and (1 - o:)"^^^'^; that is, the series is SUAOIATION OF SERIES. 415 equal to tlie product of |£ [£ into the coefficient of aj""' in the expansion of (1 —x)~'^'^'^'^'\ Hence the series is equal to I p \q \n- 1+p + q p + q + 1 \n — 2 671. By the method of Art. 660 we may investigate an ex- pression for the sum 1'' + 2'' + S' + + ?i'', wliere r is any posi- tive integer. Denote this sum by S ; then it may be shewn, as in Arts. 460 and 461, that S can be put in the form of a series of descending powers of n, beginning with n'^'*'^, and all we have to do is to detennine correctly the coefficients of the various Powersoft. Assume that aS'= Cn^-'+ Ay + ^ ^y-^ + '±z}lA^n^-^ + "'^''''^l ^^~^^ A^n^-^+ It is convenient to represent the coefficients in the manner here exhibited ; thus instead of a single letter for the coefficient T of A^'"~^ we use the symbol - A^, and so on. We shall now pro- ceed to determine the values oi A^, J,, A^^ ; and it will be found that these quantities are independent of r as well as of n. In the assumed identity change n into n^-\) thus >S'+(7^ + l)'' = (7(7^ + l)'•■^l + ^(n + l)'• + ^^l(n + l)'-^ + -^^3 ^ ^.o^+ir-4- Therefore, by subtraction, Expand all the expressions (n + 1)'*^S (h + 1)'', {ii^Vf-^, by the Binomial Theorem ; and then equate the coefficients of the various powers of n. Tlius, by equating the coefficients of n*", we have 1 = (7 (r + 1), then, by equating the coefficients of w''"\ we have ^=-^"2 "^ ^^'' rTl' ""2* 41G SUMMATION OF SERIES. Equate the coefficients of n""'^, putting for C aiid A^ their vahies ; thus we shall obtain generally 1 _ _^_ J_ ^. A,^ ^3 |i^~ !p+l '^2!j^"^|2 |p-i ^|3|;;-2 |4 |;?-3 + .-= —T + '\1 P-^ ■where the series on the riglxt-lmi\d side extends as far as the term involving Ap_-^ inclusive ; and bj 2>utting for 2^ in succession the values 2, 3, 4, ........ we determine in succession A^, A^, A^, ; and we see that these quactities are independent of 7i and r. Weshallobtain J, = ^, ^, = 0, A^=- — , A^=0,A, = — , It is remarkable that all the coefficients with even suffixes yl,, A , A , are zero ; this can be proved as folio vrs : In the original assumed identity change n into ?^ — 1 , and subtract ; thus n'- - C {ii'^' - {n ^ ly^'} + A^ {rJ - (n - l)*-} + 1 ^^ {71--^ - (n - 1)'-^} Equate the coefficients of 01^'^% putting for C and A^ their values ; thus 1 1 ^. -^o ^^3 P^l 2P \^[Pzl L5[£zl l±[P^ A. + [5[p-4 The result formerly obtained may be expressed thus, n 1 1 ^1 -4, A p+ I 2\p ' \''^ \P - 1 |3 Ijp - 2 \4: \p-3 1 5 p — 4 Hence, by subtracting and putting for p in succession the values 3, 5, 7, we shew in succession that zero is the value of Jj, A A,, EXAMPLES. L. 417 EXAMPLES OF THE SUMMATION OF SERIES. 1. Shew that the sum of the first n terms of the series of which the n*-^ term is n (n + l) {7i + 2) (n + m-l) is obtained by placing one more factor at the end of this expression, and dividing by the number of factors so increased. 2. Give the formula for summing the series of which the /**•* term is the recijyrocal oi n (^n+ \) {n + 2) {71 + tn — \). Sum the following five series to n terms, and also to infinity : J_ 1 1 1 1. 2 "^2.3"^ 3.4'^4.5'^ 1 _l _\ _ 1 2.4.6"^ 4.0.8 ^G. a. 10"^ 8. 10. r2"^ 1111 1.4 2.5 3.0 4./ 1111 + t; — r-^ + i^ — ^ — ^ + -. — TT-r. + 01 1.3.5 2.4.6 3.. 5. 7 4.0.6 ^* 2. 3. 4"^ 3. 4. 5"^ 4. 5. 6 "^0. 0.7"^ 8. Sum to n temis 1+3+G+. 10+ 9. If 71 be even, shew that 01 {71+ l)(?i + 2) + 2(n-l) + 3(n-2)4- + 1(1 + 1)- 12 10. Sum to n terms a^ + (a + iy + {a+ 2y + 11. Sum to n terms V + 2^x + 3 V + 4 V + 12. If the terms of the expansion of {a + b}" be multii)lied respectively by nr, (w-1)?-", (7z-2)r^, , n being a positive integer, find the sum of the resulting series. X 1 3. Expand y , ^ in a series of ascending j)owcrs of x, and shew that the coeflficient of x" is T. A. -« 4 IS EX.OIPLES. L. 1 4. Find the coefficient of oj^y" in the expansion of X (1 — ax) (1 —x)(l — ax — by) ' 1 5. Shew that 1 + -,- + — ^-TT-^ + —n—(r-~ + "^ I 3 3.6 3.6.9 + IG. If ^?,. denote the coefficient of x^ in the expansion of (1 + x)", where n is a positive integer, shew that i_ _ ^ . 1_ ^ _ — — — 2\ 1\ 1\ Pn-l 1 • 2 (Po +Pi) {Pi +P2) (Pn-l +Pn) - III P-2 Ih {-^y"~''Pn -.11 1 A 6 n 2 3 71 / \ \^ x^ 17. Shew by developing the identity ( :j 1 ) = 71 v^ *^^^^ n{n+\) (n+jo— 1) n {n-\) (92 + ^ — 2) l£ 1 * ~~\P ?^(7^-l) (?^-2) (?i+;?-3) is zero when n and p are positive integers and n greater than p. 18. If shot be piled on a triangular base, each side of which exhibits 9 shots, find the whole number contained in the pile. 19. Find the number of shot contained in 5 courses of an unfinished triangular pile, the number in one side of the base being lo. 20. The number of balls contained in a truncated pile of which the top and bottom are rectangular, is '^ {2y + 3 (??i + ?i - 1)^; + 6/?^?^ - 3m - on +1}, "W'here in and n represent the number of balls in the two sides of the top, and -p the number of balls in each of the slanting edges. exa:hples. l. 419 21. Shew that l*+2* + 3*+ +71' It" n* n^ n n , , v . -. , x / -. o ^ 5- ■" 2 ■" y - 30 = 30 ^'' + ^) (^^^ + ^) (•^'^' ^ ^'^ - ^- 22. Shew that (1 + xv) (1 + x'v) (1 + x\') (1 + x^v) l-x {l-x){\-x) (1 -a;)(l-a;-)(l-a;^) 23. In the expansion of {\ + x) (1 -{- ex) (1 +cV) (1 + c^x) ... the number of factors being infinite and c less than unity, the co- efficient of a,*" is (i-c)(l-0(l-c^) {i-cy 2i. If A^ be the coefficient of a;'" in the expansion of (1 + :t;)^ (1 + - j (l + ^A ( 1 4- ^3 j ad injinitumy 2^ 1072 prove that ^^ = ^_— ^ (.1^_^ + ,-l^_^), and that J, =-3^5" • 25. If n be any multiple of 3, shew that ,, , {n-2){n-?>) 0^-3)(>i-4)0z-5) _ „ l-(r.-l)+ j-^ ^-: ^ -(-!;. ].I. 1:n EQUALITIES. 672. It is often useful to know v/hiuh is the greater of two given expressions ; propositions relating to such questions are usually collected under the head Inequaliiies. We say that a is greater than b when a— h is a posilicc quantity. See Art. 9 J. 27—2 420 INEQUALITIES. G73. An inequality will still hold after tJie same quantity has been added to each Tiiemher or taken from each member. For suppose a>b, thevefore a-b is positive, therefore a =fc c — (6 ± c) is positive, therefore a =t c > & ± c. Hence we may infer that a term may be removed from one member of an inequality and affixed to the other with its sijn changed. G7-i. If the signs of all the terms of an inequality be changed the sign of inequality must be reversed. For to change all the signs is equivalent to removing each term of the first member to the second, and each terai of the second member to the first. G75. An inequality will still hold after each member has been imdtiplied or divided by the same positive quantity. For suppose a>b, therefore cc- b is positive, therefore if m be positive m [a — b) is positive, therefore ma > rjib ; and similarly i , ,. . .,. , a ■ b — (cc - o) IS i~)ositive, and — > - . ?n ^ ■ m lib In like manner we can shew that if each member of an ine- quality be multiplied or divided by the same negative quantity, the sign of inequality must be reversed. 676. If rt > b, a' > o', a" > b", then a+ a' + a" -\- > 6 + 6' + 6" + For by sui^position, a — b, a — b\ a" — b'\ are all positive; therefore a — b ^ ci —b' ■\- a!' —b" -^ is positive ; therefore a + a! + 0." -v >b^b'-rb" ■\- 677. If a>b, a' >b\ a" >K\ and all the quantities ai'e l)ositive, then it is obvious that aa'a" > bb'b" 678. If a >5, and a and b are positive, then «" >&", where n is any 2^ositive quantity. This follows from the preceding Article if n be an integer. If n V) be fractional suppose it ^ \ let a^ -h and ¥ ^Ic) then h is >ky INEQUALITIES. 421 and we have to prove that /i" > k' ; this we can prove indirectly ; 11 ]_ i for if h' = h\ then h^h^ and if U^ < k'', then hJc''. If n be a negative qiiantitv, let n = ~m, so tliat m is positive; then —,< fr;,; that is, a" 2 3 11 nominators are all of the same sign, then the fraction a,+a„ + a^+ + a 1 2 3 u 6, +6^+63+ +6„ lies in magnitude between the least arid the greatest of the fractions ^ «. 5 ^ 6.' b/ 63'- b/ For suppose ^^ , 7^ , 7-2 .... -7^ to be in ascending order of mas;- ^. ^2 ^3 ^,. nitudOj and suppose that all the denominators are positive; then ■r-^ = 5^ , therefore a, = 5, x y-' ; K h ^. Y^ > r-J^ , therefore «,> 6, x z-' ; 7-^ > ,— , therefore «, > ^3 x 7 - ; ^3 ^1 ^1 and so on ; therefore, by addition, cr, + a., + a3+ + a. > (6, + 5, + 63+ + ^.) ^ J therefore -r — i r r > T~ • Similarly we may prove that a^+a^ + a.j+ . b^ + b^ + b^+ +b„ ^b^ ^. 5 — • 422 INEQUALITIES. In like manner the theorem may be established when all the tlenominators are supposed negative. If T^ = 1^ = i" = ••• , then each of these fractions is equal to \ K K the fraction whose numerator is the sum of the numerators and denominator the sum of the denominators. See Art. 384. 680. Since (x-yY or x^ - 2xy + y"* is a positive quantity or zero, according as x and y are unequal or equal^ we have l{x^ + y^)>xy, the inequality becoming an equality w^hen x = y. Hence 1 (a + 6) > J {ah) ; that is, the arittlmetic mean of two quantities is greater than the geometric mean, the inequality becoming an equality w-hen the two quantities are equal. G81. Let there be n positive quantities, a, h, c, ... k ', then 'rt 4 6 + c -t- . . . + ^\" , , ) > abc ... k . n J unless the n quantities are all equal, and then the inequality becomes an equality. 7 7 /a+b c + dX" therefore aocd < ( — ^ . - ^ I ; , a + h c + d (h(a + h) + i(c + d)Y and -2- . -y- rC + d+ .- , ,* > (ahcd ...y \ n by extending the signification of the terms arithmetical mean and geometrical mean, we may enunciate the theorem thus : the arith- metical mean of any number of positive quantities is greater than the geometrical mean. 682, The following proof of the theorem given in the pre- cedincr Article will be found an instructive exercise. ^ „ , / T T ^x- -, ^ 1 ', a + h + c + d+ +k Let P denote {abed A:)", and Q denote . Suppose a and b respectively the greatest and least of the n quantities a, 6, c, d, h ; let a^ = 5^ = J {a + b), and let Pj = {afi^cd hy^- \ then since a^, > ah, we have P^ > P. Next if the factors in /*i be not all equalj remove the greatest and least of them, and put in theii' places two new fiictors, each equal to lialf the sum of those removed ; let P^ denote the new geometrical mean; then P^>P^. If we proceed in this way, we obtain a series P, P , P^, P^, P^, each term of which is greater than the preceding term ; and by taking r large enough, we may have the factors wliich occur in P^ as nearly equal as w^e please ; thus when r is large enough, we may consider P^ = Q ; therefore P is less than Q. 683. We w411 now compare the quantities a + , r — - and f4T 424 INEQUALITIES. We suppose a and b positive, and a not less than h. fa^h a -by* /a + b a-bY = 2 lb ) ^-iT2-(-J-) (-2-) m(n - \){m - 2) (m - 3) /a + 6\"-' /a - iV ) "" @ v"2 J la;-' /• Since is less than ■ , the series is convergent (Art. 564), SO that we have a result which is arithmetically intelligible and true. Hence if m be negative or any positive integer, it follows tnat — ^ — > I I . If ??i be positive and (ess than unity, oT + b'^/a + by^ . . , . . , . , — ^ — < I — - — J . It remains to consider the case in which m is positive and greater than unity, but not an integer. Suppose m — -, where |? is > 5', and let a = «^, l^=^b% A^a^, J3 = P^. Then a' 4- b^ . fa ^by ,. oT + Bf . fo? + ^'y is > or < I — - — J , according as — - — ls > or < ( — - — J ; 2 that IS, according as I — -^ — j is > or < — —^— ; that is, according as ( — ^ — ] is > or < . We know by what has ah'eady been proved, that the expression on the left-hand side is the gi^eater, since — is positive and less than unity ; hence fr + ft" . /a + bV" , . . . , , ^, • — ^ — is > ( — ^ — 1 when m is positive and greater than unity. 68-4. Let there be n positive quantities a, b, c, k ; then a" + 6"* + c*" + 4- ^•"' /a + 5 + c + + ^ > n \ n when m is negative, or positive and greater than unity ; but the INEQUALITIES. 425 reverse liolds when in is positive and less than unity. The in- equality becomes an equality when all the n quantities are equal. This may be proved by a method similar to that used in Art. 681. We will suppose m negative, or positive and greater than unity. Then a"* + ^z"" > 2 ("^^^X , C" + tr > 2 (~^X ] , therefore •^{KT^ (-:')"} > 9 9 (^izJmi^x • ,, „ rt" + ^z" + c" + tZ" fa + h + c + clY therefore >( — j. By proceeding in this way we can establish the theorem in the case v/here the number of quantities is ^, if jo be any positive inte.gi-al power of 2. Now let p = n-\-r, and let the last r of the p quantities be all equal, and each equal to t, say, where « + 6 -f c + {n terms) n oT + lr + 0"" + /a + 6 + c + >l ^T7 )' thei'efore n + r (nt + rt\" ) ; n + r J that is, > {n + r) C ; therefore c^ + lf + 0" + >7if' ; w^hich was to be proved. In a similar way we may establish the rest of the theorem, namely, that when m is positive and less than unity the reverse holds. The theorem of this Article may also be established by a method similar to that used in Art. 682. 685. If X and [i are positive quantities, and x and px less than unity, (1 + x^ is less than — - — — ^ . 426 INEQUALITIES. We liave in fact to sliew that (1 + x)~^ is greater than 1 — fix. Now, by Dhe Binomial Theorem, LI L^ each term of tliis series is srreater tlian the sueceediniic term, for /3 -h 7Z ;- X is less than unity, since x and Bx are each less than unity. Hence, as in Art. 558, the series is greater than 1 — ^x. G8G. Let y be a positive quantity greater than /3 ; then 1 + ya; is greater than — -- provided (I + yx) (1 — /Sx) is greater than 1 ; that is provided {y- /3)x is greater than /3yx^, that is provided y — /? is greater than fSyx. Hence we have the following result ; if x, /S, and y are positive, and y greater than yS, then by taking x small enough we can make (1 +x)^ less than 1 + yx ; this holds however small the excess of y over jS may be. 687. If X be positive log (1 + x) is less than x. For suppose y = log (1 + x), then 1 + ic = e'' ; and, by Art. 542, c* = 1 4- ?/ + '-y + I - 4- , wliich is greater than y + 1. A 1 . -P • .111 1 As an example put for x m succession z: , -, -- , : J '6 4: Ql we have log ^ < ^ , log - < - , losj — ' — < - . Hence, by li J o o ^ n n -,.. - w + 1 1 1 1 addition, log -^— < - + ^ + +- . (j^'$>. If X be positive and less than unity log (1 + x) is greater 2 than X-'- . For log (1 + a:;) = a; - — + ~ - -r + ; hence, as in Art. 558, Jj o ^ we see that log (l+aj)-(a;--^\isa finite positive quantity. INEQUALITIES. 427 689. If cc be positive and less than unity log^j ^ is greater tlian X l-x °\ / 2 3 4 1 a;* x^ x^ For log- - = -log(l -a;) = a: + - + - + — + ; hence log — cc is a finite positive qnaiitit j. 690. The following three examples will illustrate the subject of Inequalities, and furnish results of some interest. T- J. 1.3.5 (2/1-1) 3.5.7 (2/1 + 1) 1. it iL = r — - — -z ^ — and V = — !^- \ 2.4.6 2ro " 2.4.6 2)i ' then when n is infinite u is zero, v is infinite and u v ls finite. w 1. 1 3 5 27Z-1 Wehave ^'^^ 2 ' I ' 6 -2^ «^ 2 4 6 2n therefore, by Art. 376, w„<- . - . - ^-^ (2). Therefore, by multiplication, ii^^ < . Hence, by increasing n we can make ic^ less than any assigned quantity. feimilarlv, r .~- . — . - ... ^3^ • " 2 4 6 2/t ^^' 4 6 8 ^71 + '^ therefore, by Art. 376, v,^>- . ^ . ^ ^ '^ (4). o i 271+1 ^ ^ 2n + 2 Tlierefore, by multiplication, v^> '' " , that is, >?i + 1. Hence, by increasing n we can make v^ greater than am- as- signed quantity. Last, by (1) and (4) we see that 1 2/1 + 2 . _ n + \ , , , ^ . . 1 '^^n'^n> o n ^ > t^^t IS, > , and therefore, a fortiori. >- ; J An + i 2n -r i 2 and by (2) and (3) we see that u^^v,^<\. Hence w„v„ lies between ^ and 1 , and is therefore finite. 428 INEQUALITIES. IT. If 771, ?i, a tire in descending order of magnitude, then \m — a J \n — a J For take the logarithiiis of both sides ; thus we have to compare _ « /i ^ 1 + - /I + - ni log ' with 71 log I — — and the first of these is less than the second. Hence the required result follows. III. Let there be n positive quantities a, h, c, k ; then n is ( — 9— ) • Suppose a greater than &; let « = c + :^", 6 = c - a*. We have to shew that (1+-) (-^""j is>l, / X that is, that or that /I 4- z\^ ^ Now the logarithm of f t^) (1 - ^') ^^ INEQUALITIES, 429 and this Is a positive quantity ; and as the logarithm is positive the expression is greater than unity. The demonstration is then extended to the case of three or more quantities by a method similar to that used in Art. 682. The problems in the next three Articles are analogous to the' subject considered in the present Chapter. 691. Divide a given number 2a into t^vo parts, such that their product shall have the gi-eatest possible value. Let X denote one part and 2a — x the other part, and let i/ denote the product ; then we have to determine x so that ?/ may have the gi-eatest possible value. Since y = x (2a - x), we have a' - 2ax + ?/ = j therefore x = a^ ^l{a^ — y). Thus since x must be real y cannot be gi-eater than a", and x = a, vrhen y = a^. 693. Divide a given number 2a into two parts, such that the sum of their square roots shall have the greatest possible value. Let X denote one part and 2a -x the other part, and let?/ denote the sum of the square roots of the parts ; then we have to determine x so that y may have the greatest possible value. Since Jx + J {2a -x)=y, 2a-x = (y- Jxf = y'-2y Jx + x, and 2x-2y Jx + y^-2a^^', therefore ^/a; = | ± ^t_|:^ . Since Jx must be real y" cannot be greater than ia, thus 2 J a is the gi-eatest value of vy, and x- a when y = 2 J a. 693. Find the least value which ^ — can have whatever real value x may have. ^ ^ x^ + a' ^, , " A n c .. y ^ N^^2/' "li^) Put = y^ then x'-xy + «- = ; thus a; = - ± ^ . Hence y' cannot be less than \a- ; or 2a is the least value of?/. Or thus, ^ ^- = x V — \ suppose x positive, then we can X X put this expression in the form (j^-'J^ + 2a ; and as 2a is constant the least value of the whole expression will be obtainod 4:)0 EXAMPLES. LI. when the positive term (Jx ^j vanishes, that is, when x = a. 2 8 ^ /. , X + a It is unnecessary to consider negative values of x, because has the same numerical value when x has any negative value as when X has the corresponding positive value. EXAMPLES OF INEQUALITIES. Ill the following examples the symbols are supposed to denote positive quantities ; and the inequalities may, in certain cases, become equalities, as in some of the Articles of the text. 1. If a, 6, c be such that any tAvo of them are greater than the third, 2 (ab + he + ca) > a" -f 6^ + c". 2. If I' + m- + ?i' = 1, and V" + rn'^ + n'^ = 1, then W + mm' + nn' < 1. 3. {a + h - c)' + (a + c -by + {b + c - ay > ah + be + ca. 5. ab (a + b) + be {b + c) + ca (c + a)> Qabc and <2{a^+b^ + c^). 6. (a + b) (b + c) {c + a) > Sabc. 7. Shew that i:c^— 8a; + 22 is never less than 6, whatever may be the value of x. 8. Which is greater, 2x^ or a; -r 1 ? 1 1 "" 9. If n be > 1, then x + — is > 1 + - , if a; be > 1, or < . Qix n n (a ^x) ib ■¥ x) 10. rind the least value of -^ ^ . X 11. Divide an odd integer into two other integers, of which the product may be the greatest possible. 1 2. If a > b, then J{a' - ¥) + J{;lab -b')>a. 13. If a, b, c, d are in harmonical progression, a-\-d>b + e. 14. If a, by c are in hai-monical progression and n a positive integer, a" + c''>26". EXAMPLES. LI. 431 15. If a>b, shew that -rr^ ^ is >or < — —-r, — 7— , accord- ing as a; is > or < J{ab). 16. If a, b, c, or b, c, a, or c, a, b are in descending order of magnitude, a'b + ¥c + c'a > a^c + b^a + c'b ; if they are in ascending order of magnitude, a"b + b'c + c^a < a^c + b'^a + c'b. 17. {A' + B' + C' + ...){a' + b' + c' + ...)>(Aa + Bb + Cc+ ...y. 18. 3{a^ + b^ + c^}>{a+b + c){ab + bc+ca), 19. 9abc<{a + b + c)(a'+b' + c'). n-1 20. -^- (a^ + a^ + a^ + ,.. + a) > J{a^a^) + ^{a^a^) + ^(a^a^) + . . . 21. The difference between the arithmetic and the geometric mean of two quantities is less than one-eighth of the squared difference of the numbers divided by the less number, but greater than one-eighth of such squared difference divided by the greater number. n 23. \n>n^. 24. 1.3.5 ...{2n-l)abc(a + b + c). 27. 8 {a' + 6" -H c') > 3 (a -t- 6) (b + c){c + a). 2a 26 2c 28. -, + — + 1 > 3. b+ca+ca+o 29. (a + b+ cf > 27 abc and < (a' + b' + c'). 30. If p and q be each less than unity, loor„(l -p) . p 1 ^^(1 -7) 31. ^+^^-,^+ +t.+«..^+^>^. »2 »3 «4 ««-l ^n »1 1 X 32. If a and a; both lie between and 1, then > x. I -a 432 THEORY Oy NUMBERS. LII. THEORY OF NUMBERS. 01)4-. TlirougliGut the present Chapter the Avord number is used as an abbreviation for 2)ositive integer. G95. A number which can be divided exactly by no number except itself and unity is called a jyi'ime number, or shortly a prime. G96. Two numbers are said to be prime to each other when there is no number, except unity, which will divide each of them exactly. Instead of saying that two numbers are prime to each other, the same thing is expressed by saying that one of them is prime to the other. 697. If ci 7iumber p divides a product ab, and is prime to one factor a, it must divide the other factor b. Suppose a greater than p ; perform the operation of finding tlie gi'eatest common measure of a and p ; let q, q', q'\ ... be the successive quotients, and r, r, r", . . . the corresponding remainders. Thus a= jyq + r, p = rq' + q-'^ r = r'^' + r'\ . . . multiply each member of each of these equations by b, and divide by p • . - ab ^ br ^ br , br' br br „ br" therefore — = bq -\- — , o = — xq + — , — = — x + M{p), where M (p) de- notes some multiple of p ; thus j^' is divisible by p, and is therefore not a prime. 706. The number of prime numbers is infinite. For if the number of prime numbers be not infinite, suppose p the greatest prime number ; the product of all the prime numbers up to p, that is, 2.3.5.7.11 p is divisible by each of these prime numbers ; add unity to this product, and we obtain a number which is not divisible by any of these prime numbers ; this THEORY OF NUMBERS. 435 number is therefore either itself a prime number, or is divisible by some prime number greater than p ; thus p is not the greatest prime number, which is contrary to the supposition. Hence the number of prime numbers is infinite. 707. 7y a is i^rime to b, and the quantities a, 2a, 3a, (I3 - 1) a, are divided by b, the remainders will all he different. For, if possible, suppose that two of these quantities Tna and m'a when divided by h leave the same remainder ?', so that ma = nb + r and m'a = nb + r ; then i^ni - in) a={n — n') b ; ,1 p a n-n' therefore y = , : m- ni hence m — m' is a multiple of 6 (Art. 699); but this is impossible, since m and m are both less than b. 708. A number can be resolved into pn*/?ie factors in only one way. Let N denote the number; suppose N=abcd , where a, b, c, d, are prime numbers equal or unequal. Suppose, if possible, that JH also = a/?yo . . , where a, /S, y, 8, ... are other prime numbers. Then abed = a/?y8 ; hence a must divide abed , and therefore must divide one of the factors of this product ; but these factors are all prime numbers ; hence a must be equal to one of them, a suppose. Divide by a or a, then bed =l^y^ j from this we can prove that /3 must be equal to one of the factors in bed ; and so on. Thus the factors in abed cannot be different from those in a/3yS TOO. To find the liijhust poimr of a prime number a ivhich is contained in the product ,m. Let / ( — ) denote the greatest integer contained in — , \a J 00 ^ (Til \ ^11j —^ ) denote the greatest integer contained in —^ , let / ( — o I denote the gi-eatest integer contained in -. , and so on ; 28—2 4:3G THEORY OF NUMBERS. then the highest power of the prime nuniljer a Avhich is con- tained in m is / f — j + / y^A + 7 f-. j + For among the numbei-s 1, 2, 3^ ... m, there are I i — \ wliidi contain a at least once, namely the nnmbers a, 2a, oa, 4a, Similarly there are I ( - ^ ) which contain «' at least once ; thei-e are 7(— 3 j which contain a^ at least once; and so on. The sum of these expressions is the required highest power. This i:)roposition will be illustrated by considering a numerical examj)le. Suppose for instance that m=14: and « = 2 j then we have to find the highest power of 2 which is contained in [14 . Here I (^^ = 7, I ("^^ = 3, I ("^^ = 1 ; thus the required power is 11. That is, 2" will divide \li, and no higher power of 2 will divide 1 1 4. Now let its examine in what way this num- ber 11 arises. Of the factors 1, 2, 3, 4, 14 there are seven which we can divide at once by 2, namely 2, 4, 6, 8, 10, 12, 14. There are three factors which can be divided by 2 a second time, namely 4, 8, 12. There is one factor which can be divided by 2 a third time, namely 8. Thus we see the way in which 7 + 3 + 1, that is 11, arises. 710. The 2^^'oduct of any n successive integers is divisible ly [n. Let m-\-\ be the first integer ; we have then to shew that (m+l)(m + 2) (m + n). ■ 1 IS an uiteger. Multiply both nume- rator and denominator of this expression by \m ; it then becomes !m + n ^ p - , . , which we shall denote by — . Let a be any prime \rii\n^ "^ (4 " ^ number; let rj, r^, 9*3, denote the greatest integers in respectively; let 5^, 5^, s^, 1 m + n m + n ni + n a ' €i>^ ' a' THEORY OF NUMBERS. 437 1 , ,1 , - • . . m m m . , denote the greatest integers ui — , -^, 3, respectively; and CC Q/ C(/ let fi, t^, f^, denote tlie greatest integers in - , -, , -3, Cli CV Oi respectively. Then in P the factor a occurs raised to the power rj + r„ + 9*3 + ; in Q the factor a occurs raised to the power 5, + s^. + ^3 + + ^j + '^s "^ ^3 + Now it may be easily shewn that r^ is either equal to s^ + t^ or to s^ + t^ + \, and that r is either equal to 5^ + t^ or to 5^ + ^^ + 1, and so on. Thus a occin'3 in P raised to at least as high a power as in Q. Similarly any prime factor which occurs in Q occurs in P raised to at least as high a power as in Q. Thus P is divisible by Q. 711. If n he a ^:>ri/?ie number, the coefficient of every term in the expansion 0/(3, + b)", except the first and last, is divisible by n. For the general form of the coefficients excluding the first and . n{n-V) (n-r + 1) , , , last IS j — ^ , v/here r may have any value from 1 to w — 1 inclusive. Now, by Art. ,7 10, this expression is an integer ; also since n is a prime number and greater than ?*, no factor which occurs in \r can divide n ; therefore (n—l)(7i-~2) (n — r + l) must be divisible by [r. Hence every coefficient, except the first and last, is divisible by n. 712. If n he a 2^fime number, the coefficient of every term in the expansion of (a + b + c + d+ .,....)% except those of 'a^^ b", c", d", , is divisible by n. Put /5 for 6 + c + fZ + ; then {a + b + c-\-d-^ Y----{a + fi)". By Ai-t. 711, every coefficient in the expansion of (a + P)" is divisible by n, except those of a" and fS", and the coefficient of each of these terms is unity. Again, l^" = {b + c + d+ y = {h-\-yY suppose; and every coefficient in the expansion of (b + y)" is divisible by 71 except those of 6" and y". By proceeding in this way we arrive at the theorem enunciated. 438 THEORY OF NU^MBEI^S. 713. IJ whe a j^rlnie number y and N ^;ri/?i<3 to n, then N"~' - 1 is a multiple of n. (Fermat's Theorem.) l^y the preceding Article, {a^h + c+d+ + ^)'' = a" + i'' + c'' + c/"+ +^'' + J/(/i), where M (n) denotes some multiple of n. Let each of the quanti- ties a, h, c, d, ^ be equal to unity, jind suppose there are iV of them; thus iV" = iY+ J/(n); therefore i\^(iy''-^ - 1) = J/ (ti). Since ^^ is prime to n, it follows that N"~^—l is divisible by n, We may therefore say that N"~^ ^ I -ir 2^n, where j^ is some positive integer. 714. Since oi is a prime number in the preceding Ai-ticle, 71 — 1 is an e^'e?i number except when oi — 2 ; hence we may n—l n-1 write the theorem thus, (^ '^ — 1) (lY '^ +l)^M(n); therefore, n-l 71-1 n-1 either N ■^ -I or lY - + 1 is divisible by n, so that -Y '■^ =pn + ly or else = 2^?i - 1, where p is some positive integer. 715. The following theorem is an extension of Fermat's. Let n be any number ; and let 1 , a, h, c, n — \, be all the num- bers which are less than n and prime to n ; svippose there are m of these numbers ; then will x"* — 1 = J/ (n), when for x we substitute any one of the above 7n numbers, except unity. For multiply all the 7?i numbers by any one of them except unity, and denote the multiplier by x ; thus we obtain 1 . cc, ax, bx, ex, {n — l)x', these products are all different and aU prime to n. It may be easily shewn that when these products are divided by n, the re- mainders are all different and all prime to n ; thus the remain- ders must be the original di numbers 1, a, b, c, n-ly they will not necessarily occur in this order, but that is imma- terial for the object we have in view. Hence the product of the new series of m numbers x, ax, hx, ex, {^~^) ^i ^^^^ only differ from the product of the original m numbers by some mtdtiple of n ; thus x""abc (?i- \)^abc {n- 1) + J/(?i). THEORY OF NUMBERS. 439 Since two of the three terms which enter into this identity- are divisible by ahc (n — 1), the third term must likemse be so divisible, and as ahc [n- 1) is prime to n, the quotient after M{n) is divided by ahc {n— 1) must still be some multiple of n, and may be denoted by M {n) ; thus »"• ^ 1 + M{n\ and cc"* - 1 = M (n). 716. We will now deduce Fermat's theorem from the result of the preceding Article. Suppose n a prime number ; then the numbers 1, 2, 3, n — 1, are all prime to n ; thus 'm = n — \. Therefore oj""^— 1 =J/(n), where x maybe any number less than n. Next let y denote any number which is greater than n and prime to n, then we can suppose y=pn + x, where ^ is some integer and X is less than n. Therefore y"-'=.{pn + xY-' = x''~' + {n-\)x''--27n+ = x''-' + M{n)', but we have already shewn that x"~^ =\ + M (n) ', thus y"-' = 1 + Jf {n), and 2/""' - 1 = M(n). Thus Fermat's theorem is established. 717. If -n. he a prime numher, 1 + |fi — 1 is divisihle hy n. (Wilson's Theorem.) By Ai*t, 549 we have \n-\ =-{n- ly-' - {n - 1) {n - 2)"-' ^(n-l)(n-2)^^^_3^,.,_(.-l)(n 2)(„-3)^^_^^...^ 1 . J L . Jl . O by Fermat's theorem we have {n - ly-' = 1 + p^n, {n - 2)""' = 1 + p_pi, (n - 3)"-' = 1 + p^n, where p^, Poi 2^3, ^I'e positive intogers. Therefore \n-l =3f{n) + l-{)i-l) (n-\)(n-2) (n-\)(n-2)(n-3) "^ 172 TTTa ^ ' the series 1 - (n - 1) + — \- - . . ., of n - 1 terms, is equal to (1 — 1)""' — (- 1)"~', that is, to —4, since n—l is an even number. Thus \n-l = J/ («,) — 1 ; therefore 1 + |n - 1 is divisible by n. 440 THEORY OF NUMBERS. If u be 7iot a prime number, 1 + \n-l is not divisible by n. For suppose p a factor of n j then p is less than n — 1, and there- fore \71-l is divisible by p ; hence 1 + \n — l is not divisible by p, and therefore not divisible by n. 718. Tlie following inference may be dravvn from Wilson's Tlieorem : If 2/? + 1 be a prime number, j'^}" + (-!)'' is divisible by 2p + 1. By Wilson's Theorem, since 2p + 1 is a piime number, I +\2p is divisible by 2p + 1. Put ?i for 2p + 1, then \2p may be written thus, 1 (n-l) 2 {n-2) 3 {n- 3) p(n-p); ii these factoi-s be supposed multiplied out, it is obvious that we shall obtain (- iyV2'3^ p^ together with some multiple of n. Hence 1 + (- 1)'' '[p}^ must be divisible by n, and therefore pY + (— 1)'' must be divisible by n. 719. Let X denote any positive integer; then the number of positive integers which are less than x and prime to x will be denoted by L (x). Consider, for example, the positive integer 1 2 ; there are 4 positive integers which are less than 12 and prime to 12, namely 11, 7, 5, 1 : tliusZ(12) = 4. 720. If m he prime to n then L (mn) = L (m) x L (n). For let 1, a, 6, m—\ be the positive integers which are less than m and prime to m ; then, r denoting any one of these, the following n positive integers are all less than mn and are all prime to m, r, r + m, r + 2m, r + (n-l)m. And every positive integer which is less than mn and is prime to m must be of the form r +pm, where p is zero or some positive integer less than n, and r is oije of the positive integers 1, a, 6, ... 971-1. Hence we see that the number of positive integers less than mn and prime to m is nx L (m). THEOllY OF NUMBERS. 44 i Out of the positive integers which are less than mn and prime to 7Ji we must now determine those which are also prime to n. Let r have the same meaning as before. If we divide each term of the set r, r + m, r + 1m^ r + (?i — 1) m by n the remainders will all bo different ; this is shewn by the method of Art. 707 : thus the remainders must be 0, 1, 2, ... ^z— 1; though they will not necessarily occur in this order. If a re- mainder be prime to n the corresponding dividend is prime to ra ; and conversely if a dividend is prime to n the corresponding re- mainder is prime to n. It follows therefore that out of the n positive integers in the above set there are L (n) which are prime to n. And since this holds for eajch such set of integers as we have considered it follows that L{mn) = Z{m) x Z {n). Hence if I, m, n are all prime to each other, we have L (Imn) = L (Ini) x L (n) = L {I) x Z (m) x L (n) ; and a similar result holds for any number of factors which are all prime to each other. 721. To find the number of jyositive integers which are less than a given number and prime to it. Let N denote the number, and first suppose iV= a^, where a is a prime number. The oi).ly terms of the series 1, 2, 3, 4, N N which are 7iot prime to N are a, 2a, 3a, ia, " ^ j and there a N are — of these terms. Hence after rejecting these multiples of a a, we have remainini; lY- — terms, that is, N[\ — ) terms : thus ' ° a \ aj there are -V( 1 — ) positive integers whijch are. less than X and prime to N. Next, suppose N =^ a^b'^c where a, h, c, are all prime numbers. 4-t2 THEORY OF NUMBERS. Then, by Art. 720, - - l{y) = L{a^) X L{b') X L{c') X = a^(l-l)x6'(l4)> (p — 1) squares. 60. If n be any positive integer 2'^"+15n—l is divisible by 9. 61. If P^ denote the sum of the products of the first oi num- bers taken r together, I + F^ + F^ + ...+P^_^ is a multiple of [w. 62. Shew that the 100'*' power of any number is of the form 125n or \'25n + 1. LIII. PROBABILITY. 726. If an event may happen in a ways and fail in h ways, and all these ways are equally likely to occur, the probability a of its happening is 7 , and the probability of its failing is ■ T . This may be regarded as a deGiiition of the meaning of the word prohahility in mathematical works. The following ex- planation is sometimes added for the sake of shewing the consist- ency of the definition with ordinary language : The probability of the happening of the event must, from the nature of the case, be to the probability of its failing as a to h ; therefore the proba- bility of its happening is to the sum of the probabilities of its happening and failing as « to a + 6. But the event must either happen or fail, hence the sum of the probabilities of its happen- 443 PROBABILITY. ing and failing is certainty. Therefore the probability of its hap- pening is to certainty as a to a + b. So if we represent certainty by unity, the probability of the happening of the event is repre- sented by -. •^ a + b 727. Hence if p be the probability of the happening of an event, 1 - p is the probability of its failing. 728. Tlie word chalice is often used in mathematical works as s\Tionymous with probability/. 729. When the probability of the happening of an event is to the probability of its failing as a to b, the fact is expressed in pojiular language thus ; the odds are a to b for the event, or b to a against the event. 730. Suppose there to be any number of events A^ B, C, ^^^'^ ^^^® P^^" bability of drawing a red ball and a white ball is -^ . Tlie sum T. A. 29 450 PROBABILITY. of tlie six probaLilities which we have just found is unity, as, of course, it should be. We will give one example from a subject which constitutes an, important application of the theory of probability. According to the Carlisle Table of Mortality, it appears that out of 6335 persons living at the age of 14 years, only GO 4 7 reach the age of 21 years. As we may su2)pose that each individual has the same probability of being one of these survivors, we may say that , is the pro- bability that an individual aged 14 years will reach the age of 288 21 years : and is the probability that he will not reach the VOOO age of 21 years. 732. Suppose that there are two independent events of which the respective probabilities are known : we proceed to estimate the probability that both will happen. Let a b,e the number of ways in which the first event may happen, and h the number of ways in which it may fail, all these ways being equally likely to occur ; and let a be the number of ways in which the second event may happen, and b' the number of ways in which it may fail, all these ways being equally likely to occur. Each case out of the a + h cases may be associated with each case out of the a' + h' cases ; thus there are {a + h) {a! + V) compound cases which are equally likely to occur. In aa of these compound cases both events happen, in hh' of them both events fail, in ah' of them the first event happens and the second fails, and in a'h of them the first event fails and the second happens. Thus ^r-7—, — TT- is the probability that both events hapiien, (a + 6) (a + 6 ) ^ ^ ^ f f / . ; — 7-, — 7T- is the lorobability that both events fail, {a + h) {a' +b) ^ -^ ah' (is the probability that the first event happens and {ci -h h) {a + h') ( the second event fails, ah ris the probability that the first event fails and the {a H- h) (ct' + b') \ second event happens. PROBABILITY. 451 Thus if p and p' be the respective probabilities of two inde- pendent events, 2U^' ^ ^^^ probability of the happening of both events. 733. The probability of the concurrence of two dependent events is the product of the probability of the first into the probability that when that has happened the second will follow. This is only a slight modification of the principle established in the preceding Article, and is proved in the same manner ; we have only to suppose that a is the number of ways in which after the first event has happened the second will follow, and 6' the number of wa^^s in which after the first event has happened the second will not follow, all these ways being supposed equally likely to occur. 734. In like manner, if there be any number of independent events, the probability that they will all happen is the product of theii" respective probabilities of happening. Suppose, for example, that there are three independent events, and that 2^, p , p" are their respective probabilities. By Art. 732, the probability of the con- currence of the first and second events is 'pp ', then in the same way the probaljility of the concurrence of the first two events and the tliird is pp y-p" , that is, ppp" . Similarly the probability that all the events fail is (1 — p) (1 — p) (1 — p"). The probability that the fii'st event happens and the other two events fail is p (1 ~ p) (1 —p") '} and so on. 735. ^Ve will now exemplify the estimation of the probability of com2)ound events. (1) Required the probability of throwing an ace in the first only of two successive throws with a single die. Here we require a compound event to happen ; namely at the first throw the ace is to ai)pear, at the second throw the ace is not to appear. The. probability of the fii-st simple event is - , and of the second simj^le 5 5 event -^ ; hence the require 1 probability is ;t-^ . Of) 2 452 PEOBABILITx. (2) Suppose 3 white balls, 4 black balls, and 5 red balls, to be thrown promiscuously into a bag ; required the probability that in two successive trials two red balls will be drawn, the hall first drawn being replaced before the second trial. Here the probability 5 of draAmig a red ball at the first trial is — , and the probability is the same of drawing a red ball at the second trial ; hence the probability of drawing two red balls is f , .^ (3) Suppose now that we require the probability of drawing two red balls, the ball first drawn not being replaced before the second trial. This will be an example of Art. 733. Here the 5 probability of drawing a red ball at the first trial is y^ ; if a red ball be drawn at first, out of the eleven balls which remain foiu" are red, and therefore the probability that a second trial will give .4 a red ball is yj ; hence the probability of drawing two red balls is 5 4. — X — • This is the same result as vre found in Art. 731, for the probability of drawing two red balls simidtaneously ; and a little consideration will shew that the results ought to coincide. (4) Eequired tlie probability of throwing an ace vrith a single . . 5 die in two trials. The probability of failing the first tmie is ^ , 5 and the probability of failing the second time is also -^ ; hence the /^■^\2 25 probability of failing twice is { t: ) , that is, -^ . Hence the pro- 25 11 bability oi not failing twice is 1 - o?. j ^^^^ i^, _„ ; this is there- fore the probability of succeeding. (5) In how many trials will the probability of throwing an ace with a single die amount to -? Suppose x the number PROBABILITY. 453 of trials ; therefore the probability of failing x times in succes- sion is (p) , by Art. 73i. Hence the probability of succeeding is 1 - f- j ; therefore l-frj = ^ ; hence yA = -', hence 5 1 loo" 2 X lo^ -X = locr - , therefore x = , t^-^-j z: . By usinoj the values ° 6 ° 2 log 6 - log a "^ ° of the logarithms, we find a; =3*8 nearly. Thus we conclude that in 3 trials the probability of success is less than |-, and that in 4 trials it is greater than |-. (6) In how many trials is it an even wager to throw sixes with two dice 1 The probability of sixes at a single throw with two dice is ^ x - , that is, -^ ', hence the probability of not having sixes IS —^ , Suppose x the number of trials; tlien we have /35 36 35Y 1 , /35\" 1 ^, ^ log 2 D =2^ ^^""'^ (m) =2^ '^'''^'''' ^- log 36 -log 35- By using the values of the logarithms, we- find x lies between 24 and 25, which we interpret as before. (7) To find the probability that two individuals, A and B, whose ages are kno^vn, will be alive at the end of a year. Let p be the probability that A wiU be alive at the end of a year, p' the probability that B will be alive; then pj/ is the probability that both will be alive at the end of a year. The values of p and p' can be found from the Tables of Mortality in the manner exempli- fied in Art. 731. (8) To find the probability that one at least of two indivi- duals, A and B, whose ages are known, will be alive at the end of a given number of years. Let p be the probability that A will be alive at the end of the given number of years, ji>' the probability that B will be alive. Then 1 - ^j is the probability that A will be dead, and I -p' is the probability that B will be dead. Hence (1 —p) (1 -p') is the probability that both will be dead. The probability that both will not be dead, that is, that one at least will be alive, is 1 - (1 - p) (1 -pO, that is, p -^-p' -pp'. 454< PllOBABILITY. 73G. If an event may happen in different independent ways, the probability of its happening is the sum of the probabilities of its happening in the different independent ways. If the independent ways of happening are all equally probable, this proposition is merely a repetition of the definition of proba- bility given in Art. 726; and if they are not all equally probable, the proposition seems to follow so naturally from that definition, that it is often assumed mthout any remark. The following method of illustrating it is sometimes given : Suppose two urns A and B; let A contain 2 white balls and 3 black balls, and let B contain 3 white balls and 4 black balls ; required the pro- bability of obtaining a white ball by a single drawing from 'one of the urns taken at random. Since each urn is equally likely to be taken, the probability of taking the urn A is - , and the proba- 2 bility then of drawing a white ball from it is ^ ; hence the proba- bility of obtaining a white ball so far as it depends on A is 12 - X -. Similarly, the probability of obtaining a white ball so far ^ o 13 as it depends on -S is k ^ f • Hence the proposition asserts that 12 13 the probability of obtaining a white ball is-^x-+^x-, that is, 1 /2 3\ ^ ( -^ + ?7 ) . The accuracy of this result may be confirmed by the following steps : First, without affecting the question, we may re- place the urn A by an urn A', containing any number of balls we please, lyrovided the ratio of the white halls to the hlack halls he that of 2 to 3 ; and similarly, we may replace the urn B by an urn B\ containing any number of balls we please, ^jrovided the ratio of the white halls to the hlack halls he that of 3 to 4:. Let then A' contain 14 white balls and 21 black balls, and let B' contain 15 white balls and 20 black balls ; thus A' and B' each contain 35 balls. Secondly, without affecting the question, we may now suppose the balls in A^ and B' collected in a single urn ; thus there will be PROBABILITY. 455 70 balls, of wliich 29 are white. The probability of drawing a 29 14+15 white ball will therefore be — - ; that is, — ^ ; that is, 70' ' 70 ' ' 1 /U L5\ ^. ^ . 1/2 3\ 737. The probability of the happening of one or other of two events which cannot concur is the sum of their separate pro- babilities. For the complete event we are considering occurs if the first event happens, or if the second event happens; thus the proposition is a case of the preceding proposition. 738. The probability of the happening of an event in one trial being known, required the probability of its happening once, twice, three times, tfcc, exactly in n trials. Let p denote the probability of the happening of the event in one trial, and q the probability of its failing, so that q=\-2). The probability that in n trials the event will occur in one assigned trialf and fail in the other n — l trials is pq"~^ (Art. 734) ; and since there are n trials, the probability of its happening in some one of these and failing in the rest is npq""'^. The probability that in n trials the event will occur in two assigned trialsy and ftiil in the n (n "~ 1 ) other n— 2 trials, is p'q"~'' ', and there are — ^ — ~- ways in which the event may happen tNv^ce and fail n — 2 times in n trials; there- fore the probability that it will ha})pHn exactly t^^dce in n trials is T — o P^?"~^* Similarly the probability that the event ^oLl hap- . . . . . . . n(n — 1) (n — 2) „ .^ . pen exactly three times in n trials is — — :j — '—^ -p q ; and the probability that it will happen exactly r times in 7i trials is n{n- 1) (n-r + l) , „_^ i P 1 • Similarly, the probability that the event will fail exactly r .. . .-1 • n(n-l) {n-r+ l) ^^-,^. tunes in n trials is — ^ \ — p q . [r * 739. Thus if (p 4- q)" be expanded by the Binomial Theorem 456 PROBABILITY. in the series p" + njf'^q + tfcc, the terais will represent respectively the probabilities of the happening of the event exactly n times, n—\ times, n—2 times, &c., in n trials. Hence we may de- termine what is the most probable number of successes and failures in n trials ; we have only to ascertain the greatest term in a the above series. Let us suppose, for example, that p = -r , q = 7 , n = 111 {a + 6), where a, h, and m are integers ; then, by CC "T Art. 511, the most probable case is, that of r failures and n — r . , . . T . 7Z + 1 , successes, where r is the greatest mtesrer contained m — — , that ^ + 1 is, in 7)ib A rj so that r = mh^ and n — r = ma. The most a + probable case therefore is, that in which the numbei-s of successes and failures are proportional to the probabilities of success and failure respectively in a single trial. T-iO. The probability of the happening of the event at least r times in n trials is „ , liin — V) „ „ n(n-l){n-2) (r-f-1) „_^^ ■*" \7i-r ^^ ' for if the event happen every time, or fail only onee, twice, {qi — r) times, it happens r times ; therefore the probability of the happening of the event at least r times is the sum of the proba- bilities of its happening every time, of failing only once, twice, n — r times; and the sum of these is the expression given above. For example ; in five throws "with a single die w^hat is the probability of throwing exactly thr^e aces? and what is the pro- bability of throwing at least three aces ] 15 Here ^ = -,5 = -, ?z=5, r = 3; thus the probability of , , . 5.4.3 /1\V5\' 1 . 250 throwmg exactly three aces is = — ^ — ^ ( ^^ j ( ^ j , that is, ^^^ j PROBABILITY. 457 and the probability of throwing at least three aces is /IV k/IVS 5.4/1\V5V ^, ^ . 276 The following four Articles contain problems illustrating the subject. 741. A and B play a set of games, in which ^'s probability of winning a single game is p, and ^s probability is q ; required the probability of A's winning m games out of m + n. If A wins in exactly m + r games he must win the last game and m — 1 games out of the preceding m + r—l games ; the proba- bility of this is Mif*~^q''p, where M is the number of -combinations oi m + r~\ things taken m — 1 at a time ; that is, the j)robability is I m + r — 1 \ iT^ i^V- \m— I \r Now in order that A may win m games out of ra + n, he must win m games in exactly m games, or in exactly m + 1 games, , or in exactly m + n games. Hence th« probability required is the sum of the series obtained by giving to r the values 0, 1, 2, n 1 7?i + r — 1 in the expression - p*"^', that is, the required probability is „ r^ vi(m + \) o m(m + l) (m + n-V) „") p- [l+mg+ \^,^ ^ q' 4- + -^^ L^^ ^. q-y If A in order to win the set must win m games before B wins n games, A must win m games out of m -^ n—l; the probability of this event is given by the preceding expression with the omis- sion of the last term. Similarly, the probability of B'^ winning n games out of m + n - 1 is { l+n» + — V-H-^P + +— —^ ~P i .m \m— L This problem is celebrated in the history of the theory of probabilities, as the first of any difficulty which was discussed ; it was proposed to Pascal in 1654, with the simplifi.eation however which arises from supposing p and q to be equal. 45S PROBABILIXr. It a})pcrirs from tlic preceding investigation that the probability of ^'s winning r games out of 9i is '•('• + 1)^,^ ',' ■('■+ 1) <"-^) ^-^ n — r \ but this probability must from the nature of the question be the same as the probability of the happening of an event at least r times in n trials when the probability of the event is p. Thus the expression just given must be equivalent to that given in Art. 740; we may verify this as follows : Denote the expression just given by Vn, and that given in Art. 740 by ^«„, and let v^^^ and w^^, denote respectively what they become when n is changed to n+\'y then we shall shew that if u^^ = v^^ when n has any specific value, then also u .^=v ... 71 + 1 n + 1 We have u^ = u^^ {p -^ ^) ] ^-^^^ '^„ {p "•" (?) gives two series, and when the like terms in these two series are united we obtain {n + l)n (?' + !) r n+i~r , n (u - '[) (r + 1) \7i+ 1 — r PT + — z — P9 ti+i n — r , - , . n(7i—l) r _ „+,_^ therefore u _^, =io (p + q)A ^ p q : , , . -, n{n-V) r ^ „^j_^ and obviously v^^, = v,. + ^^ |— i" :~" I' 2 This shews that u ^ —v^/\iu =v . Now obviously w is equal to v„ when n = 7' ; therefore u^ is equal to v„ for every value of n greater than r. For some more remarks on this problem the student is re- ferred to the History/ of ProhahilitT/, page 98. 742. A bag contains n-\-l tickets which are marked with the numbers 0, 1 , 2, 7i, respectively. A ticket is drawn and replaced : required the probability that after r drawings the sum of the numbers drawn is s. The number of drawings which can occur is {71 + l)^ for any one of the tickets may be drawn each time. The number of ways PROBABILITY. 459 in wMcli the sum of tlie drawings will amount to s is the coeffi- cient of x' in the expansion oi {x" + x' + x^ + + x'J ; because this coefficient arises from the different modes of forming s by the addition of r numbers of the series 0, 1, 2, n. Thus the pro- bability required is found by dividing this coefficient by {n+ 1)\ The above coefficient may be obtained by tlie Multinomial Theorem ; or we may proceed thus : (x''+x'+x'+ +^.")- = |l^^'j'"=(l-a;-7(l-x)-'-; 1 n ."^'V 1 ...'' + ia-''('"~^)-.^''+2 r(r-l)(r~'2) and (I —X ) = i — rx H — ^ — ^r- x ; — -. — -; -t. + ^ ^ 1.2 1 . J . o We must therefore find the coefficient of x' in the product of these two series ; it is r{r+\) (r + s-1) r (r + 1) (?• + g - ?i - 2) r \s — n r(r-\) r(7- + l) (r+s-27^-3) ■^ 1. 2 • ' is-271-2 this series is to stop at the {i + 1)*^ term, where i is the gi'eatest g integer contained in ; then the required probability is ob- tained by dividmg this series by (ii + 1)''. It is not difficult to determine the probability that after r drawings the sum of the numbers drawn shall not exceed s ; see History of ProhahiUty, page 208. 743. A box has three equal compartments, and four balls are thrown in at random : determine the probability of the different arrangements, assuming that it is equally likely that any ball will fall into any compartment. Since it is equally likely that a ball vnW fall into any com- partment there are 3 equally likely cases for each ball ; and on 460 PROBABILITY. the wliole tliere are 3^ equally likely cases. Now there are four possible arrangements. I. All the balls may be in one cGmpartmeiit ; this can happen in 3 ways. II. Any three of the balls may be in any one of the com- partments, and the remaining ball in either of the remaining compartments ; this can happen in 4.3.2 ways. III. Any two of the balls may be in any one compartment, and one of the remaining balls in one of the remaining compart- ments and the other in the other ; this can happen in 6.3.2 ways. IV. Any two of the balls may be in any one compartment, and the other two balls in either of the remaining compartments ; this ca,n happen in 6 . 3 ways. Thus the probabilities of the different arrangements are re- spectively -^ , ^-r , -rpr , -^ ', tlio sum of tliGSO fractions is, of oi oi oi oi course, unity. In the preceding solution the point which deserves particular attention is the statement that there are 81 equally likely cases ; for when this is admitted all the rest follows necessarily. If this is not admitted and the student substitutes any other statement in the place of it, he will be really taking another problem instead of the one intended. In fact in a problem which relates to permuta- tions, combinations, or probabilities, it is not unfreqnently found that different results are obtained because different meanings have been attached to the enunciation ; especial care is necessary in these subjects to ensure that whatever meaning is given to the enun- ciation should be consistently retained throughout the solution. "We will next consider the general problem of which the present is a particular case. 744. A box is divided into m eqnal compartments. If n balls are thrown in promiscuously, required the probability that there will be a compartments each containing a balls, h compartments each containing [i balls, and so on, where aa + 6/5 + cy + =n. PROBABILITY. 4Gi Since any ball may fall into any, compartment) there are on'* cases equally likely to occur. We shall first shew tliat the num- ber of different ways in which the n balls can be divided into a+6 + c + parcels containing a, /3, y, balls respectively is \n [\^rm'uY---'vi'±\i ^ ^' "'^' For consider first in how many ways a parcel of a balls can be , , . ^ -. , . . n(n—\) (n — a-\-\) selected from n balls : the result is . ways; [tt Then consider in hov/ many ways a second parcel of a balls can be selected from the remaining n - a balls ; the result is {ii-a){u-a-l) (n-2a + l) ^. ., , ,. , , » -^^ . feunilarlv a t/tuxl parcel oi [a a balls can be selected from the remaining n-2a balls in (n-2a) (n-2a-l) (n - 3a + 1) ,,^ ., , vrays. Ave might tlien at a first infer that the number of ways in v/higli three parcels of a balls each can be selected from n balls is — ^^ T'l" i " > \a \a \a and this is correct in a certain sense ; but each distinct group of three parcels has in this way occurred [3 times, and we must there- fore divide by |3 in order to obtain the number of different ways in which three parcels of a balls each can be selected from n balls. Similarly the number of diffei'ent Avays in which a parcels of a balls each can be selected from n balls is — ^^ ",|"' ^.- . By proceeding thus we obtain the proposed result. Now the number of ways in which the parcels can be arranged in the m compartments is m {in —\){m — 2) {m —s+ \), where s -a + h + c + Hence, the probability required is Nm (m - 1) (m - 2) {m-s + 1) ■ ^" • For example, suppose six balls throvrn into a box which has three compartments. The seven possible modes of distribution 4;G2 PROBABILITY. are, 6, 0, ; 1, 5, ; 2, 4, ; 3, 3, ; 1, 1, 4 ; 1,2,3; 2,2,2; and their respective probabilities are fractions whose common denominator is 243, and numerators 1, 12, 30, 20, 30, 120, 30. 745. If I) represent a person's chance of success in any trans- action, and 7?i the sum of money which he will receive in case of success, then the sum of money denoted by pm is called his expectatio7i, This is a definition of the meaning we shall attach to the word expectation, and might of course be stated arbitrarily without any further remark j it is however usual to illustrate the propriety of the definition as follows. Suppose that there are ??i + ?i slips of paper, each having the name of a person wi-itten upon it, and no name recurring ; let these be placed in a bag, and one slip drawn at random, and suppose that the person whose name is drawn is to receive Jla. Now all the expectations must be of equal value, because each person has the same chance of obtaining the prize ; and the sum of the expectations must be Vv'orth £a, becaiise if one person bought up the interests of all the persons named, he would be certain of obtaining £a. Hence, if £x denote the expectation of each person, we have (m + 01) x = a ; a thus X = . m + n Also, it is evident that the value of the expectation of tvv'o per- sons is the sum of the values of their respective expectations ; and so for three or more persons. Hence the value of the expectation ma ^_ , , , . of m persons is . Now suppose that one person has his ^ m + n ^ ^ ^ name on on of the slips ; then liis expectation is the same as the sum of the expectations of in persons, each of whom has his , . , . . . 'ina name on one slip; that is, his expectation is . But his ^ ' ^ m + n Till chance of winninoj the \)v\zq is , since he has m cases out of ° ^ m + n^ m+n in his favour ; thus his expectation is the product of his chance of success into the sum of money which he will receive in case of success. PIIOB ABILITY. 4G3 746. An event has happened which must have arisen from some one of a given number of causes : required the probability of the existence of each of the causes. Let there be n causes, and suppose that the probability of the existence of these causes was estimated at P^ , F^,...P^ respectively, hrfore the event took place. Let p^ denote the probability of tlie event on the hypothesis of the existence of the first cause, let j9., denote the probability of the event on the hypothesis of the existence of the second cause, and so on. Then the probability of Pp the existence of the r''^ cause, estimated after the event, is _-,/ , where ^Pp stands for P^p^ + I*„p„ + ... + P,j:>„. From our first notions of probability we must admit that the probability that the r^^ cause was the true cause is p)roportionol to the antecedent probability that the event would haj)pen from this cause, and may therefore be rejiresented by CP^p^. And since some one of the causes must be the true cause we have ^{^i2\ + -^2P2 + • • • -^ ^\^'P,} = Ij therefore C = ^-p- ; therefore the PrPr probability that the r^^ cause was the true cause is ^jy- • 747. The preceding Article vvdll requii'e some illustration before it will be fully appreciated by the student. Let there be, for example, two urns, one containing 7 white balls and 3 black balls, and the other 5 wliite balls and 1 black ball ; suppose that a white ball has been dravv^n, and we wish to know what the probability is that it came from the first urn, and what the probability is that it came from the second urn. It must have come from one of the two urns, so that the sum of the required probabilities is unity. Instead of the given urns let us substitute two others wliich have the whole number of balls the same in each urn, and such that each urn has its white and black balls in the same ju'oportion as the urn which it replaces. Thus we may suppose one ui-n with 21 white balls and 9 black balls, and the other with 25 white balls and 5 black balls. Each urn now contains 30 balls, and the chance 404 PROBABILITY. of each ball being drawn, is the same. Since, by supposition, a white ball is drawn we may suppose the black balls to have been removed, and all the white balls put into a new urn. Thus there would be 46 white balls ; and the probability that the wliite 21 ball dra-svn was one of the 21 is -ry. , and the probability that it 25 7 5 was one of the 25 is — ^ . ISTow here z>i = ttt i and ^o = ;r ; thus ^ — = TTT , and — — — = -— . Thus the result aoTees with that 2h + 2^2 4b ^:>i+^2 46 '^ given by the theorem in Art. 746, supposing that P^ and Po ^i'® equal. Next, suppose that there had been 4 urns, each having 7 white balls and 3 black balls, and 3 urns, each having 5 white balls and 1 black ball. In this case, by proceeding in the manner just shewn, we may deduce that the probability that a white ball which was drawn came from the group of 4 similar urns is 4 X 21 -j — ^ — ^ — ^ ; and the probability that it came from the group 3 X 25 of 3 similar urns is ^ ^ — tt^ . N^ow let us apoly the theorem 4 X 21 + 3 X 25 ^^ -^ of Art. 746 to estimate the jDrobability that the white ball came from the first grouj^ and the probability that it came from the second group. Since there are 7 urns, of which 4 are of the first 4 3 7 kind and 3 of the second, we take P,^ = - , and F^-t^ , also ^^ = — , 5 and p^ = - . Til us 7 ^1-1 7 ^ K' ^2- _35'^-4^35' 7 ''lO'^'y'' 6 7"" 10"^7'' 6 and these results agree v/ith those which we have akeady indicated. 748. It is usual to call the quantities F^, F^, ... P^ of Ai-t. 746 the a 2^riori probabilities of the existence of the respective causes; and Q^, Q^, ... Q^^ the a posteriori iDrobabilities. Students PROBABILITY. 4G5 are sometimes perplexed in endeavouring to estimate P , P , . . . P„ • the safest plan is to observe that the product P^p^ denotes the jjrobability that the event will happen as the result of the r^^ cause • and the correctness of the product is the important part of the solution, because P^ and p^ do not occur separately in the results. The whole proposition may be best understood if arranged in the following order. First suppose the different causes all equally- probable before the observed event; let ot^ denote the probability of the occurrence of the event on the hypothesis of the existence of the r^^ cause ; then the probability of the r^^ cause, estimated after the occurrence of the observed event, is -?^ . This seems nearly self-e\ddent, and if any doubt remains it may be removed by the mode of illustration given in the first part of Art. 747. Secondly, suppose that the terms in 2ot can be arranged in grou2:»s ; suppose there to be /m^ terms in the first group, and that each term is equal to p^ , suppose there to be /x., terms in the second group, and that each term is equal to p^, a^d so on, the last group consisting of /a„ terms, each equal to ;7,,. Then ^cr may be written ^ixp, where the series %ixp consists of n terms. Thus the proba- bility of the ?•**' cause is --^ . Also the probability of the first group of causes is the sum of the separate probabilities of the members of that group, that is, ^^ . Similar expressions hold 2/xp for the probabilities of the other groups. Thus we finally arrive at the results given in Art. 746, where, in fiict, ^ixp ^ix.p 749. When an event has been observed, we may, by Ai-t. 740, estimate the probability of each cause from which that event could have arisen ; we may then proceed to estimate the pro- bability that the event will occur again, or that some other event will occur. For by Art. 73G we multiply the probability of each cause by the probability of the happening of the required event on T. A. 30 - . Hence, assuming that before the observed event the three 4G6 PROBABILITY. the hypothesis of the existence of that cause, and the sum of all sucli products is the probability of the hap^^ening of the requii-ed event. For example, a bag contains 3 balls, and it is not kno^vn how many of these are white ; a white ball has been drawn and replaced, what is the probability that another drawing will give a white ball 1 There are three possible hypotheses : (1) all the balls may be white, (2) only two of the balls may be white, (3) only one of the balls may be white. We have first to find the probability of each hypothesis by the method of Art. 746. On the first hypothesis, the observed event is certain, that is, the probability of it is 1 ; on 2 the second hypothesis, the probability of the observed event is - ; o on the third hypothesis, the probability of the observed event is 1 3 hypotheses were equally probable, we have after the observed event, probability of first hypothesis =l-hl + -+^[ = ^, . 2 C 2 1) 1 probability of second hypothesis '=o~=~il + Tj + Qr = Q> probability of third hypothesis = - -f- - 1 + ^ -h ^ [ = ^ • The probability that another drawing will give a white ball is 1 .12 ^ X 1, so far as it depends on the first hypothesis ; it is ^ x -, so far as it depends on tlie second hypothesis ; and it is ^ x - , so far as it depends on the third hyjDothesis. Hence the requii'ed pro- bability is - + ^ + — ; that is, ^ . Suppose that in the enunciation of this problem instead of the words " it is not known how many of these are white " we had the words '' it is known that each ball is either white or black." We PROBABILITY. 4G7 may understand tlie new enunciation as equivalent to tlie old, and so give the same solution as before. We may however, and perhaps most naturally, understand the new enunciation differ- ently, namely that the probability that each ball is wliite is to be taken as ^ before the observed event. In this case we cannot as- sume that the thi-ee hypotheses are equally probable before tlie 13 3 observed event j the probabilities must be ^ , -^ , and - respec- o o o tively by Art. 734. Then after the observed event we shall obtain 111 J, -, and - respectively for the probabilities. And the proba- bility that another dra^\'in2f will save a white ball is - + - + r— • . J o & 4 3 12 750. "We give another example. Suppose a bag in wliich the ratio of the number of white balls to the vdiole number of balls is unknown, and it is equally probable, a priori, that the ratio is any one of the following quantities x, 2x, 3x, nx ; suppose a white ball to be dra"svn and replaced : required the probability that another drawing will give a white ball. Here n hypotheses can be formed. On the first hypothesis the probability of the observed event is x, on the second hvpothesis it is 2x, on the thii'd 3x, and so on. Hence the probability of the X 2 first hypothesis is —7., x r ; that is, -— . The •^^ xCi + 2 + + n) ' n{ti+ 1) 2x2 probability of the second hypothesis is — — -.. . The probability ■^ "^ " ^ n [n-\- \) 2x3 of the thii-d hvpothesis is — ;- . And so on. Hence the " ^ n (n + 1) . . . 2x probability that another drawing will give a white ball is — - — -— ^ 2.^ X 2' , , , , . 2a; X 3" on the first hypothesis, — -, ^-— on the second hypothesis, -— — -^^ •'•^ ?i (?i+ 1) ?i(?i + 1} on the third, and so on. Hence the required probability is ^"^ h+2'+ +7A; n {n + 1) ( 30—2 4G8 PROBABILITY. , . 1x n0i + l)(2;i+l) . . xi^n+l) that is, ■— TT . — ^ TT j that is, ^ ^ . ' n{ii + 1) 6 When 71 is very great this ai:>proximates to -^ . If tho ratio of the number of the white balls to the whole number of balls is equally likely, a priori, to have aiiT/ value between zero and unity, then x is indefinitely small and nx — 1, so that the .2 requii'ed probability is ^ . 751. The following problems will illustrate the subject. (1) A bag contains m white balls and n black balls ; if ^9 + 5- balls are drawn out, what is the probability that there will be p white balls and q black balls occurring in an assig7ied order ? We suppose 2^ less than m and q less than ?i. j and the balls are not replaced in the bag after being drawn out. Suppose, for example, that the first ball is required to be white, the second to be black, the third to be black, the fourth to be white, and so on in any assigned order. Then the required pro! -la- bility is the product of m n n- 1 m—l m+ n^ wi-f- n — 1 ' 7/1 + 71 — 2 ^ m + 71 — 3 ^ therefore the required probability is m {m - 1 ) (m - 2) ... (m — p + 1)71(71- 1) (71- 2) ... (n — q + 1) {m + 7i) (m + 71— 1) (on + n — 2) ... (in + 71-2^ - q+ 1) ' and it will be observed that so long as j9 white balls and q black balls are required, the probability is the same whatever miay he the assigned order in which they are to occur. (2) The suppositions being the same as in (1), what is the probability of ^j> white balls and q black balls occurring in a7iy order whatever ? Let N represent the number of different orders in which p white balls and q black balls can occur ; then the required j)roba- bility is obtained by multipljdng the probability found in (1) by \p + q mOBABILITY. 4G9 The problems (1) and (2) are introductory to one which we shall now consider. (3) A bag contains m balls whicli arc known to be all either white or black, but how many of each kind is unknown ; suppose p white balls and q black balls have been drawn and 7iot replaced ' find the probability that another drawing will give a white ball. The observed event here is the drawing o£ p white balls and q black balls. To render this possible, the original number of white balls may have been any number from m — q to p inclusive, and the number of black balls any number from q to m— p inclusive. Let us denote the hypothesis of m — q white and q black by II^ , and the hypothesis of m — q — 1 white and q + 1 black by II^ , and so on. Then H^ gives for the probability of the observed event jf. A^-2)(^-') PP +{1-P) (1 -P) PP' + (1 -P) (1 -/) * Similarly, if the assertion be also made by a third person whose probability of speaking truth is p", the probabilities of the truth and the falsehood of the assertion are respectively ppY „„,, (1 -p) (1 -/) (1 -p") . pp'p" + {i -p) (1 -p) (1 -p") pp'p"+(\ -p){i-p) (i-Pl ' and so oii if ijiore persons join in the assertion. PHOBABILITY. 471 753. We will make a few remarks on the preceding Article. "When we say that the probability of ^'s speaking truth is p, we mean that out of a large number of statements made by A, the ratio of the number that are true to the number that are not time is that of jo to \—p; thus the value of p depends on the correct- ness of J.'s judgement as well as on his veracity. The result in Art. 752 gives the probability of the truth of the assertion, so far as that tmtli depends solely on the testimony of the witnesses considered ; there may be from other sources addi- tional evidence for or against the assertion. Thus the person who is estimating the probability may himself have a conviction more or less decided in favour of the assertion which is independent of the testimony he receives from the witnesses. It has been proposed to combine this conviction with the testimonies which are con- sidered in the problem. Thus, if there be two witnesses with pro- babilities p and p' respectively of speaking the truth, and a third person estimates the probability of the truth of the assertion at p" from his own independent sources of belief, then to him the odds in favour of the truth of the assertion are »/;/' to (1 -?.)(! -/)(!-/')• Still the result is considered unsatisfactory by some writers, who object with great reason to the solution on the gi'ound that it omits all consideration of the circumstance that it is the same occurrence to which the several testimonies are offered. In the following problem this circumstance is expressly considered. 754. Two persons, whose probabilities of speaking the truth SiXQ p and p respectively, assert that a specified ticket has been which is the j)robability of therr selecting the specified ticket among the undrawn tickets. Thus the probability of the (1-/?) (1 -v') observed event on the second hypothesis is , , . Thus ^^ n{n-l) the odds in favour of the truth of the assertion are ^-^— to ■'^ / -,. , or jm to \ — ^-^ . n n [n— 1) n — 1 755. The question in Art. 752 is respecting the truth of concurrent testimony ; we may now consider the truth of tra- ditio7iary testimony. A says that B says that a certain event took place : required the probability that the event did take place. Let p and p' be the probabilities of speaking the truth of A and B respectively. The event did take place if they both speak truth, or if they both speak falsehood ; and the event did not take place if only one of them speaks truth. Thus the odds that the event did take place are pp' + (1 -J)) (1 -p') to p (1--;/) +y (1 -p). 756. If there be n witnesses, each of whom has transmitted a statement of an occurrence to the next, and if p be the probability of speaking the truth of each witness, the probability of the truth of the statement is to the probability of its falsehood as the sum of the odd terms of the expansion of (p + qY is to the sum of the even terms, q being put equal to 1 —p after the expansion has been effected. For the statement is true if all the witnesses speak truth, or if two, or four, or any even number speak falsehood. 757. Suppose that certain arguments are logically sound, and that the probabilities of the truth of their respective premises EXAMPLES. LIII. 473 are known : required the probability of the truth of the conclusion. For example, suppose that there are three arguments, and let p, 'p', if denote the respective probabilities of their premises. The conclusion is valid unless all the arguments fail. The chance that they all fail is (1 -ji) (1 - p) (1 -p") ; hence the chance that they do not all fiiil is ^ - (^ - 1^) (}■ ~ p') {^ - p"), which is, therefore, the required probability. 7o8. Of such an extensive subject as the Tlieory of Proba- bility only an outline can be given in an elementary work on Algebra. The student who is prepared for further investigation will find a list of the necessary books in the article Probability in the English Cyclopcedia ; to that list may be added the work of Professor Boole on the Laws of Thought. For a discussion of the first principles of the subject the student may consult De Morgan's Formal Logic, Chapters ix. and x., and Venn's Logic of Chance. We may also refer to the History of the Mathematical Theory of Probability, from the time of Pascal to that of Laplace; this work introduces the reader to almost ^very process and every species of problem which the literature of the subject can furnish. EXAMPLES ON PROBABILITY. 1 . The odds against a certain event are 3 to 2 ; and the odds in favour of another event independent of the former are 4 to 3. Find the odds for or against their happening together. 2. Supposing that it is 8 to 7 against a person who is now 30 years of age living till he is 60, and 2 to 1 against a person who is now 40 living till he is 70 : find the probability that one at least of these persons will be alive 30 years hence. 3. A party of 23 persons take their seats at a round table : shew that it is 10 to 1 against two specified individuals sitting next to each other. 4. The chance that A can solve a certain problem is j ; the . . 2 chance that B can solve it is ^ : find the chance that the problem o will be solved if they both try. 474; EXA^klPLES. LIII. 5. Find the cliancc of drawing two black balls and one red from an urn containing five black, three red, and two white. 6. Find the probability that an ace and only one will be thrown in two trials with one die. 7. Find the probability of throwing one ace at loast in two trials with one die. 8. Find the odds against throwing one of the two numbers 7 or 11 in a single throw with two dies. 9. Two purses contain the same number of sovereigns and a different number of shillings ; one purse is taken at random and a coin is drawn out : shew that it is more likely to be a sovereign than it would be if all the coins had been in one purse. 10. There are four men, A, B, C, D whose powers of rowing may be represented by the numbers, 6, 7, 8, 9, respectively ; two of them are placed by lot in a boat, and the other two in a second boat. Find the cliance which each man has of being a winner in a race between the boats. 11. In one throw with a pair of dice find the chance that there is neither an ace nor doublets. 12. If from a lottery of 30 tickets marked 1, 2, 3, fom- tickets be drawn, find the chance that 1 and 2 Avill be among them. 13. A has 3 shares in a lottery where there are 3 prizes and 6 blanks ; B has 1 share in another where there is but 1 prize and 2 blanks. Shew that A has a better chance of getting a prize than B in the ratio of 1 6 to 7. 14. Two bags contain each 4 black and 3 white balls; a person draws a ball at random from the first bag, and if it be white he puts it into the second bag and then draws a ball from it : find the chance of his drawing two white balls. 15. A coin is thrown up n times in succession : find the chance that the head will present itself an odd number of times. 16. When n coins are tossed up, find the chance that one and only one will turn up head. EXAMPLES. LIII. 475 17. Supposing the House of Commons to consist of m Tories and n Whigs, find the probability that a committee of 'p-Vq selected by lot may consist of ^^ Tories and q Whigs. 18. Find the chance that a person with two dice will throw aces at least four times in six trials. 19. Find the chance of throwing an ace with a single die on'Ce at least in six trials. 20. If on an average 9 ships out of 10 return safe to port, find the chance that out of 5 ships expected at least 3 will arrive. 21. In three throws with a pair of dice, find the probability of having doublets one or more times. 22. Find the chance of throwing double sixes once or oftener in three throws with a pair of dice. 23. In a lottery containing a large number of tickets where the prizes are to the blanks as 1 to 6, find the chance of drawing at least 2 prizes in 5 trials. 24. If four eard« be drawn from a pack, find the probability tliat there will be one of each suit. 25. If four cards be drawn from a pack, find the probability that they will be marked one, two, three, four, of the same suit. 26. If J.'s skill at any game be double that of j5, the odds again-st J.'s winning 4 games before B wins 2 are 131 to 112. 27. Two persons A and B engage in a game in which ^'s skill is to ^'s as 2 to 3. Find the chance of ^'s winning at least 2 games out of 5. 28. Three white balls and five black are placed in a bag, and three persons draw a ball in succession (the balls not being re- placed) until a white ball is drawn. Shew that their respective chances are as 27, 18 and 11. 29. In each game that is played it is 2 to 1 in favour of the winner of the game before. Find the chance that he who wins tho first game shall win three or more of the next four. 47G EXAMPLES. Liir. 30. A certain stake is to be won by the first person who throws ace with a die of 7i faces. If there be 2^ persons, find the chance of tlie r*^ person. 31. There are 3 parcels of books in another room and a parti- cuhir book is in one of them. The odds that it is in one particular parcel are 3 to 2 j but if not in that parcel it is equally likely to be in cither of the others. If I send for this parcel giving a description of it, and the odds I get the one I describe are 2 to 1, find my chance of getting the book I want. 32. In a purse are ten coins, all shillings except one which is a sovereign ; in another are ten coins all shillings. Nine coins are taken from the former purse and put into the latter, and then nine coins are taken from the latter and put into the former. A person is now permitted to take whichever purse he pleases : find which he should choose. 33. One urn contained 5 white balls and 5 black balls ; a second urn contained 10 white balls and 10 black balls; a ball, of which colour is not known, was removed from one urn, but which is not known, into the other. A drawing being now made fi'om one of the urns chosen at random, what is the chance that it will give a white ball 1 34. Find the chance of throwing 15 iii one throw with 3 dice. 35. rind the chance of throwing 17 in one throw with 3 dice. 36. Find the chance of throwing not more than 10 with 3 dice. 37. When 2n dice are thrown, prove that the sum of the num- bers turned up is more likely to be 7?^ than any other number. 38. When 2n + 1 dice are thrown, prove that the chance that the sum of the numbers turned up is 7?i + 4 equals the chance that the sum of the numbers turned up is In + 3, and that the chance is greater than the chance that the sum is any other number. 39. Out of a set of cards numbered from 1 to 10 a card is drawn and replaced : after ten such drawings what is the proba- bility that the sum of the niunbers drawn is 24 ? EXASIPLES. LIII. 477 40. Counters numbered 0, 1, 2, n, are placed in a box ; after one is drawn it is put back, and the process is repeated. Find the probability that m drawings will give the counter marked s. 41. There are 10 tickets 5 of which are blanks and the othei-s are marked 1, 2, 3, 4, 5 : find the probability of drawing 10 in three trials, the tickets being replaced. 42. Kad the probability in the preceding Example if the tickets are not replaced. 43. From a bag containing ?i balls p balls are di-awn out and replaced, and then q balls are di^awn out. Shew that the proba- bility of exactly r balls being common to the two drawings is \p \q \n — p\n — q \n\r p — r \q — o' \n — p —q + r 44. Eight persons of equal skill at chess draw lots for part- ners and play four games ; the four wimiers draw lots again for partners and play two games ; and the two winners in these play a final game : find the chance that two assigned persons will have played together. 45. In a bag are m white balls and n black balls. Find the chance of drawing first a white, then a black ball, and so on alternately until the balls remaining are all of one colour. If m balls are drawn at once, find the chance of drawing all the white balls at the fii^t trial. 46. In a bag are n balls of m colours, 2h heing of the first colour, p^ of the second colour, . . . p,,,^ of the m^^ colour. If the balls be drawn one by one, find the chance that all the balls of the first colour will be first drawn, then all the balls of the second colour, and so on, and lastly all the balls of the m^^ colour. 47. A bag contains n balls ; a person takes out one and puts it in again; he does this n times : find the probability of his hav- ing had in his hand every ball in the bag. 48. Two players of equal skill, A and B, are playing a set of games. A wants 2 games to complete the set, and B wants 3 games. Compare the chances of A and B for winning the set. 478 EXAMPLES. LIIl. 49. If tliree persons dine together find in how many different ways they can be seated in a row. When they have dined toge- ther exactly so many times, taking theii* places by chance, find the probability that they will have sat in every possible arrangement. 50. iV is a given number ; a lower number is selected at ran- dom, find the chance that it will divide N. 51. A handful of shot is taken at random out of a bag: find the chance that the number of shot in the handful is prime to the number of shot in the bag. For example, suppose the number of shot in the bag to be 105. 52. If n = a'^, and any number not greater than ?^ be taken at random, the chance that it contains a as a factor s times and no . 1 1 more is -m . 53. Two persons play at a game which cannot be drawn, and agree to continue to play until one or other of them wins two games in succession : given the chance that one of them wins a single game, fiiid the chance that he ^vins the match described. For example, if the odds on a single game be 2 to 1, the odds on the match will be 1 6^ to 5. 54. A person has a pair of dice, one a regular tetrahedi'on, the other a resrular octahedron : find tlie chance that in a sinsjle throw the sum of the marks is greater than 6. 55. There are three independent events of which the pro- babilities are respectively ^^j, j^-a 2^3' ^'^^ the probability of the happening of one of the events at least ; also of the happening of two of the events at least. 56. A certain sum of money i& to be given to one of three persons A, B, C^ who first thi'ows 10 with tliree dice : sujiposing them to throw successively in the order named until the event has happened, find their respective chances. 57. The decimal pai-ts of the logarithms of two numbers taken at random are found from a table to 7 places ; find the pro- EXAMPLES. LIII. 47^ bability that the second can be subtracted from the first with- out horroKxing at all. 58. A undertakes with a pair of dice to throw 6 before B throws 7 ; they throw alternately, A commencing. Compare their chances. 59. A person is allowed to draw two coins from a bag con- taining four sovereigns and four shillings : find the value of his expectation. 60. If six guineas, six sovereigns, and six shillings be put into a bag, and three be drawn out at random, find the value of the expectation. 61. Ten Russian ships, twelve French, and fourteen English are expected in port. Find the value of the expectation of a merchant who will gain £2100 if one of the first two which arrive is a Russian and the other a French ship. 62. From a bag containing 3- guineas, 2 sovereigns, and 4 shillings, a person draws 3 coins indiscriminately : find the value of his. expectation. 63. Find the worth of a lottery-ticket in a lottery of 100 tickets, having 4 prizes of £100, ten of £50, and twenty of £5. 64. A bag contains 9 coins, 5 are sovereigns, the other four are equal to each other in value : find what this value must be in order that the expectation of receiving two coins omt of th-e bag may be worth 24 shillings. 65. From a bag containing 4 shilling pieces, 3 unknown English silver coins of the same value, and one unknown English gold coin, four are to be drawn. If the value of the drawer's chance be 15 shillings, find what the coins are. 6G. A and B subscribe a sum of money for which they toss alternately beginning with A, and the first who throws a head is to win the whole. In what proportion ought they to subscribe 1 If they subscribe equally, how much should either of them give the other for the first throw ? 480 EXAMPLES. LIII. G7. There are a number of counters in a bag of wliicli one is marked 1, two 2, &C. up to r marked r; a person draws a number at random for wliicli lie is to receive as many shillings as the num- ber marked on it : find the value of his expectation. 68. A bag contains a number of tickets of which one is marked 1, four marked 2, nine marked 3, ... up to r^ marked r ; a person draws a ticket at random for which he is to receive as many shillinijs as the number marked on it : find the value of liis expectation. GO. A man is to receive a certain number of shillings; he knows that the digits of the number are 1, 2, 3, 4, 5, but he is ignorant of the order in which they stand : find the value of his expectation. 70. From a bag containing a counters some of w^hich are marked with numbers, b counters are to be drawn, and the draw^er is to receive a number of shillings equal to the sum of the numbers on the counters which he draws : if the sum of the numbers on all the counters be n, find the value of his expectation. 71. There are two urns; one contains 8 white balls and 4 black balls, and the other contains 12 black balls and 4 white balls ; from one of these, but it is not know^n from which, a ball is taken and is found to be white : find the chance that it was di'awn from the urn containing 8 white balls. 72. Five balls are in a bag, and it is not kno\vn how many of these are white ; two b?.ing drawn are both white : find the probability that all are white. 73. A purse contains n coins and it is not known how many of these are sovereigns ; a coin drawn is a sovereign : find the probability that this is the only sovereign. 74. A bag contains 4 white and 4 black balls; two are taken out at random, and without being seen are placed in a smaller bag ; one is taken out and proves to be white, and replaced in the smaller bag : one is again taken out and proves to be again white, find the probability that botli balls in the smaller bag are white. EXAMPLES. LIII. 481 75. Of two purses one originally contained 25 sovereigns, and the other 10 sovereigns and 15 shillings.; one purse is taken by chance and 4 coins drawn out which prove to be all sovereigns : find the probability that this purse contains only sovereigns, and the value of the expectation of the next coin that will be drawn from it. 76. A bag contains three bank notes, and it is known that there is no note which is not either a £5, a £10, or a £20 note; at three successive dips in the bag (the note being replaced after each dip) a £5 note was drawn. Find the probable value of the contents of the basr. 77. It is 3 to 1 that A speaks the truth, 4 to 1 that JO does, and 6 to 1 that C does : find the probability that an event took place which A and B assert to have happened and which C denies. 78. A speaks truth 3 times out of 4, B 4 times out of 5 ; they agree in asserting that from a bag containing 9 balls, all of dif- ferent colours, a white ball has been drawn : shew that the proba- .96 bility that this is true is — . 79. Suppose thirteen witnesses, each of whom makes but one false statement in eleven, to assert that a cei'tain event took place; shew that the odds are ten to one in favour of the truth of their statement, even although the a priori probability of the event bo 10 + 1 80. One of a pack of 52 cards has been removed ; from the remainder of the pack two cards are drawn and are found to be spades : find the chance that the missing card is a spade. 81. Two persons walk on the same road in opposite directions during a + 6 + c minutes, one completing the distance in a minutes and the other in h minutes : find the chance of their meetincf. 82. Find how many odd numbers taken at random must ha multiplied together, that there may be at least an even chance of the last figure beincf 5. Given log,„ 2 = N^OIOS. T. A. 31 482 MISCELLANEOUS EQUATIONS. LIY. MISCELLANEOUS EQUATIONS. 759. Equations may be proposed which require peculiar arti- fices for their solution ; in the following collection the student will find ample exercise : he should himself try to solve the equations, and afterwards consult the solution here given. 1. 1 = H . x + l x+ 4: X + 2 x + 3 Here x+l + - — zi + re + 4 + r = x+2 + ^ +x + 3 + - -r Uj n- -I -r . — ,Aj 1- j^ I ^ ~r t^ T- ty -r ^ ) x+l x + 4: x + 2 x+0 14 2 3 so that q- H 7 = Z7 + therefore that is x+l x + 4: x + 2 x + 3' 12 3 4 x + l x + 2 x+ 3 x+ 4* X X x^+3x + 2~ x^ + lx + l2' therefore either x = 0, or cc'' + 3aj + 2 = ic^ + 7a; + 12 ; firom the latter 4a; = — 10 ; therefore a; = — 2^, 1 1 1 1 {x + af-h' {x + hf-a^ x'-{a + by x'-{a-bf' XT 1 f 2a; ) 1 1 Here t < -^ — ; tt-o > = -5 — 7 — -tt? + therefore therefore x + a + bXx'-(a-byj x'-{a + bf x^-{a-bf' 1 x-{a + b) _ 1 x + a+b x'-{a-by x'-{a + by' x-{a + b) _ 1 a;^ - (a - bf ~x-{a + b) ' therefore {x-{a + b)Y = x^-{ob- 5)^ j therefore 2a; {a + b) = {a + b)" +(a~ hf ; therefore X = r t a + o f.s-"'-- MISCELLANEOUS EQUATIONS. 483 V3 ~x)' He. 3g4«).10(|-i).eaU|-^ = „ then 3 ^f + ^) =. lOy, so that y' -^ + ^^ 1 9^ therefore 2/ = 2 or -; therefore x^—12 — 6x or 4aj; therefore a; = 6 or -2 or 3 ±^(21). (6x' + lOx' + I) (5a^+ 10a' + I) (x'+10x' + 5){a^+10a' +5) ~^^' Here x'^\Ox^-\-bx " 5a" + 10a' + 1 ' adding and subtracting the numerator and denominator of each fraction, we have f x + 1 Y _ /I + ^Y \x-\) ~ VI^/ ^ therefore = : therefore x — - , x—\\—a a 5. ■ {x-iy+{2x+2>f^21x^+^. Since (a? - 1) + (2a; + 3) = 3a; + 2, divide both sides by 3a; + 2, which gives a; = — ^ for one vahie of x ; and we obtain o (x- ly -(x-l) {2x + 3) + (2a; + 3)^ - Oa;'- 6a; + 4, that is 3a;'+ 9a; + 13 = 9a;'' -6a; + 4, therefore 6a;'^ — 15a; = 9; , , , 5a; 25 25 3 49 therefore x _— + -=- + -=-; 5 7 1 therefore £c - y = ± 7 ; therefore a = 3 or - ^ . 4 4 ^ 31—2 484) MISCELLANEOUS EQUATIONS. Here 31^ =- + ^ + 11} = 29^ ^ + „- -U, (ic+l x + i j { x + 2 x+3 ) therefore 31 (^ . AU 29 l-A. ^ '^ (x+1 x + i) {x + 2 x + 3 . . 1111 therefore ^ + -. = ^ -i ^ 5 a;+l a; + 4 £C + 2 x+Z' therefore 1111 x + l x+2 x+?> x + 4:' therefore {x + 1) {x + 2) ^ {x + 3) (ic + 4) ; therefore 3ic + 2 = 7a; + 12 ; therefore 4aj = — 10 ; therefore x = — 2 J. l{x+l)(x-B) l {x + 3){x-5) 2 {x+5)(x~'7) 92 5(£c+2)(a;-4)'^9 (a;+4)(a;-6) 13 (a; + 6) (a;- 8) ~ 585' It IS clear that the numerator and denominator of each fraction involves the expression x^ - 2a;, put therefore (x - 1)^ = y j then the equation becomes 1 y-4 1 y-lG _ 2^ y-36 _ 92 5 ^■^'^9 2/ -25 ~r3y-4:d "585' ^"^^ 5*^9 13 ~ 585^ subtracting corresponding terms, we have 1 5 1 9 2 13 5 2/- 9 "^9^-25 13y-49~ ' that 13 H — = : y/-9 2/--'5 7/-49 ' therefore that is MISCELLANEOUS EQUATIONS. 485 1111 2/_9 2/ -49 2/- 41) y-25' -40 24 2/-0 2/-25' therefore 3 (?/ - 9) + 5 (y - 25) - 0, that is 8y=152j therefore 2/=^^ and cc = 1 ± x/(19). X + 3a ,. ,a + 3x _^ a;'^ aj + 3a c- c + 3a: ,, , . a;- + Sax^ c^ + 3c^x Here — — —^ = — — , that is — j— = —r r : ^Sa + 3x ^,x+6c J + 3Jx x^ + 3cx^ adding and subtracting the numerator and denominator of each (x^^-Jy (l + Jy fraction, we have — y y~ — — i r' 'y (a;^ - a^)^ (c^ - x''f ,, - a;- + a^ c- ^ x^ x c therefore -^ ^ = — I ; therefore - = - ; a;2 _ ^2 c- - a;^ ax therefore a; = ± ^(ac). ( ,/^ X t a;— v/(2aa;) + a „ Here \x + J{2(tx) + ^t} + -^^ — { = 2, therefore x + J(2ax) + a+ rrx — ^ — 2 ; '^ ^ ^ x + J{2ax) + a therefore {x + J{2ax) + a]' - 2 {x + ;^/(2aa:) + a} + 1 = ; therefore x-\- a+ J{2ax) = 1 ; therefore {x + af -2 {x + a) + 1 = 2ax ; therefore a;'' - 2a; + 1 = 2a - a^ ; therefore a; = 1 ± J(2a - a^). 486 10. Here that is Let then therefore therefore therefore therefore 11. Here or therefore Let then therefore therefore therefore MISCELLANEOUS EQUATIONS. {x + a) {x + 2a) {x - 3a) {x - 4a) = c*. (x + a) {x - 3a) (x + 2a) (x - 4a) = c\ {x' - 2ax - 3a') {x' - 2ax - 8a') = c\ x^ — 2ax = ya^, (y-3)(2/-8) = |; 121 c* 25 11 (4c'' + 25a')\ 2/ = — ± x" 2ax 2a' Ua' (4c'' + 25a^)^ ^ — =fc- '2 X = a^ 1 J13a' ^ (4cV25a")- + 1 1 1 + {2x-l){3x-2) {3x-2){4:X-3) 1 6a:;- 4 = 8x'-6x + l. = {2x-l){ix-l), = {2x - 1) {ix - 1) ; 3;r-2 (2£c-l)(4x-3) 2 = (2ic- l)'(4ic-l)(4aj-3) = {2x - ly [2 (2a; - 1) + 1} {2 {2x - 1) - 1}. y = 2x-\, /(4/-l) = 2; ^jl 2 _ J^ 2_33_ ^ "4 ^64"G4 '^4~G4' 2/^-^(1=^^33); x = 1+2/ 1 1 ± v/{§ (1^^/33)}' MISCELLANEOUS EQUATIONS. 487 ^^' \x-q)\x + ^) ■^U+t3;U-'J/ x*-36* a; + 6 /a; - 4y a^ - 6 /x + 9y_ x + Q x-Q ic - \a; + 1/ a; + 6 \a; - yy a - G x+ Q* — :Hr!{(j^)'-}-:^i{'-e^:)"}. . £c — 6 36a; 03 + 6 16a; *^'''* '^ a;"T6 "" (^^::97^ " ^^ (x^^ ^ therefore a; = is one value ; and for tlie other values we have /a;-6\' 16 /a;-9\' ,, ^ x-Q 2 a; -9 ( ;. =77-. 7 , therefore ■ -. = ± 7 ; \a; + 6/ 36 \a; + 4/ ' a; + 3 a; + 4 ' therefore 3 {x'- 2a; - 24) = ± 2 (a;' - 3a; - 54) ; these quadratics can now be solved in the ordinary way. a;* + lax +ac ax x^ + 2cx + ac (x + a) (x+c)' Let (a; + a) (x + c) = xy^ ,, a;^ + 2aa; + ac a then —^ — = - ; x + lex + ac y ,, I. 2 (a;^ + aa; + ca; + ac) a -vy therefore — ^ — - — r = - — — , Ix {ci — c) <^ — y {x -\- a) {x •¥ c) a + y or thus X {a — c) Oj — y y « + 2/ a — G ci — y^ therefore y' — y(^ ~ ac— a^ ; c 1 therefore y = - ± - J{c^ + 4ac - 4a^) = a supjwpe ; thus a;'' + a; (a + c) + ac = a'a ; therefore a;^ + a; (a + c — a) = — ac ; therefore x = — =*= o VIC* + ^ ~ ")* - iac\. 'A A 488 MISCELLANEOUS EQUATIONS. 14. 2(:. + «)(^ + <') + («-<'r = ^^£T|^- (x + cY Here (x + af + {x + cf = , f(^^ ^ a) + {x -^ c)} ' Let x + a--^y{x + c).. (a), «- ^'^^%i^ ^^^- From (a) x + c + a- c ^ 1/ {^ -^ ^) > a — c therefore x + c y-i' therefore (/3) hecomes 3/^ + 1 = ^^^ ; therefore v' = -^— + 1 = - ; therefore y= I )*; c c \c / 1 1 yc — a a c — ac^ , .\ a therefore x = = — r j- = {acy -^ 1-2/ c* - a"* a** - c 1 • 4 {x + a + cY -'r{x + h + dY n ^ ^' Let a + h = a-h (i^ therefore a = \ [a + h + c ■¥ d), c+d = a-l3j ' /3 = ^(a + b-c-d), let a + c = ttj + /?j| therefore a^ = ^(a -¥ h + c + d) = a, h + d^a^-(3j' P^^ila-b + c-d). Hence by assuming x + a = i/, (1) may be put into the shape (y + py + (7j-(3f _m 9/ + 107/(3' -^Syl^' _m or 2/' (^^ -m)-^l Oif {iiP' - mp^') = 5 (mj^^* - nfi') (2), M'hich is a common (piadratic equation. j^ m /3^ {a-\-h — c — dy n^ f^^ {a-h + c- d'f ' (2) takes the form y' = 5/3-/3,' j therefore y = (5)^ {(3^,)^ or x = y-a = ^[0^ J {{a - d'f - (b - c)''} -(a+b + c + d)]. MISCELLANEOUS EQUATIONS. 489 16. x^ + a' + y' + h' = sj^{xia + y)-h{a-y)}, x'-a'-y^+h^ = J2{x{a-y) + h{a + y)]. Adding and subtracting, ic* + &" = J'2(ax + by) (a), y- + a^ = J'2{xy-ab)-j multiplying together, (x^ + b') (y^ + a^) = 2 (ax + by) {xy - ab), or (ax + byY + {xy - abf = 2 {ax + by) {xy - ab) ; therefore ax + by = xy - ab ; therefore y = a y . Substituting in (a), 2 ,o ,^ r bx + b^) ,^x^ + b^ therefore, neglecting the impossible root, x — b-aJ2y therefore x = a J2 + b, and y = ci' — — 1 = b J 2 + a. X — ^ 17. {d" ^y' + c')^ + {x-y + cf^ = 2 {ixy)^ (1), '-^W (2). y X c ^ Since {x-y -^ cy = x~ + y^ + c^ - 2xy + 2xc - 2yc, and from (2) xc — xy — yc = (a) ; therefore {x-y +cf = x^ + y^ + c* ; therefore (1) becomes, {x-y + c)- = ixy = 4c {x - y) from (a) ; therefore (^x — y—c)' = ^; therefore y — x-c, but y = — '■ — : therefore x'^' - c^ = cx : ther(;fore x^ - ex + - = — : ic + c ' 4 4' c c therefore a; = ^ (1 ± ^.5), and y = - (- 1 ± Jo). 490 MISCELLANEOUS EQUATIONS. 18. 2{x''-\-xy + y'-a') + J^{x'-y'')-0 (1), 2{x' -xz + z' -h') -^ J^{x' - z') = (2), 2/^-c^+3(2/«^-c^) = (3). Multiplying (1) by 2 it becomes Z{x^yy + {x-yy +2 J^{x' -y') = ice ; therefore J^{x + y)-¥ x-y = ^2a. Similarly from (2) J2> {x-z) +x + z =^ 2b. By subtraction we obtain on the left-hand side (^3 — l){y + z), and on tlie right-hand side =^2{a-b) or ± 2 (a + 6) j thus we have four values for y + z: choose any one of these and denote it by m. From (3) 2^/ + Gyz' = 8c', that is (y + zf +{y- zf = 8c' ; therefore {y - zf = d>c^ - m^ ; therefore 2/ - - = (^c" - m^f ; therefore y = \{m^ (8c' - m')% and z^\{m- (8c' - m'f). And x{J^+l]^^2a-y{J^-l} = =fc 2a - '^^f^ {m + (8c' - iiff) ; thus x is known. 19. 3a; + 3?/-;s = 3 (1), ^^^,f.,^J±lh (2), x^ + y^ + z"" = ZxTjz + j~~ (3). From (1) ^{x-^y + z) = iz+^ (a), From (2) x' + y' + z' ^ 2z' + 1 -"^-^ (/S), 17?; 4-44 From (3) 2 (x' + y' + s' - Sajy^) = ^ (y) ; MISCELLANEOUS EQUATIONS. 491 then multiplying (a) and (/?) together and subtracting (y), we have x^ + y^ + z^ + 3 {x^y + xy^ + xz^ + x^z + y^z + yz^ + Qxyz = ^z^-V2z'+(jz-l; or {x^y -^zf = {2z-\Y ; therefore x + y = z — \. From (1) a; + 2/ = o + 1 ; therefore 2; - 1 = ^ + 1 j therefore z = Z ; therefore a; + 2/ = 2, £c+2/=2;H t. = ^ ; therefore 2 (a;' +?/')-(« + y)' = 5 - 4 = 1 j therefore a; - ?/ = ± 1 ; therefore a; = 1 J or - , and 3/ = - or 1 J. ^^^ (ac + l)(a;^+l) _ (6^^+l)(a:y + l) , a;+l 2/+1 ^ ' (f^c+1) (?/- + !) ^ (c" -t- 1) (a;?/ + 1) 2/-1- 1 a;+ 1 ^^ From (1) -= -. ^ (a), ^ ^ xy+l y+ I ac+l From (2) ^^ -=^ — -. ^; ^ ' a-y + l a; + 1 «c+ 1 therefore i^±MQJ) = l^-J^'^f J) . (xy + ly (ac + ly Subtracting denominators from numerators, we have i^-yY i^^-^y ,, ^ x-y a-c ,r,\ ~ ^ = 7^ Tw ; therefore ^ = ± (^) ; {xy + 1)' {ac + \y ' a;y + 1 ac +1 ^'^ . 1 /» , , s rt - c , , , c - a therefore — 2,' = (xy + 1) , or {xy +1) ^ ; therefore using the first value and calling — — y- = w, a — c «c + 1 we have y (1 + riix) = x — m ; therefore 2/ — i — x — m rax 492 MISCELLANEOUS EQUATIONS. x^ + l xy+l a^ + 1 ^ Now from (a) jc+l y+l ' ac+l' x^ — rax - + 1 X :^ + 1 a' + 1 I + mx a' + 1 a:' + 1 therefore ^ ^ ^ - ^^ ^ ^ . ^_^^^ ac+ 1 * 1 +maj + ir-??i ' h i 1 +mx therefore {ac + 1) (1 + mx + x-m) = {a'' + l){x + l) ; cr I +ac + x {a- c) + x (I + ac)- {a- c)={a^+ I) + x (a^ + 1) j therefore a; (a - c) - aa? (a - c) --- a (a - c) + (a - c) ; 1 /. 1 + a therefore a(l-a) = l + a; therefore x = -— — ; 1 +a a—c x — m 1 - a 1 +ac _1+^ •^ ~ 1 + mx ~ ' (1 +a) (a-c) 1 - c * (l-a)(l +ac) Similarly, if we use the negative sign in (/?), we have -= — 1 - c and -z for the correspondins; values of x and ?/. 1 + c 21. (27/- l)(a;^ + 4x + 3)^ - {2x - 1) (?/* + iy-\- 3)^ = (a; -y) (re + 2/ -25^2/ + 4).. (1), ,a:y-iy V \2x-lJ x+l ^ '' From (1) (2y - 1) (x" + 4ic + 3)^ - (2aj - 1) (3/' + 43/ + 3)^ = x^ — y^ — 2x^y + 2xy'^ + ix — i^ = y'{2x--\)-x'(2y-l) + 2{2x-l)-2 (2y - 1> = (2/»4-2)(2a;-l)-(a^^+2)(22/-l); therefore (2y - 1) {x' + 2 + J{x' + 4.X + 3)} = {2x -l){f-+2 + J{y' .-¥ iy + 3)}, x' + 2^-J{x'^Ax^Z) y--\-2 + J{7/ + iy + ^) . . ""' 2^31 2y^l ('^>' Kow x* + ix-¥?> = {x^^2x + 1) {x^ -2x + Z) = 'Uv if u = x^ + 2x + \ and v = x'' —2x-¥2> ; tliercfore u + v = 2 {x^ -{■ 2) and i(.-v = 2 {2x - 1). MISCELLANEOUS EQUATIONS. 493 Hence (a) assumes the form ^3 u — v 1 where u^ — y-+2y-\- 1, and v^ = 'i/ -2y + 3 ; t]n,s ^'^±^^ = >i±4^ ; therefore ^ = '^ ; rc" + 2x + 1 _ 2/^ + 2;/ + 1 ^ a;^"^2xT"3 " y'-2y+'i ' 'ad fraction, we have therefore adding and subtracting the numerator and denominator of each therefore Syx^ + 4y - a;' — 2 = 2:^?/^ + 4aj - 2/' - 2 ; therefore 2yx {x — y)-(o(^ — y') — i (x — y) = ()j therefore x = y ; or 2xy = aj + y + 4, so that y = -^ zr . Substituting the value y = x in (2), we have V (^^) " ^' ^'' ^^^ " ^ ' *^^®^*^^^^'® a? = 11 and 2/ = 1^. . . .„ cc+4 . 2/ + 1 3(a;+l) 3 Again, if 2/ = ^r- — - , then ^ - - 2a;- 1' xy-l (x+lf x + l' 2y-l 9 . y + 1 3 thus ^ — ^ == -7; z-r-. , and '^ - 2aj-l (2a;-l)=' a;+l 2a;-l' Hence equation (2) becomes v w+i) ~ 2^^ " 2^^ ' ^"^ V v^nj " 2^^ ' 1 12 therefore = — — ^ , or 4a;^ - 1 Ga; = 1 1 : a; + 1 (2a; - 1)^ ' ' therefore 4a;' - 1 6a; + 1 6 = 27 ; therefore 2a; - 4 = ± 3 ^3 ; therefore a;=i(4±3V3); ^^^ 2/ = |^ = 2 (r^Ts) ' 494 EXMIPLES LIV. MISCELLANEOUS EXAMPLES. 1. Solve J{l+x')-J{l-x') = Jil-x*). 2. Solve x"" {h-y) = ay {y - ?i), y^ (a-x) = bx (x — n). 3. If x"" + xy + y'^ = c% 2 . . 2 1,2 X -r XZ + Z =0 , y' + yz + z'=- a% prove tLat xy-¥yz + zx= Vj^ (2cr6^ + 2h'c' + 2c'a' -a'-h*- c')\ ; and shew how to solve tlie equations. 5. Determine c so that bx + iy = c may have ten positive in- tegral solutions excluding zero values, and c may be as great as possible. 6 If — r- = -^ r and x, y, z be unequal, then each xil-yz) y(l-xz) ^ — xy member of this equation will be equal to — t^j ;-^ , to x-¥y -^z, Z\\.— xy^ 1. 1 1 1 and to - + - 4- - . X y z 7. Shew that if n and N are very nearly equal, ( — I = -1^ + — \ "^^ry nearly, \n) N+n 4?^ ^ ^' -, (N'-nY and that the error is approximately g /^y . \3 • 8. A man's income consists partly of a salary of j£200 a year, and partly of the interest at 3 per cent, on capital, to which he each year adds his savings ; his annual expenditure is less by £95 than five-fourths of his income : shew that whatever be the origi- EXAMPLES. LIV. 495 nal capital its accmniilated value will approximate to £6000. If the original capital be £1000, shew that it will be doubled in about thii'ty years ; having given log 2 = -301030, log 397 - 2'598790. 9. If 71 be a positive integer, and c = , ,^2 j shew that , , _, (7i-2){n-3) „ (7^-3)(7^-4)(7l-5) 3 \-(n-l)c+ ^ 'j- c- - ^ -^^~ —^ c^+ ... 1 m— 1 (m + 1)"* 10. If aj be any prime number, except 2, the integral part of (1 + V^)') diminished by 2, is divisible by 4a;. 11. If any number of integers taken at random be multi- plied together, shew that the chance of the last figure of their product being 5 continually diminishes as the number of integers multiplied together increases. 1 2. Two purses contain sovereigns aiid shillings ; shew that if either the total numbers of coins in the two purses are equal, or if the number of sovereigns is to the number of shillings in the same ratio in both, then the chance of drawing out a sove- reign is the same when one purse is taken at random and a coin drawn out as it is when the coins are all put in one purse and a coin drawn out. If neither of these conditions holds, the chance is in favour of the purse taken at random whenever the purse with the greater number of coins has the smaller proportion of sovereigns. LY. MISCELLANEOUS PROBLEMS. 760. We have already given in previous Chapters collections of problems which lead to simple or quadratic equations ; we add here a few examples of somewhat greater difficulty with their solutions. 1. Each of three cubical vessels A, B, C, whose capacities are as 1 : 8 : 27 respectively, is partially filled with water, the 49 G MISCELLANEOUS PROBLEMS. quantities of water in them being as 1 : 2 : 3 respectively. So much water is now poured from A into B and so much from B into C as to make the depth of water the same in each vessel. After this 128i cubic feet of water is poured from C into B, and then so much from B into A as to leave the depth of water in A twice as great as the depth of water in B. The quantity of water in A is now less by 100 cubic feet than it was originally. How much water did each of the vessels originally contain ] Let X - the number of cubic feet in A originally j therefore 1x — the number of cubic feet in B originally ; and 3a; = the number of cubic feet in C originally. Now when the depth of the fluid is the same in all, it is clear that the quantities vary as the areas of the bases of the vessels, that is, are as 1 : 4 : 9. Therefore, since ^x is the total quantity, the 6iB OX quantity in ^ = -z — ^ = ~= , and the quantities in B and C are I2x , 21 X ^, , —— and -^r- respectively. Again, when the depth in A is twice that in B, the quantity in A is half as much as that in B. Now A contains cc-lOOj therefore B contains 2(a;-100), and C contains — 128-|^. therefore 3 (^- 100) + ^ - 1284= 6jc; therefore ^ = 300 + 128^; 900 7 therefore x = 350 + — x - = 500 ; therefore the quantities in A, B, C at first were 500, 1000, 1500 cubic feet respectively. 2. Three horses A, B, G start for a race on a course a mile and a half long. "When B has gone half a mile, he is three times MISCELLANEOUS PROBLEMS. 41)7 as far ahead of A as lie is of G. The horses now going at uniform speeds till U is within a quarter of a mile of the winning post, C is at that time as much behind A as A is behind B, but 1 th the distance between A and B is only yz of what it was after B 1 ■■'• had gone the first half mile. C now increases his pace by -^^j- of Do what it was before, and passes B 17Q yards from the winning post, the respective speeds of A and B remaining unaltered. What was the distance between A and C at the end of the race 1 Let llaj = the distance in yards between B and C at the end of the first ^ mile, 33^; = the distance in yards between B and A at the end of the first ^ mile. When B has gone 1^ miles B is 3x ahead of A, and 6x ahead of C ; therefore while B went | mile or 1320 yards, A went 1320 + 30x yards, and C went 1320 + 5a; yards. Hence, after C increases his pace, the speeds of A, B, C will be 54: proportional to 1320 + 30a;, 1320, and — (1320 + 5a;) respectively. Do ' Since C passes i> when he is 176 yards from the post ; therefore wliile B was going 440 — 176 or 264 yards, C went 264 + 6x ; 54 therefore 1320 : ^(1320 + 5a;) :: 264 : 264 + 6a;, therefore 1 320 + 30a; = -^ (1320 + 5a;) ; therefore x (1590 - 270) = 1320 ; therefore a; = 1 ; also it will be found that C's increased pace is equal to A's ; therefore there will be the same distance between them at the end of the race as there is when B is ^ mile from the ^\dnning post, namely 3a; or 3 yards. 3. A fraudulent tradesman contrives to employ his /alse balance both in buying and selling a cei*tain article, thereby gaining at the rate of 11 per cent, more on his outlay than he would gain were the balance true. If, however, the scale-pans in T. A. 33 498 MISCELLANEOUS PROBLEMS. which the article is weighed when bought and sold respectively, were interchanged, he would neither gain nor lose by the article. Petermine the legitimate gain per cent, on the article. Let IV and to^ be the apparent weights of the same article when bought and when sold. Let 2^ — the prime cost of a luiit of weight, X = the legitimate gain per cent. ; then an article which cost pw is sold fov w(p + y— j ; therefore by the question iv^ (p + -^J - icp = ^ --^-|^ — (1). Again in the supposed case the cost of the article =pw^ and the selling price = ^;w? f 1 + j^j ; therefore pt^i =pw f 1 + -^ j , (2). rrom(l), ^.,(n.^)=^.(l+^);. from (2), w(l + ^^j = lo^ ; / X \^ .r + 1 1 therefore (^1 + __j = 1 + 1^^__ ; therefore re' + 100:c = 1100, so that (a; + 50)' - 3600 ; therefore a; + 50=±60j therefore a = 10. 4. A person buys a quantity of corn, which he intends to sell at a certain price ; after he has sold half his stock the price of corn suddenly falls 20 per cent., and by selling the remainder at this reduced price, his gain on the whole is diminished 30 per cent. ; if he had sold fths of his stock before the price fell, and the diminution in the price had been in the proportion of £20 on the prime cost of what he before sold for £100, he would have gained by the whole as many shillings as he had bushels of MISCELLANEOUS PROBLEMS. 499 corn at first. Find what the corn cost him per bushel, and what he hoped to gain per cent. Let X — the cost price, in pounds, per busliel, y = the gain per cent, he exj)ected ; theii X |*(i + ( 1 + — — : j = the price per bushel for wjiich he sold half his corn ; ( 1 + jttt: 1 = the price per bushel for Avhich he sold the other half; therefore the average price per bushel — — (l + y^] ', therefore Hs gain per bushel = — M + ^ - j - x. If he had sold the whole as he sold the first half, the gain per bushel would have been ykk '> therefore by the question y^\1 + ^ j " ^ " 10 UR) ' y 1 therefore -^tt = -^ : therefore y = 50. 500 10' ^ Now the prime cost of what he at first sold for 100 is - 100 that is -— — , and if he were to lose .£20 on tliis, the loss per cent, o would be — , that is 30. ^UO Thus in the supposed case the average selling price of a bushel X 111 31a; gain on a bushel = t x -^^ x = — — , and this by the question Six 1 4 equals one shilling ; therefore -^^ = ^^ ; therefore a: = 7-- . oU JO ol 32—2 500 MISCELLANEOUS PROBLEMS. 5. A and B having a single horse travel between two . mile- stones, distant an even number of miles, in 2j-| hours, riding alternately mile and mile, and each leaving the horse tied to a mile-stone until the other comes up. The horse's rate is twice that oi B I B rides first, and they come together to the seventh mile-stone. Finding it necessary to increase their speed, each man after this walks half a mile per hour faster than before, and the horse's rate is now twice that of A^ and B again rides first. Find the rates of travelling, and the distance between the extreme mile-stones. Let 2x = the distance they travelled in miles. Kow at first A walks 4 miles and rides 3 miles while B walks 3 miles and rides 4 miles, or A walks 4 while B walks 3 and rides 1 j that is, since the horse's rate is double of ^'s, while B walks 31 miles; therefore -4's and ^'s rates at first may be represented by 8y and 1y respectively. Again, A walks iK — 3 and rides a; — 4, while B walks a; — 4 and rides £c - 3 ; therefore A walks x-Z while B walks cc - 4 and rides 1, that is, while B walks cc - 4 and A walks ^ ; therefore 7 A walks X- ^ while B walks £c - 4 j but A walks 8^/ + o while B walks ly + -^; therefore ^—^ = j- , from which y = 1^-33^ • Now the total time A took in hours is 82/+2 2(82/4-2) therefore ^ + t^tt^-.^ 2f j ; 5 3aj-10 7y "^IgTTI ,, ^ 5 3a: -10 188 1 therefore ^ -f -; =-r = -^rr^ x 7 4a; -14 63 4a; -30' therefore MISCELLANEOUS PROBLEMS. 501 4]rc-140 94 1 4a;- 14 9 2a;- 15' therefore 9 (82a;' - 895a; + 2100) - 376a; - 1316 ; therefore 738a;' - 8431a; + 20216 = 0, from which a; = 8 ; therefore y = ny therefore the distance =16 miles ; the rates of travelling at first = 4 and 3^ miles per hour respectively. 6. A and B set out to walk together in the same direction round a field, which is a mile in circumference, A walking faster than B. Twelve minutes after A has passed B for the thii^d time, A turns and walks in the opposite dii^ection until six minutes after he has met him for the thii^l time, when he returns to his original direction and overtakes B four times more. The whole time since they started is three hours, and A has walked eight miles more than B. A and B diminish their rates of walking by one mile an hour, at the end of one and two hours respectively. Determine the velocities with which they began to walk. Let X = the number of miles per hour of A at the first,^ y = the number of miles per hour of B at the first. In 3 hours A has gone a; + 2 (a; - 1) or 3x - 2 miles, and B has gone 2?/ + (?/ - 1) or 3y — 1 miles ; therefore by the question 3a;-2-{3?/-l) = 8; therefore x-7/=3, that is, the relative speed of A and ^ is 3 miles per horn- ; therefore A will gain a circumference on J5 in J of an hour, and will therefore be passing B for the third time at the end of the first hour. Also since the relative speed of A and B is the same in the last hour as in the first, and since A passes B for the fourth time at the end of the third hour, therefore he will pass him all the four times within the last hour ; the first time being exactly at the commencement of the third hour. 502 EXAMPLES. LV. Now in 12 minutes after the fii*st hour the distance between 1 2 A and Ij is ^ (a; — y — 1) = > miles ; therefore the time of fii-st 2 meetmg =p-7-(a; + y — 1); and the time of meeting ttvice more = 2 -i- (x + y - I). In G minutes the distance between them ^Yni^ + y — ^)'} therefore if A now turns, the time of over- taking B . . 1 5 2 11, -,, , therefore -= + . + — r +t7T + o7^(^ + 2/-1) = 1j therefore u^-142«=-48; therefore w-7-±l ; therefore u = S or 6; therefore x + y=9ov7; and a; - ?/ = 3 ; therefore a; = 6 or 5, y = 3 or 2. 761. The equations in the preceding Chapter and their solu- tions, and the solutions in the present Chapter, are due to the Rev. A. Bowei', late Fellow of St John's College. Should any student desire more exercises of this kind, he is referred to the collection of algebraical equations and problems edited by Mr W. Rotherham of St John's College. MISCELLANEOUS EXAMPLES. 1. Exhibit [71 J{a^ + h') — a J{ni^ + n^)}' + h^ni^ as a square. 2. Extract the square root of 6 + JG + ^14 + ^^21. 3. Find the radix of the scale of notation in which the num- ber 1G640 of the common scale appears as 40400. 4. Shew that 7 + o + i-7r+?rs + ......at? inf. = 2. 4 8 16 015 EXAMPLES. LV. 503 5. At a contested election the number of candidates was one more than twice the number of persons to be elected, and each elector by voting for one, or two, or three, ... or as many persons as were to be elected, could dispose of his vote in 15 ways: required the number of candidates. 6. In how many ways may the sum of £24:. 15s. be paid in shillings and fraijcs, supposing 2G francs to be equal to 21 shillings? 7. Find the sum of n terms of the series 1 z z' l + z (l-^z){l+z') (l + z){l+z'){l+z') :^ "^(1+^) (l+z') {l+z') (1+;^')"*' 8. Shew that 1 + 2x^ is never less than x^ + 2x^. 9. If an equal number of arithmetic and geometric means be inserted between any two quantities, shew that the arithmetic mean is always greater than the corresponding geometric mean, 10. If a; be any prime number, except ^2, the integral part of (2 + JSy - 2^"-' + 1 is divisible by 12a;. 11. Shew that if n=pq, where ^ and q are positive integers, \n is an integer. 1 2. Shew that t + t^ + o -^ lop^ n is finite when n 1 2 o n is infinite. 13. If p be the probability d, priori that a theory is tnie, q the probability that an experiment would turn out as indicated by the theory even if the theory were false, shew that after tlie experiment has been performed, supposing it to have turned out as expected, the probability of the truth of the theory becomes P p^q-pq' 14. Of two bags one (it is not known which) is known to contain two sovereigns and a sliilling, and the other to contain one sovereign and a shilling \ a person draws a coin from one of 504 EXA3IPLES. LV. the bags, and it is a sovereign, wliicli is not replaced. Shew that the chance of now drawing a sovereign from, the same bag is half the chance of doing so from the other. Supposing the drawer might keep the coin he draws, find the value of the expectation. 15. All that is known of two bags, one white and one red, is that one of them, but it is not known which, contains one sovereign and four shilling ]>ieces, and that the other contains two sovereigns and three shilling pieces ; but a coin being drawn from each, the event is a sovereign out of the white bag and a shilling out of the red bag. These coins are now put back, one into one bag, and the other into the other, but it is not kno^ni into which bag the sovereign was put. Sliew that the probability of now drawing a sovereign is in favour of the red bag as compared with the white bag in the ratio of 1 3 to 9. 16. If n be the number of years v/hich any individual wants of 8G, find the value of an annuity of £1 to be paid during his life ; adopting De Moivre's supposition, that out of 86 persons born, one dies every year until they are all extinct. LYI. CONVEEGENCE AND DIVERGENCE OF SERIES. 762. In Chapter xl. vv^e have discussed the subject of the convergence and divergence of series. The chief general result which has been obtained may be expressed thus : an infinite series is convergent if from and after any fixed term the ratio of each term to the succeeding term is greater titan some quantity which is itself numerically greater than unity ; and divergent if tliis ratio is unity or less than unity, and the terms are all of the same sign. There is one case to which this result does not apply, which it is desirable to notice, namely the case in which the ratio is greater tha,n unity but continually approacliing unity. See Arts. 559, 560 and 561. The statements of those Ai-ticles are here reproduced, but in a different form, as for our present pur- pose it is convenient to regard the ratio of a term to the suc- ceeding term instead of to the 2J7'eceding term. CONVERGENCE AND DIVERGENCE OF SERIES. 505 763. We sliall now investigate theorems which will supply tests of convergence and divergence for the case to which the former tests do not apply. In the infinite series which we shall consider we shall suppose that all the terms are positive, at least from and after some fixed term if not from the beginning. 764. A series is convergent if from and after some fixed term the ratio of each term to the succeeding term is never less than the corresponding ratio in a second series which is known to he convergent. It is obvious in this case that the proposed series is not greater than a certain convergent series ; and is therefore con- vergent. 765. A series is divergent if from and after some fixed term the ratio of each term to the succeediyig term is never greater than the corresponding ratio in a second series which is Icnown to he divergent. It is obvious in this case that the proposed series is not less than a certain divergent series ; and is therefore divergent. 766. Let u^ denote the n^^ term of a series ; then if from and after some fixed value of n the value of n i -^ 1 j is always greater than some positive quantity which is itself greater than unity, the series is convergent. Supj^ose that from and after some fixed value of n the value of n ( — 1 ) is always gi-eater than y, where y is ix)sitive and u v greater than unity. Then — ^ — 1 is gi-eater than - ; and there- n fore — ~ IS gi'eater than 1 + — . u ^, ° n Now, by Art. (^^(^, a positive quantity p greater than unity can be found, sucli that when n is larcje enough I ) is less 506 CONVERGENXE AND DIVERGENCE OF SERIES. u than 1 + - . Hence, wlien n is large enongh, — ^ is greater n ^<„ + , than f^^^Y- But, by Art. 562, the series of which the n"-^ term is — is convergent when p is positive and greater than imity • hence by Art. 761 the series of which the n*** tenn is u is converf^ent. 767. Let u^ demte tJie n^^ term of a series; then if from and after some fixed value of n tlie value of n ( — ^ jwsitive and greater than unity, the series is divergent. - — 1 IS never u For here after some fixed value of n the value of — "- is 1+1 equal to 1 + - or is less than 1 + - . But, bv Art. 562, the series ^ n Qi " of which the ro-^ term is - is divergent; hence, by Art. 765, the n series of which the n^^ term is u^^ i& divergent. 768. The rules given in Arts. 766 and 767 will often enable us to decide on. the convergence or divergence of series in the case noticed in Art. 762 in which our former roles do not apply. There is one case to which the new rules will not apply, which it is desirable to notice, namely that in which from and after some fixed value of n the value of n ( —^ — 1 ) is alwavs positive and gi'eater than unity, but continually approaching unity. We shall proceed to investigate theorems from which we shall deduce tests for this case. 769. It is obvious from the nature of a logarithm that if n increases indefinitely, so also does log n. But it is important to observe that log n increases far less rapidly than n increases ; in fact — ^ — can be made as small as we please by taking n large n enough. For suppose n = e"", so that Xogn^x; then as n increases CONVERGENCE AND DIVERGENCE OF SERIES. 507 indefinitely, so also does x. Now — ^— = — /p . 1 tliis is less than .. , tliat is less than X- X ^ X x' ' |2 |_3 |2 [3 and it is obvious that this can be made as small as we please by taking n large enough. These remarks will be found useful in studying the remainder of the present Chapter, "We shall adopt the following notation for abbreviation : let log n be denoted by \ (n) ; let log (log n) be denoted by X' {n) ; let log {log (log ?i)} be denoted by X^ (n) ; and so on. 770. The series of which the general term is nX {n) X'{n) A'" {n) {KT^^ {n) ] ^ ^^^ is convergent i/\) he greater than unitij^ and divergent if "^ he equal to unity or less than unity. AYe suppose n so large that )C^^ (n) is possible and positive. The truth of this theorem when r = has been shewn in Art. 5G3 ; we shall prove it generally by Induction, By Art, 563 the series of which (1) is the general term is convergent or divergent simultaneously with the series of which the general term is (2), mU (m") X'' (7/i") \'-{/jt") {A''+^ (wi")}^ where m is any positive integer. I. Suppose p greater than unity. Let 7n be any positive in- teger greater than the base of the Napierian logarithms ; then X (m") is greater than n. Hence it follows that the general tenn (2) is less than ^ -(3); nX{n)X'{n) X"-' {n) {X" {n)\^ thus by Art. 764 if the series of which (3) is the general term is 50S CONVERGENCE AND DIVERGENCE OF SERIES. convergent, so also is that of which (2) is the general term, and so also is that of which (1) is the general terni„ Therefore if the series of which (3) is the general term is convergent when r has any si)ecLfic value, it is convergent when r is changed into ?' + 1 . But since p is greater than unity, by Ai*t. 5G3 the series of which (3) is the general term is convergent when r= 1, and therefore when r = 2, and therefore when r = 3, and so on. Thus the series of which (1) is the general term is convergent. II. Suppose ^9 equal to unity. Let ??^ = 2 which is a positive integer less than the base of the Napierian logarithms ; then \ {m") is less than oi. Hence it follows that the general term (2) is greater than 1 nX{n)X'{n) y-'(oi)y (n)' Hence by proceeding as in I. we can shew that the series of which (1) is the general term is divergent. III. Suppose p less than unity. Then the general term (1) is greater than it would be if p were equal to unity, at least when n is large enough, and therefore ct, fortiori the series is divergent. A simple demonstration of this theorem by means of the Integral Calculus is given in the Integral Calculus, Chapter iv. 771. Let 11^^ denote the general term of any proposed series. If from and after any value of 7i the value of un\ (n) \' (n) ^ {n) {X'^i {n)Y is always finite, 2> being any fixed quantity greater than unity, the proposed series is convergent. For in this case the terms of the proposed series have a finite ratio to the terms of a series which has been proved to be con-- vero-ent. If from and after any value of n the value of u^^ n\ (n) X'(n) X*" (n) X"^' (n) is always finite or infinite, the proposed series is divergent. For in this case the terms of the proposed series have at least a finite ratio to the terms of a series which has been proved to be divergent. CONVERGENCE AND DIVERGENCE OF SERIES. 503 772. The theorem of Ai-t. 771 may be used in cases in which the tests already given of convergence and divergence do not ^Pply ; but it will in general be more convenient to use the rules which we shall demonstrate in the next Article. 773. Let V ^ stand for i\i — "^ — 11/ tJcen if from and after X'-'-u + i / some fixed value of n the value of X (n) (P^ — 1) is always greatet than some positive quantity which is itself greater than unity the series of ivhich the n'^ tervi is u^ is convergent; and if from and after some fixed value of n the value of X (n) (P^ — 1) is never 2yosi- tive and greater than unity the series is divergent. I. Suppose that from and after some fixed value of n the value of X(n)(^P^— 1) is always greater than y, where y is posi- tive and ffi*eater than unity. Then P, - 1 is greater than , , , ; ^ -^ & . X{n) 7/ 1 'V therefore - — ~ is greater than In 1 — tA-t • Let . =^-L_ . then -^ = '-^ i^J^Y. " n{X{n)Y' v,^^^ n \ X{n) J Now X(n + 1) = X {n) + x(l + -j; therefore X (n + 1) is less 1 v than X (n) -\ — by Ai-t. 687 : and therefore — '— is less than n + \ (1 + -) |l + ^ , . t ; and therefore when n is large enough \ n/ { nX[7i)) — ^ is less than (1 + -] \l + , , , >, provided <7be greater than p : see Aii;. 686. Thus -^ is less than 1 + - + ~j-~. + „^ , . : ^ v^^., n 7iX(n) nX{n) and when n is taken large enough the last of the four terms jnst V given is incomparably smaller than the third ; and therefore — — n +1 1 r is less than 1 + - + . ^ , , provided r be greater than q, n nA in) 510 CONVERGENCE AND DIVERGENCE OF SERIES. This result holds however small may be the excess of q above p, and however small may be the excess of r above q : hence since y is greater than unity we may suppose that y is greater than r, and yet have 'p positive and greater than unity. Since y is greater than r we have — "- greater than — ^ . But, by Aii;. 770, the series of which the general term is v^ is con- vergent when 'p is positive and greater than unity ; hence, by Art. 764, the series of which the ^i*'^ term is u^^ is convergent. II. Suppose that from and after some fixed value of n the vahie of \{n){F^-\) is never positive and gi-eater than unity. Then P^ — 1 is positive and not greater than , , . or is negative. u . 11 In both cases -"^^ is less than In v Let v = — : then — ^ = ■ ^ ^ ^ ^ . " nX[n) v^^j Qi X(n) Now A. (;i + 1) = X (?i) + X ( 1 + - j ; therefore X {ii + 1) is greater 11 V . than X (n) -\ ^r—c, by Art. 688 : and therefore — ^ is sfreater than ( 1 + ) s 1 H r-r^ — r> 9x / x \ j ^-^id therefore when n is \ nj I nX{rt) 2nX{n))' V . 11 large enough — "^ is gi-eater than In h v^^j 71 nX(7i)' u . V Thus when ?^ is lar^je cnou2;h - ""^' is less than — ^ . But, by n+2 n+l Art. 770, the series of which the general term is v^ is divergent ; hence, by Art. 765, the series of which the 9i*^ term is ic^^ is divergent. 774. The theorem of Aii;. 773 does not apply to the case in which X (oi) (P^ — 1) is always positive and greater than unity, but continually approaching luiity ; another theorem may then be used which also is inapplicable in a certain case. A series of theorems can thus be obtained each of which may be advanta- CONVERGENCE AND DIVERGENCE OF SERIES. 511 geously tried in succession if all that precede it are inapplicable. The theorems will be found in the Integral Calculus, Chapter iv. ; thej might be demonstrated in the manner of Art. 773, but as they will not be required for elementary purposes we need not consider them here : as an exercise for the student the theorem which is next in order to that of Art. 773 is given as the last Example in the set at the end of the present Chapter. "We shall illustrate the rules which have been demonstrated by applying them in the next three Articles. 775. The name hyper geometrical has been given to the series g./? a(a4-l)^(^ + l) ^, , g (a+ l)(a+2) /3(^+ 1)(/3 + 2) ^^3 , , ^"^l.y 1.2.y(y+l) ^ "^ 1. 2 . 3 . y(y + l)(y + 2; ' we shall now determine when the series is convergent, and when divergent. Denote the series by u^ + it^-\- u^ + u^-\- ...] tlius ('1 + 1V1 + } n^^ [n + \){n + y) \ uj \ n {n + a){n + ^)x A _^g\ /^ _^ ^\ X thus, by Art. 762, if a? is less than unity the series is convergent, and if X is greater than unity the series is divergent. Put x = 1 ', then / .S / |-;^ \ oiJ\'^ n) thus if y — a — (3 is positive the series is convergent, and if y — g — ^ is negative the series is divergent : see Arts. 766, 767. Ify — g — /5 is zero we must use Art. 773 ; we have then 1 + 71/ \ nj this can be made as small as we please by taking ?i large enough, and therefore the series is divergent. 512 CONVERGENCE AND DIVERGENCE OF SERIES. ^^ „ 1 , w 01^ + an^~'^ + hn^'"^ + cn^~^ ■¥ . . . , 77G. Suppose that — ~ = -r — j^-j-- — , where yfc is a positive integer, and no exponent is negative ; and a, h, c, ... A, B, G, ... are any constant quantities : we shall shew that the series of which the ?*"' term is u^ is convergent, i£ a — A—1 is positive, and divergent if a — ^ — 1 is negative or zero. „ (« - A) n' + (h-B) oi'-' + (c- 0) n*-2 + . . . Here i\-- thus iia—A-\ is positive the series is convergent, and if a— ^1 — 1 is negative the series is divergent : see Arts. 766, 767. If a —A — \ is zero we have _ n'' + (b-B)n''-' + ... ^ '~ 7i'' + An''-' + Bji'-% ,..' we may still in some cases determine whether the series is con- vergent or divergent without using any new rule, for instance if b —B — A is negative the series will be divergent by Art. 7G7. But it will be more convenient to use Art. 773 ; we have then XUUP n - ^(n){{b-B-A )n ^-^ + {c-C-B)n^-^-,...} . this can be made as small as we please by taking n large enough, and therefore the series is divergent. 777. We shall now examine the expansion of (1 +x)"' by the Binomial Theorem and determine whether it is convergent or divergent when a; = 1 or — 1. Let u^ denote the r*^ term in the expansion of (1 + xf ; then ^r + l + '^'r + 2 + "^r + a + _ Cm — r + 1 (m — ?• + 1) (m — 7') g \ '^ { 7' r (r + 1) j We must then consider the series included between the brackets. I. Suppose x:=l. Let r be numerically not less than m; then the terms of the series between the brackets are alternately positive and negative. EXAMPLES. LVI. 513 If m is positive, or negative and numerically less than imitj, each term is numeiically less than the preceding term and the series is convergent bv Art.. 558, If m = — \ the series between the brackets takes the form -1 + 1-1 + , which is convergent according to the definition of Art. 554. If m is negative and numerically greater than unity each term of the series between the brackets is numerically gi'eater than the preceding term and the series is divergent. II. Suppose a; = - 1. Then the series between the brackets is r -711 — 1 (r — m—l){r-7n) (r — 7?i — l)(r — m)(?'--m+ 1) "^ '/ T~^ ^ 7 TV7 ' — c^ + . . ., 7* r(r+l) r(r -f l)(r + 2) Let r be numerically not less than on ; then the terms of this series are all of the same sign. In Art. 775 put a=l, (3 = r — m'-l, and y = ?• : hence we find that the series is convergent if m is posi-- tive, and divergent if m is negative, EXAMPLES OF CONVERGENCE AND DIVERGENCE OF SERIES. 1. Shew how to determine whether the product of an infinite number of factors Wj, u^, u^, W4 ... is finite or not, 2. Shew that the valine when n is infinite of \n n' {x + l)(x+2) ...(x + n) is finite except when a; is a negative integer. 3. Shew that when x is unity the value of u^ in Art. 775 increases indefinitely with ?2, if a + ^ - y - 1 is positi\e. 4. Shew that when x is unity the value of i'^ in Art. 775 is finite when qi increases indefinitely ifa + yS — y — 1 is zero. 5. Shew that when x is unity the value of u„ in Art. 775 is indefinitely small when n increases indefinitely ifa + ^ — y— 1 is negative. T. A. 33 514 EXAilPLES. LVI. 6. Determine whether the following series is convergent or divergent, x being positive : ax + —\- — ^ x' + — ^— tP ic + ... 7. If iin = ""^Ti shew that the series is divergent. 8. Determine whether the fuUowino- series is convei'^rent or o o divergent, x being positive : , X 1 xr I.?) x' 1.3.5 X* 1 -] 1 . — H . 1 . h ... 1 2 3 2.4 5 2.4.6 7 9. Determine whether the following series is convergent or divergent, (^ being a j^ositive proper fraction : .^ ,i8(l-/3) , (l-f^)/3(l-^ )(2-/J) (•2^(^)il+ft)fta-l3) {2-(3){3-/3) - r'.2'. 3^ ?i 10. If Un = -, , X , where 2^ ^''^^ Q ^^^^ positive, determine whether the series is convergent or divergent. 1 1. Shew that if from and after some fixed value of n the value u . of n log — — is always greater than some positive quantity which is itself greater than unity the series is convergent. 12. Shew that if from and after some fixed value of n the value of 71 log — — is never positive and greater than unity the series is divergent. 1 3. Determine whether the following series is convergent or divergent, x being positive : a + x (a+ 2xy (a + SxY ~T~"^"~|2 "^ [3 ■^•" 14. Give an investigation of the results of Art. 775 without using Art. 773. EXAMPLES. LVI. 515 15. Give an investigation of the results of Art. 776 without using Art. 773. y„ u^ n'^ + a'nfi + hny+ ... , ^ .. 1 o. If = —-r — =- where a.p, y, ... are positive w„+i n'^ + A'nfi + J^ny+ ... t-; /? r constants in descending order of magnitude, and a, 6, ... A, B, ... are any constants, determine whether the series of which the ?i*^ term is u^ is convergent or divergent. 17. Shew that the two series u^ + Ui + u,^ + u^+ ... and — i + — + are both convergent or both divergent jw^^, u^, u^y ...being all positiv^e quantities. 18. Let P, stand for \(n){P^-l) ; then if from and after some fixed value of n the value of \^ (n) (F^ - 1) is always greater than some positive quantity which is itself greater than unity the series of which the n^^ term is w„ is convergent ; and if from and after some fixed value of n the value of A.^ (n) (P^ — 1) is never posi- tive and greater than unity the series is divergent. LYII. CONTINUED FRACTIONS. 778. The most general form of a continued fraction is J, a3=fc... Here ctj, a^, a^, ... and S^, h^, h^, ... may denote any quantities, whole or fractional, positive or negative. The simple fractions h h h -' , — , — , ... maybe called components of the continued frac- ^1 ^2 ^3 tion. Either sign might be taken where ± occurs ; but we shall consider only two cases, namely that in which every sign is + , and that in which every sign is — . We shall thus have two classes of continued fractions, which we shall call the first class and the second class respectively. 33—2 516 CONTINUED FRACTIONS. In Chapters xliv. and XLV. we confined ourselves to continued fractions of the first class in which every component had unity for its numerator, and a positive integer for its denominator : but we shall now give some propositions relating to the more general form. 779. The fractions obtained by stopping at the fii'st, second, thii'd, . . . component are called the first, second, third, . . . convergents. Thus the first convergent is -^ : the second convergent is — — ^- , that is — ^-i-r- j and so on. «3 780. In Ai'ticles 781... 785 we shall treat of continued frac- tions of the first class; in Arts. 7 8 6... 79 3 we shall treat of con- tinued fractions of the second class : in all these Articles we shall assume that every component has both its numerator and its deno- minator positive, 'n 1) 'n 781. Denote the successive convergents by — , — , — , ... ^ 9, ^2 9s Then we can shew as in Art. 604 that the successive convergents may be obtained by these laws : Hence ^^' - ^" = - hi^l^"- (^^ 'P^--^ 9n^l 9n 9n + X \9'n ^n-l/ ' and ''"^^ = »+iy>t-i which is a proper fraction. Thus 9'n + l «'n + l9'n + K + iqn-l ^-^ ^ is numerically less than — -- ^-^^ , and is of the con- 9'n + l qn qn 5'n-I trary sign. Now ^-^ _-i^ = — ^ ^^ \ = -^ : and this is positive. Hence q, q^ «i ccxd^ + b^ q,q^ we see that the following series consists oi positive quantities which are in descending order of magnitude : 'Pl_P2 Pi p^ Pi p^ P^ p* Ps P^ ?i q-, 93 9a* 9z 9< q^ q: qs q,' CONTINUED FRACTIONS. 517 This result involves the following facts for a continued fraction of the first class : The convergents of an odd order continually decrease^ and tho conver gents of an even order continually increase. Every convergent of an odd order is greater ^ and every conver- gent of an even oixler is less, than all following convergents. 782. Now suppose the number of components infinite. It may happen that by taking n large enough we can make the dif- ference between the n^^ convergent and the next convergent less than any assigned quantity ; or it may happen that however large n may be the difiference between the n^^ convergent and the next convergent is always greater than some fixed quantity. In the former case the value towards which the odd convergents continually decrease, and the even convergents continually in- crease, may be called the value of the infinite continued fraction : and we shall say that the infinite continued fraction is definite. In the latter case the infinite continued fraction cannot be said to have a single value ; but it may be considered to represent two values, one being that to which the odd convergents tend and the other that to which the even conversjents tend. 783. If from and after some fixed value of r the value of Q O -T~^ is greater than some fixed positive quantity, the infinite con' • tinned fraction is definite. Let y denote the fixed positive quantity. By successive applications of the result in Ai*t. 781 we have Now ^+l^r-l^ ^r.n9^r-l K^lQr-l 1 and this is less than ^j since -'"/■•'- is in-eater than v. 1 + 7 i>r^x ^ ' 518 CONTINUED FRACTIONS. Hence ^-j^±i —^-^ is numerically less than — ,„ ,. , , where ?„., 9n / {i + yy-^y' c is some constant ; and by taking n large enough this may be made less than any assigned value. Therefore the infinite con- tinued fraction is definite. We shew here that the condition stated is sufficient to ensure that the infinite continued fraction is definite j we do not assert that the condition is necessary. 784. An infinite continued fraction of the first class in which every component is a iwoper fraction loith its numerator and V' denominator integral must he an incommensurable quantity. For if possible suppose the continued fraction commensurable, ■n and denote it hy -j , where A and B are positive integers. Thus 7? h -7 = — ' — , where p, denotes the infinite continued fraction be- ginning with the component -^ . Thei'efore p^ — — ^-^ ^ ; the numerator of this fi-action is an integer, which we will denote by C : and C must be positive for p^ is positive. In like manner, if p^ denote the infinite continued fraction beginning with the component — we find that p^ — Yi^ where D is also a positive integer. And so on. Moreover -7 , -^ , ^ , • • ■ must all be proper fractions. For ABC -J is less than — , and this is a proper fraction ; — is less than — , and this is a proper fraction ; and so on. Hence A, B, C, D, ... fonn a series of positive integers, which are in descending order of magnitude, and yet infinite in number : this is absurd. Hence the infinite continued fraction cannot be a commensurable quantity. 785. If some of the components of the infinite continued frac- tion are not proper fractions, hut from and after a certain component CONTINUED FRACTIONS. 519 all the others are proper fractions the injinite continued fraction is incomTnetisurable. For suppose that -^^^ and all the subsequent components are proper fractions, then by Art. 784 the infinite continued fraction beginning with -^^^ is incommensurable ; denote it by x. As in Art. 781 we have and the value of the infinite continued fraction will be obtained by chan^mcj a^ into a„ + £c ; so that it is 7 ^ t-: , tnat isj ^ — ^l^n~\ _ rjTj^-g cannot be commensurable unless — = -^^^ , and this bv aid of the value af — leads to -^^^ =-^1^ ; and so we On 5'n-l <7n_2 should arrive at ^ = ^ . This is impossible, as we cannot have 5, = or 5 = 0. 1 2 786. A contimced fraction of the second class in which the denomiriator of every comjwnent exceeds the numerator by unity at least, has all its convergents positive proper fractions which are in ascending order of magnitude. The first convei^g^it -^ is a positive proper fraction by hypo- thesis. The second converirent is V ; ^"^^ ^^ ~ i^ ^ proper a ^^ «. a — b . .. fraction, and a^ exceeds h^ by unity at least, a^ - - is positive and greater than b^ ; and thus the second convergent is a positive proper fraction. The third convergent may be denoted by ^ a where — ' stands for — ^ , so that — ' is a positive proper fraction a. 0, a. «,- a 526 CONTINUED FRACTIONS. for the same reason that the second convergent Is : hence for the same reason the third convergent is a positive proper fraction* Tlio fourth convergent may be denoted by ^ , where ^ de- «>--" 2 notes a fraction of the same form as the third convergent, which is therefore a positive proper fraction : hence the foui*th convergent is a positive proper fraction. And so on. Again; as in Art. 781 we shall find that the successive con- vergents may be obtained by these laws : p —a r> , — hp „, a —a q ,~hq „. I IX III n — 1 ni. n — i' J-n nin — l nJ.ti — 3 Hence P^-t-^ Lti£^ (P. _ Pj^) ■ thus -^^^ — — is of the same si^n as — — -^-^^ , 5',. + , ^n S',. ^„-l Now i-2 —li = 2_i_ 1 ^_i__a . r^j^(j tj^ig ig positive. ^2 ^1 »1^2-^2 «1 ?l!7« Hence it foUows that -^ . ^ , ■^, form a series of positive 9i ^. 5'3 proper fractions in ascending order of magnitude. 787. If the number of components is infinite the convergents form an infinite series of proper fractions in ascending order of magnitude ; and so the terms will never exceed some fixed value which is unity at most. We may say then that an infinite cou' tinued fraction of the second class in which the denomincctor of every component exceeds its numerator by unity at least is definite. 788. We shall now shew that p^ and §'„ in Art. 786 increase with n. ^o^ Pn~Pn-i = K-'^)Pn-i-KPn-2', ^^w «„-! is at least ag large as h^^ ; therefore/*^ is gi-eater than p^_^ ^^ Pn-i is greater than p^_2 ; and so on : and p^ is obviously greater than p^. Thus p^ is greater than p^^_^. Similarly q^ is greater than q^_^. 789. If in an infinite continued fraction of the second class every component has its numerator not less than unity and its denominator greater than its numerator hy unity, the 'Value of tJi^ infinite continued fraction is unity. CONTINUED FKACTIONS. 521 Here we have always a„ = 6„+ 1 j therefore, by Art. 786, so that ^„ - pn.i = K (iK-x - Pn-a)- Now p^ = 6j, p^ = ap^ = {b^ + 1) 6, ; thus p^-Pi = hfi^. Hence we obtain in succession Therefore, by addition, 2\ = b^ + bfi^ + hfi,J)^+ +hfij).^...b^. Similarly we have q,, - q,,,^ = 6„ {q^_^ - q„_^). Now q, = b^ + 1, q^ = (6j + 1) (b^ + 1) - 5^ ; so that q^-q^= bfi^. Hence we obtain in succession %- % = ^AKy $4 - ?3 = ^iKKKy > 9[n- qn-i ^ KKK • • • ^n- Therefore, by addition, Thus, g =p + 1 : and ^ = -^^, = r • ^^w » by our hypothesis is not less than 7i, and so may be made as great as we please by taking n large enough ; therefore -^ may be made ■In to differ from unity by less than any assigned quantity : and we may therefore say that the value of the infinite continued fraction is unity. 790. It will be seen that the investigation of the preceding Article establishes rather more than is contained in the enuncia- tion which we used for simplicity. The essential conditions are that a = 6 + 1 for all values of n • and that j) should increase indefinitely with oi. It is sufficient for the latter condition that b should be never less than unity, but not necessary. The ne- cessary and sufficient condition is that the infinite series of which the m*^ term is bfijb^ ••• ^^ should be divergent; this would be secured for example if 6„ = r : see Art. 562. 522 CONTINUED FRACTIONS. 791. Jf the denominator of any comiwnent exceeds its nu- merator hy more than unity while tlie denominator of every com- ponent exceeds its numerator hy unity at least the value of the infnite contvnued fraction is. less than unity.. Suppose, for example, that ct^ — h^^-p where p is positive and greater than unity. The infinite continued fraction is equivalent to -^ , where p is a positive quantity which represents a,— ' 2 'I the infinite continued fraction beginning with the component — r a^ Now. p cannot exceed unity by Art. 787 : hence ■-, '- is a positive proper fraction ; and therefore as in Art. 786 we see that J— is a positive proper fraction. 792. An infinite ■co7itinued fraction of the second class in which every component is a proper fraction with its numerator and. its denmninator integral, and in which the value of the infinite continued fr auction beginning with any component is less than unity cannot he a commensurable quantity. For if possible, suppose the continued fraction commensurable, and denote it by — , where A and B are positive integers. Thus A. — = — - — where p^ denotes the infinite continued fraction be- A «! - Pi ginning with the component — . Therefore pi = — — ^ ^ ; the a numerator of this fraction is an integer, which we A\ill denote by C ; and G must be positive for p^ is positive. In like manner, if Pa denote the infinite continued fraction beginning with the component — we find that p, = 7; , where D is also a positive integer. And so on. CONTINUED FRACTIONS. 523 BCD „ ' Moreover — , — , -^ , ... must all be proper fractions hj hypothesis. Hence A, B, C, D, ... form a series of positive integers, which are in descending order of magnitude, and yet infinite in number : this is absurd. Hence the infinite continued fraction cannot be a commensurable quantity. Article 785 applies here also, with the condition of the enun- ciation in Art. 792. " 793. We have supposed in the preceding Article that the infinite continued fraction beginning with any component is less than unity. By Arts. 789, 791, this will always be secured except in the case in which from and after- some fixed component the denominator of every component exceeds the numerator by iinity. 794. For an example of an infinite definite continued fraction of the first class, suppose that eveiy conaponent is ~, where a and h are positive. Denote the continued fraction by a; ; then X = • so that X = -^ ; 2a+ jr therefore x^ + 2ax — 6 = 0; therefore x = — a^ J{cv^ + h) : the upper sign must be taken, since the infinite continued fraction is posi- tive. Thus, by transposition, we obtain h J{a^ + 6) = a + 2a + 2a+ ... This foiTnula gives various modes of expressing a square root in the form of a continued fraction. For example, take ^\7, We may put 17 = 16 + 1, or = 9 + 8 ; and so on. Thus, ^17 = 4 + L^ = 3 + ?-— -. o + ... o + ... 524 CONTINUED FRACTIONS. 795. For an example of an infinite definite continued fraction of the second class, suj^pose that every component is ^, where a and h are positive, and 2a exceeds h by unity at least. Denote the continued fraction by x ; then X = -^ — so that X = ~ ; « b 2a -X "2(1 2a-.., therefore x^ - 2ax + 6 = 0; therefore x^a± J (a' - h). The lower sign must be taken, for with the uj)per sign we have a result greater than a + a—b, that is greater than 2a — b, that is greater than unity; but the infinite continued fraction cannot be greater than unity, by Art. 787. Thus, by transposition, we obtain J(a^~h)^a ■ , 2a-. 2a-.., 796. In Art. 781 we have i\ = «« Pn. 1 + K Pn.i, ^n = «« 3'n_l + h Where B,J^±pS^, (a_^.^ 1) (^-f-^H-l) ' l(y + 27-) ' ^ 2(y + 2r+l) '' ' and A^, A„, ... may be obtained from B^, B^^ ... respectively by changing y8 into ^ + 1 and y into y 4- 1. A A Thus -— - , ~ , ... may be considered to be all equal to unity when r is indefinitely great j and so by Art. 679 we may consider C'g,. to be also unity. Since z.^^ may be considered to be unity ^j^ z^^^ becomes sim- ply K- Thus z is transformed into an infinite continued fraction. 802. For a particular case put -^ instead of a; j then suppose CONTINUED FRACTIONS. 529 tliat /? = a, and that a increases indefinitely : thus the denominator of z becomes 2 4 9 "^ "iT^"^ 1.2.y(y + l)"^1.2.3.y(y+l}(y4-2)^"" wliich we will denote by f{y) 'y the numerator may be obtained from the denominator by changing y into y + 1 . Also Z:2r-i becomes — (y + 2r-2)(y + 2r-l)' x^ and ^or becomes — -, ^ — ,. . . ^ . . '"^ (y + 2r-l) (y + 2r) Thus finally ^X , . ' is transformed into an infinite continued /(y) fraction , where »,„= , — -j^. r. ^ , P, ^ (y+m-l)(y + m) 1 + P2 This result may be obtained independently in the manner of Ai-t. 801 ; for we have /(y) -/(y + 1) = 7^^/(r -^ -) ^ '^'''' /(y) -, a^' /(y + 2) 1 •^ ^^^ — =1-1 "^-^ — ^ ; and so on. /(y+l) +y(y+l)/(y + I)' 803. In the result of the preceding Ai-ticle put ^ ^<^^' y ^^^ ? for X. Then it will be found that "^-^XA becomes —n 3 and that p becomes t-| — ? • By multiplying by y and simpli- t^^ 4m - 1 ^ — e~^ fying the fractions we ultimately obtain for ^ -y an infinite continued fraction of the first class in which the fii'st component is % , the second is ^ , and cfenerally the r*'' is . 1 ' 3 2r - 1 For y put where wt and n are positive integers ; then by n T. A, 34 530 EXAMPLES. LVIl. m m gn g n simplifying the fractions we obtain for ~ an infinite con- tinned fraction of the first class in which the first component is — , the second is -— , and generally the r^'" is — — . n du (2r - 1) ?i When r is large enough (2r -])n exceeds vi* ; hence by Art. 785 the infinite continued fraction beginning with a suitable component is incommensurable; and therefore the whole continued fraction is incoramensurable. Hence e"^ is incommensurable for all integi'al values of m and n. EXAMPLES OF COXTlNtJED FRACTIONS. 1 1. Find the value of 5 10- 1 10-... r 1 1 ^^ r 1 1 'J* 2. Shew that {w + j^ ^ [-{^-9 ^5 — ' H 2. 3. In a continued fraction of the first class every component is - : shew that p^^^ = hq^. (J/ 4. In a continued fraction of the first class every component ig ~ : find the values of »„ and q^, a 5. In a continued fraction of the first class if an = K= n, shew that Pn+ q,i = |n-f 1 . G. In a continued fraction of the first class if 5^^^ = ! +C5^^, shew that jp„ - 6„+i jt?„_i = A(- 1)", q^ - 6„^i g'^.i = £{- 1)" ; where A and £ are constant whatever n may be. 7. In an infinite continued fraction of the first class the n* component ia -^^ j : shew that i?n "*" (^' + l)i^«-i == (^ 1)"''"^ EXAMPLES. LVII. 531 8. Shew that e~' can be transformed into an infinite con- tinued fraction of the first class in which the first component is v , the second is ^ , and generally the r*^ is -^^ — ^ . 9. Shew that log 2 is equal to an infinite continued fraction of the first class in which the fii'st component is y , the second is 1 (r-lY ^ , and generally the r^^ is ^^ — — -^ . 10. Obtain from Art. 801 an infinite continued fraction of the first class for - log (1 + ct-). L\^III. MISCELLANEOUS THEOREMS. 804. The present Chapter consists of some mLscellaneouf? theorems on the following subjects : abbreviation of algebraical multiplication and division, vanishing fractions, permutations and combinations, and probability. 805., In multiplying together two algebraical expressions it is sometimes convenient to abridge the written work by expressing only the coefficients. For example, suppose it required to multiply 2x* + x^-dx + l by x^+ 3x-2 ; we may proceed thus : 2+0+1-3+1 1 + 3-2 2+0+1-3+1 G+0+3-9+3 _4_0-2+6-2 2 + G-3 + 0-10+9-2 Thus the required result is 2x'' + 6a;* - 3x* - 10a' + 9a; - 2. A similar abridgement of the written work may be made in division. This mode of operation has been sometimes called the method of detached coefficient*, 34—2 532 MISCELLANEOUS THEOREMS. 806. Syntltetic Division, Tlie operation of division may how- ever be still more abridged by a method which is due to the late Mr Horner, and which is called synthetic division. Suppose it required to divide Ax-" + Bx"^-' + Cx"^-"- + Dx-"-^ + Ex"^-* + . . . by a" + a^x"~^ + a^x"~- + a.jxT'^ + a^x''~* + . . . ; let the quotient be denoted by then it is our object to shew how A^^ A 2, A^, ... may be deter- mined. If we multiply the quotient by the divisor we obtain the divi- dend ; this oj^eration may be indicated as follows, expressing only the coefficients, A+A^+ A+ J + A^ + ... 1 + a, + CT„ + a^ + a, + ... 12 3 4 A+ A^-h A^+ A^+ A^+ ... a^A + a-^A-^^ + a-^A^ + a^A^ -\- ... M+Mi + M2+--- M+Mi + --- aA + ... 4 ^+ B+ C+ i>+ E + .... • here the last line is supposed to be obtained in the usual way by adding the vertical columns between the horizontal lines. Now A, B, C, ... are known, and we have to find A^^, A^, A^, ... ; for this jiurpose we reverse the above operation and perform the following : -a^ — a^ A + B + c+ E + E+... — a^A - a,^,- «,^. -Ms---- a,^A — Ml M -Ml---- — a A — ... 4 A + ^i. + ^4.+ ^a + .1.+... MISCELLANEOUS THEOREMS. 538 Here each vertical column expresses the same result as the cor- responding vertical column of the former operation, but expresses it in a form more convenient for our object. For example, the fourth vertical column of the foinner operation gave and the fourth vertical column in the present operation gives D - a^A^ - a^A^ - a^A = A^. Tlie method then may be described as follows i (1) If the first term of the divisor have a numerical coeffi- cient, divide every coefficient of the dividend and divisor by this coefficient ; the resultins: coefficients are those intended in the following rules. (2) Write the coefficients of the dividend in a horizontal line, with their proper signs, putting when any term is wanting. This gives the horizontal row A + £ + G + JD + U + ... (3) Draw a vertical line to the left of this series of coefficients, and wi'ite in a vertical column the coefficients of the divisor with their signs changed, putting when any term is wanting. This gives the vertical column — a^—a^ — a^ ... no notice being taken of unity, which is the coefficient of the first term of the di\dsor. (4) Multiply each term of this vertical column by the first coefficient of the quotient, and arrange the results in the fii^st oblique column. This gives the oblique column —a^A—a^A—a^A—... the first term of which is to be placed under £. (5) Add the terms in the second vertical column to the right of the vertical line ; this gives the coefficient of the second term of the quotient. That is, B — a^A = A^. (6) With the coefficient thus obtained form the next oblique column. This gives — a^A ^ — a. A ^ — a.^Aj^ — the first term of which is placed under C. (7) Add the terms in the thiixl vertical column to the right of the vertical line ; this gives the coefficient of the third term of the quotient. That is, C — o^A^ — a.^A = A^. (8) Continue these operations until the work terminates, or as many terms are found as are required. 534 MISCELLANEOUS THEOREMS. 807. For example, divide 4a;* + 3a;' - 3a; + 1 by a;' - 2a; + 3; 4 + 0+ 3- 3-J- 1 8 +16 + 14- 26- 92 -12-24-21 + 39 + 138 2 -3 4 + 8+ 7-13-46-53 Thus the quotient is 4a;'+ 8a; + 7 - 13a;~' - 46a;~'- SSa;"' Or if we wish to stop at 46a;~', we have 4a;'+3a;»-3a;+l , , „ ^ .^ _, .. _. 53a;-' - 138a;-' - — , ^ , „ = 4a3'+8a; + 7-13a; '-46a; ' ^— — . ar - 2a; + 3 a;* - 2a; + 3 If we wish to stop at — 13a;"', the oblique column — 92 + 138 must 46 — 39a;-' be suppressed, and the result is 4a;' + 8a; + 7 — 1 3a;-' — ^ — . ^ ^ a; - 2a; + 3 If we wish to stop at 7, the oblique column —26 + 39 must also be suppressed, and the result is 4a;* + 8a; + 7 — ^ — ^^ s- . 808. We may observe that the principle which is exemplified in Ai-t. 332 is often of use in effecting algebraical reductions. For example, suppose it required to prove the following identity : = I2abc {a + b + c). "We see that if « = 0, the expression which forms the left-hand member of the proposed identity vanishes ; we therefore infer that this expression is divisible by a. In the same mamier we infer that the expression is divisible by b and by c. Thus abc is a factor of the expression. And since the exj)ression is of the fourth degi'ee, there must be another factor w^hich is of the Jirst degi-ee; and since the expression is symmetrical with respect to a, by and c, this factor must be a + b + c. Hence the expression must be equal to habc (a + 6 + c), where k denotes some numerical coefficient which retains the same value for all values of a, 6, and c. To determine h we may ascribe to a, b, and c any values we find convenient ; for example, we may suppose b = a and c-a, and we find that h = \2. Thus the proposed identity is demonstrated. MISCELLANEOUS THEOREMS. 535 The following identity may be demonstrated in the eamo manner : (a + h-^c + d)*+(a + h-c-dy ■¥{a + c-h-'d)* + {a-^ d-h-cy - (a + b + c - dy - (a + b - c + a)* - (a- b + c + dy - (-^ a -h b + c + d)* =^ld2abcd. 809. Vanishing Fractions, A fraction in which the numerator and the denominator are both zero on Bome supposition as to the valne of any quantity involved, is then called a vanisldng fraction. For example, the numerator and the denominator of the fraction x^ — a^ -7 T vanish when x = a ; the fraction theii takes the form ^ , a;* - a* ^ and we cannot strictly say that it has any definite value. But we can find the value of the fraction when x has any value drfierent from a ; and we can shew that the more nearly x approaches to a the more nearly does the value of the fraction approach to a certain definite value. For put x = a + h ; then by the Binomial Theorem the fraction becomes a^ + -^a'^h-^ a~^ /t' + . . 1 1 t 1 6 , ^a ^-g'^ ^h+ ... 1 ■fl^'^'^. Tn __^^_ a* + -a * h~-^a * /r + . . .. — a 1 -^ ^ -I y Now as h diminishes the numerator and the denominator of the 1 _2 1 ? last fraction approach to the values i^a ^ and - «~* respectively ; and by taking h small enough, the numerator and the denominator may be made to difier from these values by as small a quantity as we please. Thus the fraction can be made to approach as near as ^ A. we please to , that is, to k a"^^ . This result is expressed by saying that ^ d^ is the limit to which the fi-action approaches as X approaches to a. We may also arrive at this result without using the Binomial 536 MISCELLANEOUS THEOREMS. Theorem. For suppose .r = ?/'- and a = Z>'^; then the proposed 4 1.4 fraction becomes ^3— -rs ; so long as y is not absolutely equal to h y - we may divide both nvimerator and denominator hj y-h, and so put the fraction in the form ^ — | ~^. — . As y approaches to ^ y + yb + 0' ih h this fraction approaches to — , and the fraction may be made o to differ as little as we please from — by making y - h small enough. Thus the limit of the fraction as y approaches to 6 is — ; that is, the limit of the fraction as x approaches to a is o ^^^ • Questions respecting vanishhig fractions and limits belong properly to the Dififerential Calculus, to which the student is therefore referred for more information. 810. We will now give two Articles, which form a supple- ment to the Chapter on Permutations and Combinations. They are due to H. M. Jeffery, Esq. of Cheltenham. 811. To find the number of combinations of n things taken 1, 2, 3, n at a time, when there are p of one sort, <\r where Pj, Pg, Pg, P^, have the meanings already stated. For consider only two events A and B ; let n denote the whole number of equally probable cases, n^ the number in which A occurs, no the number in which B occurs, n^^ the number in which both A and B occur. To find the number of cases in which MISCELLANEOUS THEOREMS. 541 neither A nor B occurs we proceed thus : from n take away ria and lip ; we have thus taken away too many cases, because the cases, in number riap, in which both A and JB occur have been taken away twice ; restore then 7iap. Therefore the whole number of cases in which neither A nor B occurs is n- (n^ + n^) + n„^. Hence the number of cases in which one at least of the events occurs is no. + n^ — Uap . Therefore the probability of the occurrence of one at least _ n^ + n^ - Ug^ _ n^ + ?z^ _ ^ _ p _p n n n ^ ^' Similarly any other case may be treated. III. Supposing that there are n events, required the proba- bility that an assigned ^n of them will happen, and no more. Suppose that the events of which the proVjabilities are a, /?, y, are to happen, and the events of which the proba- bilities are X, /x, V, are not to happen. Then if the events are independent the required probability is a^y (l-X)(l-/x)(l-v)......; that is, a^y to m factors - 1 - ^A. + ^X/x - ^SX/xv + ^. This we may denote by Q^- Q^^^ + Q,^^,-Q^^, + , where Q^ is the probability of the occurrence of the m assigned events, Q^^.^ is the simi of the probabilities of the occurrence of every collection of m + 1 events which includes the m assigned events, Q„^+2 ^^ *^® ^^^^ ^^ *^® probabilities of the occurrence of every collection of m + 2 events which includes the m assigned events, and so on. lY. As before we may shew that the theorem in III. is true eA en when the events are not independent, Y. Required the probability of the occuiTence of any m of the events and no more. With the previous notation this is -^Q -%Q +20 -2 and so on 1 this will be the case -svlien the events are all similar. YI. In II. we have found the probability that at least one event shall happen, and in Y. the probability that just one event shall happen ; by subtracting the second result from the first we obtain the probability that tivo events at least shall happen. Then again we know from Y. the probability that just two events shall haj^pen ; by subtracting this from the probability that two events at least shall happen we obtain the probability that three events at least shall happen. And so on, MISCELLANEOUS EXAMPLES. 1. Having given a? =^ 5y + c« + duy y = ax-\-cz+ du^ ^ = ax-¥hy-V du, il=^ ax + hy + cz, 1- xi, X 1 ^ ^ c d shew that 1 = ^. fc- - — r + \+a 1+6 1+c \+d^ K, y, z, u being supposed all unequaL fC ■?/ z 2. If = a, = h, and — — - c, find the relation be^ ^ + « ^ + x x + y x^ if i? tween a, h and c\ and shew that _ -^ _. a (1 - he) 6 (1 - ca) c (1 - aS) ' 3. Find the relation between a, h and c, having given X a y h z G , - + -=7- + - =- +-, xyz = ahcK a X y c z atid a^ + y* + »" + 2 (a6 + ac + 6c) = 0. 4. Find the relation between a, h and c, having giVeli 1/ z z X ^ X y « y a; « y X 5. Eliminate cc, y, z between the equations 9f(^ + z)^a% y\x + z)=^h^, z'{x + y) = c% xyz = alc: EJtAMPLES. LVIII. 54)3 6. Eliminate a and 6 from the equations V~y'~ Zx + 2y' ""^ -{^-y)> o, ^b -z, 7. Eliminate x and y from the equations x + y = a, x^ + y^ = h^y x^ + t/ = c'. 8. Eliminate x from the equations a \«/ a \x/ c \xj X \a/ 9. Eliminate x, y, z from the equations X y z X y z ^ y z X z X y a: yJ ' 10. Eliminate a; and y from the equations ax + by=:Oj x + y + xy = 0, x^ + y^-1^0. 11. Eliminate x and y from the equations y"-x^=ay-px, ixy = ax + l3y, x'^ + y-^l. 1 2. If («: 4- 2/)' - ic'xy, (y + ;?)' = 4a V, (« + ^)' = 46«<»a?, shew that a' + 6^ + c^ =fc 2abc = 1. 13. Eliminate a from -^ ^, - „ ^ --- — a^^i . a +x^ a" + 2/ a + « 14. Eliminate x and y from 4 (ic^ + y'') r:= aic + 6y, 2 {of - t/*) = a£C - 6?/, a:^/ = c*. 15. Shew that unless ahc + 2a'b'c = aa" + hb" + cc'^, the fol- lowing equations cannot be simultaneously true : a = xx', b = yy\ c = zz\ 2ot!= yz' + zt/^ 26' = zx' + xz\ 2c' ^ xy' + ya/. 16. Find the number of permutations which can be fonned i-vith the letters composing the word examiiiation taken 3 at a time. 17. Find thd chance of a one, a two, and a three, of the Same suit, lying together in a pack of cards which consists of m suits, and has n cards numbered 1, 2, 3, in each suit. 544 EXA^IPLES. LVIII. 18. A rectangular garden is surroiinded by a walk and is divided into mn rectangular beds by m — 1 walks parallel to two sides and n-1 walks parallel to the other two sides. Find the number of ways, no two of which are exactly alike, in which a person can walk from one corner to the opposite corner so as to make the distance equal to half the perimeter of the rectangle. 19. If a; be a proper fraction, shew that x tjc yu JO '>o xL* 20. If ic be a proper fraction, shew that 1 = {l ^ x) {\ + x") {I +x'){\ + x') ... {\-x)(l-x^){l-x') 21. Eliminate x, y, z from the equations {^-y){y-'^){^-^)=p\ (a^ + 2/)(y + -)(- + -) = ?'» {x' + f) (f + z") iz' + rc^) = r^ {x' + y') {y' + z') (z' + x') = s'\ 22. Shew that if aX+ hY+cZ^ 0, and a^X+b^T+ c^Z = ; where X= ax + a^x^ + «„, Y ~hx + h^x^ + h.^, Z=cx + c^x^ + c.^ ; then {a_^(bc^-h^c) + b„(ca^- cm) -hc.,(ah^-r(fi)Y X'+Y' + Z'^ [bc^ — b-fif + {ca^ — c^a)''' + [aOi — a^by 23. If a^, a.,, ... a„, and b^, b„, .. 6„ be two series of posi- tive numbers, each arranged in descending order of magnitude, shew that -,-+—+...+ - - is less, and — + v^- 4- . . . + ^ is ^1 K K b^ 6„_, b^ greater, than if the denominators 5^, 6„, ... 6^ were arranged in any other order under the numerators a^^ a^, ... a^ ' 24. If a be less than b, shew that a series of which the general tei-m is — I I —, — ~ is equal to the los^arithm of ( 7 ) 25. If a be less than b, shew that (-yj is increased by adding the same quantity to a and b. ( 545 ) illSCELLANEOUS EXAMPLES, 1. Simplify x-[2i/+ [Sz -:^x-{x + 2/)}] + 2x-(j/ + Zz). 2. Divide a^x'^ + {2ac - ¥) x* + c' by ax* - hx^ + c. 50;"+ 2a^- 15a; -6 3. Keduce to its lowest terms 7a;' -4a;' -21a; 4- 12* . . .^ 7>x — a ^ x-^-Za ^, a — x ^ 2a -hx 4. Add - — -^- to f^ — - ; take ^r-^ — r, from -. _ . ox -f 3a 7x+va 2a + dax + x' a' - a; -. ^ , 4a; + 1 5a; -1 _ 5. Solve — z-t; ,r— = a; — 2. lo 3 C. Solve 10a;-4y=ll, 3a;+ 2y-U|. 7. A, wlio travels 3^ miles an liour, starts 2^ liours before B Vv^lio goes the same road at 4 J miles an hour : find when B over- takes A^ 8. A bill of XI 00 was paid with guineas and half-crowns, and 48 more half-crowns than guineas were used : find how many of each were paid. 1 9. Find the square root of a* + 2a^ - a -f- ^ •' 10. Solve ^^-lV3a;-l) = |. 11. If a = 1, & = o , c - 3, d = -^, find the value of a -[2a -35- {4a -56- 6c -(7a -86 -9c -10(7)}]. 1 2. Multiply x' + (2a + 36) a; + 6a6 by a;' - (2a + 36) x + 6ab ; and divide 14a;*-lla;V-C6a;y-7a^'y+49a;?/^+15y* by 2x'-3xy-5y\ 1 3. Find the L. cm. of a;* + 5a; 4- 6 and x'' + Ox + 8. ,, ^, 2a;-<-3a . 23a;*-f 18aa;+ 17a' 14. Take r- from 3x + 4a 12.cV31aa;-i-20a'* l5. Solve r -K —^ a 5 . T. A. 35 546 MISCELLANEOUS EXAMPLES. 16. Solve 7a;-9y = 23, 9x-79/ = 57. 17. Find the time between 9 and 10 o'clock when the hour- hand and the minute-hand of a watch are together. 18. A, after doing three-fifths of a work in 30 days, calls in B, and with his help finishes it in 10 days : find in how many days each could do the work alone. 19. rind the square root of 4a;' — 12a:y + 9?/' + ixz- 6i,z + z'. 20. Solve r ^-1. X— 1 X — 6 21. If a = 1, 5 = 2, c = - , c? = ^ , find the value of a -[3a -5b- {7a _ 96 - lie - (13a - 156 - 17c - 19c^)}]. - 22. Multiply cc' + (3a -2b)x- Qab by x^ + (36 -2a)x- 6ab ; r.nd divide x* - 5Qx + 15 hj l^ix + x^, 23. Find the G. c. m. ofx'-i, £c' + 1 Oo; 4- 1 6, and a;' - 7ic - 1 8. 24. Snnplify ^^3^3^^^ x^^-^. _„ _, . x-1 llrc-3 3x-d 25. Solve -^+-^^-^^=2^. , x + i; x-y 11 ^ „ ^ 26. Solve — ~ -I- — ~ -= tp- , 5x-3y = K). 6 ^ VI 27. A person starts from Ely to walk to Cambridge, which is distant 16 miles, at the rate of 4| miles per hour, at the same time that another person leaves Cambridge for Ely, walking at the rate of a mile in 18 minutes : find where they meet. 28. In a concert-room a certain number of persons are seated on benches of equal length ; if there were ten more benches one person less might sit on each bench j if there were fifteen fewer benches two persons more must sit on each bench : find the num- ber of benches, and the number of persons seated on each. 29. Find the square root of a" - 4a;' -I- 6a;* - 8a;' -f- 9a;' - 4a; + 4. 30. Solve lla;'-lli = 9a;. MISCELLANEOUS EXAMPLES. 547 31. li a = l, h = 2, c = 3, d = 4c, find the values of 32. Multiply -3 + a' by '2 — X] and find the value of the pro- duct when x — 'l. x*-U5x-^2i 33. Reduce to its lowest terms 24.x'-\i5x^ + l' 34. Add tofi^ether , , ■. and ;; 5- . 35. Solve {x-3Y-2,{x-2f+Z{x-\f-x^=^Si-x. 36. Solve 5y- 3a: = 2, 8y-5a;-l. 37. A farmer bought equal numbers of two kinds of sheep, one at £3 each, the other at £i each. If he had expended his nionev equally in the two kinds he would have had 2 sheep more than he did ; find how many he bought. 38. The sum of £111 is to be divided among 15 men, 20 women and 30 children, in such a manner that a man and a child may together receive as much as two women, and the women may together receive £60 : find what they respectively receive. 39. X/ 'W X 'U o Find the square root of — „ + -f-5 + ^ 7 ^ - y^ ix^ y '2x ^ J «. /-< 1 tuX iX — O - ' 40. Solve ■ — - + r- = 9. x—i x+l 41. If a= 1, 5= 3, c= 5, J= 7j find the values of 42. Shew that a{a-x)(a-2x)={a-b){a-b-x)(a + 2b-2x) + b{b-x)(3a-2b-2x), 43. Find the g. c. m. and the l. cm. of x^ — x* + x^ ~x' + x -I and x^ — 1. xl* -\- 5x + 6 x* + Qx + 5 44. Simplify xi^ + bx x'' + dx _ „. 1 4 9 36 45. Solve — -r + K r + x+l 2a; — 1 3a; — 1 6x* - 1 35—2 548 MISCELLANEOUS EXAMPLES* 46. Solve 2a; + 3y-8^+35 = 0, 7x- 4i/ +z-S =0, 12x- bij -Sz + 10 = 0. 47. Find how many gallons of water must be mixed with 80 gallons of spirit which cost 15 shillings a gallon, so that bj selling the mixture at 1 2 shillings a gallon there may be a gain of 10 per cent, on the outlay. 48. A and B can together do a work in 1 2 days ; A and C in 15 days; B and G in. 20 days : find in how many days they will do the work, all working together. 49. Find the square root of a-c + 2 J (ah -^-hc-ca- 1/), 3 50. Solve 0; = 4- 4- 4-a; 51. Simplify {a -^-h + c) {x + y + z) + {a + 1) ~ c) {x + y - z) -\-{b + c-a) {y + z-x) + (c + « - 6) {z+ x-y). 52. If s = ^ , shew that {(s -a) + {s -h)f = {s- af + {s ^hf +'^ {s - a) {s-h)c. 53. Find the g.c.m. of a;* - 2icy + 5a;y - 207?/' + 4?/* and a;' - Zx^y + QxY - ?>x^f + 5y\ ^. ci' ..J, x-a x-h (a-i>y 54. Smiplify r + 7 — ^ — r-^- — rr. X — x~a [x- a){x-0) 55. Solve (3a;- 1)' + (4a; -2/ = (5a; -3)'. ^ , x + 3 y-3 „ x~3 y-3 ^ 56. Solve ^+^ — 5 = 2, ^ ^+4 — o=^- cc-3 y + 3 ' 2a; +3 2^ + 3 57. A, B, C are employed on a piece of work. After 3 days A is discharged, one-third of the work being done. After 4 days more B is discharged, another third of the work being done. C then finishes the work in 5 days. Find in how many days each could separately do the work. MISCELLANEOUS EXAMPLES. 649 58. A person walks from ^ to ^, a distance of 7^ miles, in 2 hours 17^ minutes, and returns in 2 hours 20 minutes. His rates of walking up hill, down hill, and on a level road being 3, 3^, and 3\ miles respectively, find the length of level road between A and JS. 59. Find the cube root of 8a;'' -12x' + 6x' - 37a;' + 36a;' - 9a;* + 54a;' - 27a;' - 27. /./^ r, ^ (x + o) (x + Tub) (nix + a) (x + b) 60. Solve ; M— TT = 7 r-p ~r . (a; — ma) [x — o) {x — a) (inx — b) 01. Simplify 24JX-J (x-\)Y^x-\ {x- 2)}{x - |(s>- IJ)], and subtract the result from (a; + 2) (a; + 3) (a; + 4). 62. Divide ( -2 + -1 - 2 ) by . \ar x^ J '^ a X 63. Find the g.c.m. of 5a;' - l^x'y + 1 \xif - 6y' and 7a;» - 2Zxy + 6?/^ ^. _,. ... x'-x + l 2x{x-\y 2xUx'-lY 64. Simplify —^ + , ^ a , + 8 4 — r- • ^ '^ x^ ■\-x+\ a; + a;^ 4- 1 a; V a; + 1 aK cj 1 ^ a;-2 a;4-4 - a;-l 65. Solve ^ = — — - 2 , . a;-o x — 6 x — d a; — 4 nn ci ^ OS -2a y-ib . a; + 2« y + 56 66. Solve — + ^ — ^ = 2, = ^— ^z • x- da y — ob x + a y + 3b 67. A man bought a house which cost him 4 per cent, on the purchase money to put it in repair. It then stood empty for a year, during which time he reckoned he was losing 5 per cent, upon his total outlay. He then sold it again for £1192, by which means he gained 10 per cent, on the original purchase money : find what he gave for the house. 68. A certain resolution was carried in a debating society by a majority wliich was equal to one-third of the number of votes given on the losing side ; but if with the same number of votes 10 more votes had been given to the losing side, the resolution would only have been carried by a majority of one : find the number of votes given on each side. 550 MISCELLANEOUS EXAMPLES^. GO. Solve Jx- Ja + J{x + a-b)= Jb. ox 155 X 78 70. Solve {x -2)(x-3) = — ^,,^^— . 71. If a = 2, 6 == 3, c = 6, c? = 5, find the value of ^{{a + c-byd} + ^{{b + d){5d-ic)] + ^{{c-a){d-h)], 72. Shew that x{y + zY + 1/ (z + xy + z{x + yy - ixyz ={y ■¥z){z + x) {x + y). 73. Find the g.c.m. of 5x'-19a;' + 55a;-425 and ^x^ -I5x' -Z^x+Qd. bc(x — ay ca(x — by abix-cf 74. Simplify \— — ^ + -j — \ ^ + ^ - {a -b){a- c) (b - c) (6 - a) (c-a) (c-b)' 75. Solve J{(x - ay + 2ab + b-] = x-a + b, 76. Solve ax+cy + bz=cx + by + az=bx + ay+cz=a^+b^+c^—3ab6^ 77. A and B start together from the same point on a walking match round a circular course. After half an hour A has walked three complete circuits, and B four and a half. Assuming that each walks with uniform speed, find when B next overtakes A. 78. On a certain day mackerel were being sold at a certain price per dozen ; on the next day twice as many mackerel could be bought for one shilling as dozens could be bought for a sovereign on the day before : the whole price of 20 mackerel bought 10 on one day and 10 on the other being 2s. 2d., determine the price of a mackerel on each day. 79. If re = ^{a + J^i^+b') + ll{a - J a' + b'), shew that x^ + 2>bx~2a=0. 80. Solve (a;' + 8a;'+16a;-l)^-a;=3. 81. Shew that (^ + 5' + r)^ = 4 (p3 ^ ^3 _^ ^3^ 3^^^^ (p + J + ^)4. 6gV+ QrY+ QpY- 3/- Zq'- 3r*. 82. li X = ax-\-cy + bz, T=cx + by + az, ^=bx + ay + cz, shew that X'+ T' + Z'- YZ-ZX-XY = (a* + W-\-c' -bc-ca- ab) (of ^y^ -{-z^-yz -zx - xy). MISCELLANEOUS EXAilPLES. 551 83. Find the g.c.m. of 7x* -lOax^ + ^a'x' - 4:a^x + ia* and 8x* - 1 3ax^ + 5a'x- - 3a^x + 2>a\ 1 1 2x 4a;» 8x' 84. Simplify 85. Solve \-x \ + x \ + x^ \ + x* l + x^ 4a;' + 4a;' + 8:B+l ^x' + 2x+\ 2x' + 2x + 3 X + 1 86. Solve x + y+z = a + b + c, ax + ht/ + cz = be + ca + abj {b - c) X + {c - a) y + {a - b) z = 0. 87. Tlie present income of a railway company would justify a dividend of 6 per cent. , if there were no preference shares. But as £400000 of the stock consists of such shares, which are guaranteed 7i per cent, per annum, the ordmary shareholdei-s receive only 5 per cent. Find the amount of ordinary stock. 88. The road from a place J. to a place B first ascends for five miles, is then level for four miles, and afterwards descends for six miles, the rest of the distance ; a man walks from ^ to -S in 3 hours 52 minutes ; the next day he walks back to ^ in 4 hours^ and he then walks half way to B and back again in 3 hours 55 minutes : find his rates 'of walking up hill, on le^el ground, and down hill. 89. Find the value to five places of decimals of {161 + /yi9360}-i 90. Solve + ^, = 2. X + a- c X + b — c 91. Find the value when a;= 5 of 3a; - [5y - {2x - {3z - 3y) + 2z-{x-2y- z)]]. 92. Shew that {y -z)' + {z-xY -{-{x-yy '■ =2{{y-zY {z-xY + {z-xr {x-yy + {x-yY {y -zY} z=2{a? -^-y^ + z^ -yz-zx- xy)'. 93. Find the g. c. m. of x^ + {5m - 3) a;' + (6m* - 1 5m) x-l 8/u' and x^ + {m~ 3) x' - {2m' + 3m) x + (jm\ 552 MISCELLANEOUS EXAMPLES. il-X-l) 9-1:. Shew that ■ •- — - = a + h + o. \b cj \c aj \a b, 9.. Solve (-^--^-^-)-^-^ = 0. 96. ^olve x^ -^ y^ + ^^ ^ ^xyz, x-a = y-h-z-c. 97. A bag contains sixpences, shillings, and half-crowns ; the three sums of money expressed by the different coins are the same : if there ai'e 1 02 coins in the bag find the number of sixpences, shillings, and half-crowns. 98. A person walks from -4 to -5 at the rate of 3^ miles per hour, and from -5 to (7 at 4 miles per hour ; in returning he calculates that he can complete the distance in the same time by Avalking uniformly at 3 J miles per hour, but being detained 14 minutes at B he has to walk to ^ at 4 miles per hour to finish it in the same time : find the distance from A to B, and from B to C. 09. If A' = ax + cv/ -f- hz, Y= ex +1)7/ + az, Z -hxA- ay A- cZi ishew that X^+Y^ + Z'- :^XrZ = (a' +b' + c'~- Zahc) (x' + y'Uz'- 3xyz). lOOo Solve re' -223a: -Hi 2432 = 0. 101. Solve '{-ix + 2)' - {3x^iy = (2x + '^y - (x-sy. 102. Find three consecutive numbers whose product is equal to fifteen times the middle number. 103. Solve a; + 2/ - 9, - + ~-^-. x y 2 104. If ic varies jointly as y and z j and y varies directly as X + z ; and if a: = 2 when z = 2, find the value of z when x=0. 5 2 105. Sum to 18 terms 1 4- 7^ 4- ^r + ... D 6 106. Sum to 6 tei-ms and to infinity 14-74-3.^-... MISCELLANEOUS EXAMPLES. 553 107. If the number of combinations of 2?i things taken n—1 together be to the number of combinations of 2 {n — 1) things taken n together as 132 is to 35, find n. 108. Shew that T - ~ 2"-^ + ""' ^''\~ "^^ 2"*-' -...+(- 1)" ^ 1. 1 |2 109. In the expansion of (a^ + a^ + ... + a J" if n is a positive integer, and m greater than ?i, shew that the coefficient of any term in which none of the quantities a^, a^, ... «^ appears more than once is \n. 110. Given log2= -3010300 and log 3= 4771213, find the integral values between which x must lie in order that the inte^Tal part of (1*08)* may contain four digits. 111. Solve {a{b + x-a)]U{b{a + x- h)}h = {x{a + h- x)]K 112. If a and /5 be the roots of the equation ax' •{■hx + c = Oj o form the equation whose roots are -^ and - . p a 113. Solve - + ^=^, xy = S. y X Z ^ 114. If a;-4 : a;-2 :: aj-1 : a; + 3, finda;. 115. Sum nine terms of an arithmetical progression of which 18 is the middle term. 1 1 1 116. Sum to n terms ^^ -r, + ^r — ^ — tt -j- - — 3 — 7^ + ... 1 + ^2 3 + 2 ;^2 7 + 5 ;^2 117. Prove that the number of ways in which jo positive signs and n negative signs may be placed in a row so that "So two negative signs shall be together is equal to the number of com- binations of p + 1 things taken n together. 118. Determine the coefficient of a;'' in the expansion accord- . ^ ,. , . (n-m+l)a;(l-a;)-a;'"^' + a;''^» mg to ascending powers of x of -^ — 7^ r^ , (1 — X) ' where m and n are positive integers of which m is the less. 119. Deteimine whether the series whose 11^^ temi is ^(n^ + 1) — 71 is convergent or divergent, 554; MISCELLANEOUS EXAMPLES^ 1 20. Find the value of A {^^3 " (j^f } . Given log 105 = 2-0211893, log 5303214 = 0-7245391, log 3768894 = 6-576214. 121. Solve {i + 5x-x')^^ = 2^x'^+{x' + 3x-4:)K 122. Find the relation between the coefficients of the equa- tion ax^ +hx + c = 0, that one root may be double of the other. ^^ool I I x + y 7 123. Solve - + - = xyl2x + y + o 124. Divide 111 into three parts so that the products of each pair may be in the proportion of 4., 5, and 6. 1 25. Find the number of terms of an arithmetical progression of which the first term, the sum, and the common difference are siven : find the conditions which must hold if there be two such numbers. 126. Find the sum of the reciprocals of n terms of a geo- metrical progression of which the first term is a and the common ratio r. 127. Shew that the number of ways in which mn things can be divided among m persons so that each shall have n of them is {<' 128. Shew that the coefficient of ai""^''"^ in the expansion (1 +xY of ^^ ^ is 2" ^{(n + 2r) {n + 2r + 2)+ ?i}, r being or any posi- {l—x) tive integer. 129. Find the coefficient of x* in the expansion of {l+2x--3x' + xy. 23 ^3 ^3 130. Shew that 1 + -^ + ,-^ + — 4- ... = 5e. 1^ l_ L± 131. Solve J{x'-8x + 15)-hJ{x' + 2x-15)=J{ix'-lSx+lS), 132. The numerically greater root of ax^ — hx-\-c = has the same sign as - ; and the numerically less root the same sign as - . d c MISCELLANEOUS EXAMPLES. 55^ 133. ^olve x + y + z = a + h + c, - + f^+- = 3, ^ a c x'^ + y^ + z^ = a^ + b' + c*. 134. Two persons A and B divide equally a sum of money consisting of half-crowns, sliillings, and sixpences ; the values of thg several parts being respectively in the proportion of 15, 4, and 1. It is found that each has 60 coins, A having two half-crowns more than B. Determine the sum and the coins each had. 135. The p^^ term of an arithmetical progression is - , and P the q^ term is - : shew that the sum ofpq terms is 136. If a, h, c be in arithmetical progression, and a, /3, ' in harmonical prosn:ession, and - + - = — h - , shew that aa, hB, ^ * ' y a a c ' ' ^ cy are in geometrical progression. 137. Eind the number of words beginning and ending with a consonant which can be formed out of the word equation, 138. If a^ be the coefficient of x" in the expansion of (1 + £c)% (-iy\2n shew that aj - a' + a^ — a^ + ... '= \n\7i 139. Determine whether the following series is convergent or divergent: l + -^^+-p+-^ + ... 140. If y = x-'% + %-... shew that x= y + ^^ + f:r+ ... 141. Solve (rc-3)'+3a;-22 = ^(£c'-3a;+7). 142. The number of soldiers present at a review is such that they could all be formed into a solid square, and also could be formed into four hollow squares each four deep and each con- taining 24 more men in the front rank tlian when formed into a solid square : find the whole number. 17 143. Solve ^x^-xy-\ 2y' = 0, a' + 2/ = y^ . 556 MISCELLANEOUS EXAMPLES. 144. If the speed on a railway is 20 miles an hour it is found that the expenses are just paid. If the speed is more than 20 miles an hour the increase of the receipts is found to vary as the increase of the velocity, while the increase of the cost of working is found to vary as the square of the increase of the velocity ; at the rate of 40 miles per hour the expenses are just paid : find the velocity at which the profits will be greatest. 145. Shew that the number 'p^ + 10/>j -i- lO^^ + ... + 10";:>^ is divisible by 1 3 if the following expression is, 146^ If 5 be the sum of an odd number of terms in geome- trical progression, and s' the sum of the series when the signs of the even term s are changed, shew that the sum of the ■ squares of the terms will be equal to sa\ 147. If there be n straight lines lying in one plane, no three of which meet at a poiaat, the number of groups of n of their points of intersection in each of which no three points lie in one of the straight lines is -^ \n — \. 148. Shew that 2^ 4^ . 8^'^ . 16^'^ , . . = 2. 149. Find the coefficient of cc* in the expansion of 150. Shew that if the logarithms of n quantities with respect to n bases in geometrical progression be all equal they will also be equal to the logarithm of the ratio of any one among these quantities to the preceding quantity, with respect to the common ratio of the progression as base. 1S1 «i 2(3;«-4) , 9(g-l) 2(3r«-2) 3(a;-2) 152. Shew that if a quadratic equation be satisfied by more than two values of the unknown quantity the equation is an identity. Apply this property to establish the identity c^ {x -h){x — c) b^ {x - c) (x -a) c' {x~ a) (x-b) _ , (a -b){a- c) {b - c) (b - a) (c -a)(c -b) MISCELLANEOUS EXAMPLES. 557 153. Solve (a;' + 2/')- = 6, {x' -y')'^--=\. y **' 154. Bronze contains 91 per cent, of copper, 6 of zinc, and 3 of tin. A mass of bell-metal (consisting of copper and tin only) and bronze fused together is found to contain 88 per cent, of copper, 4'8T5 of zinc, and 7 "125 of tin. Find the proportion of copper and tin in bell-metal. 155. Shew that the sum of the products of the first n natural , . 0i-l)?i6i + l)(3/i + 2) numbers taken two and two together is -^ ^-r ^^ -. 156. Four numbers are taken, the fii'st three in g. p., and the last three in h. p. ; again four numbers are taken, the first three in H. p., and the last three in g. p. : shew that if the first two numbers are the same in each set the last of the first set will be less than the last of the second. 157. Find the number of different arrangements that can be made of bars of the seven prismatic colours, so that the blue and the green bars shall never come together. 158. If (5 ^2 + 7)"* = ?i + a, where m and n are positive in- tegers and a less than unity, shew that a (?i -f a) = 1, if m be odd, 159. Find the coefficient of x* in the expansion of (l-2a;-f-3a;'-4a;'+...)"^. 160. If the whole number of persons born in any month be j^jr orthe whole population at the beginning of the month, and the number of persons who die -^r:— , find the number of months 600 in which the population will be doubled. Given log 2 = -3010300) log 3 = '4771213, log 7 = -8450980. 161. Solve a;*-f- 1=2(1-^0-)*. 162. A and B run a race round a two mile course. In the first lieat B reaches the winning-post 2 minutes before A . In the second heat A increases hia speed 2 miles an hour, and B diminishes his by the same quantity; and A then reaches the winning-post 2 minutes before B. Find at what rate each ran in the fii-st heat. 558 MISCELLANEOUS EXAMPLES. ,^o r. 1 X + 37J+5 3a; + 7/ + 4 , 1G3. Solve ^— ^ + ^^^— p = 4, x + y + I X + y — i (x + 2yy+{7j+2xy = 5(x + yy + hj. 164:. Solve — -=-^— = — -=x + y + z. y+z+l z+x x+y-l 1 Q5. Shew that the number p^ + 10;^^ + 10>^ + . . . + lO"/?, ^ divisible by 101 if the following expression is, p^ + lOpi - (p, + lOp,) + {p, + 10;?J - .... 16G. If a, 5, c be three Quantities such that a is the arith- metical mean between h and c, and c the harmonical mean between a and h, shew that & is the geometrical mean between a and c : and compare a, b, c. 167. In a plane there are m straight lines which all pass through a given point, n others which all pass through another given point, and j) others which all pass through a third given point : supposing no other three to intersect at any point find the number of triangles formed by the intersection of the straight lines. n{n — \) 1 68. If a, = r-{r-\)n + {r - 2) [2 _^^_3^.(.-l)(n-2)^ to r terms, shew that «r = (" ^)"^n-r i^ ^ be less than n-1, f/^ = if r be greater than n — \, and a^_j = (— 1)". 169. Find the coefficient of ic^ in the expansion of (1 + 2aj - 3x= - a;')l 170. Given log^„ 2 = -30103, find log^^ 50. ' i" ' 1 4 _ ■ - 171. Solve x^ + x ^ = -^ (ic + a '). 1 3 \ -/ ■ . ■ • 172. If a and ^ are the roots of the quadratic ax^ + hx + c = 0, form the quadratic whose roots are (a + (By and (a — jSy, 173. Solve 8 J{x'-y')=x + 9y, X* + 2x-y + y^ + x = 2,x^ + '2xy + y + 506. MISCELLANEOUS EXAMPLES. 559 174. A and B engage to reap a field in 12 days. The times in which thej could separately reap an acre are in the proportion of 2 to 3. At the end of 6 days, as they find they cannot finish the work in the stipulated time, they call in C and finish it with his help. The time in which A and C together could have reaped the field is to the time iij which JB and C together could have reaped it as 7 is to 8. Find in how many days the field would have been reaped if C had worked from the first. 175. A tradesman has eight weights, two of 1 oz. each, two of 5 oz. each, two of 25 oz. each, two of 125 oz. each : shew that he can weigh with a pair of scales any integral number of ounces from 1 up to 312. 176. Find four numbers in geometrical progression so that their sum may be 15, and the sum of their squares 8-5. 177. Out of 2n men who have to sit down, half on each side of a long table, p particular men desii-e to sit on one side and q on the other : find the number of ways in which this may be done. 178. Shew that the coefficient of x^' in the expansion of (9a' + Qax + 4:xy is 2'' {3a)-"-'. 179. Shew that the series u^ + u^ + ... + u^ + ... is convergent if from and after a certain term the value of {u^Y is always less than some finite quantity which is itself less than unity, and divergent if the value is unity or greater than unity. 180. Shewthiitl--—-^-—-——^^----- 1 -_... 2{n+l) 2.3{n+iy 3.4:(n+lY = log f 1 + - J , Hence shew that ( 1 + - ) increases with n. 181. Solve 9a;' + 4a:' = 1 + 12a;\ 182. Three persons A, £, C, whose ages are in geometrical progression, divide among them a sum of money in amounts proportional to the ages of each. Five years afterwards when O is double the age of A they similarly divide an equal sum ; A now received. .£17. IO5. more than before, and £ £2. 10 s. more than before. Find the sum divided on each occasion. 5G0 MISCELLANEOUS EXA]^IPLE53i 183. o 1 M-^ M^ 1 61 3 1^1- Solve - + - - 1 = rJ,jl 1 oAA - 7« 184. If a: = cy + 6;2;, 3/ = a;2; + ex, z = hx + ay y shew that .r 7/ 5; ^l(\-a') J(l-b') J{l-c')' and find the relation between a, h, and c. 185. Shew that in the scale with radix nine, eveiy number which is a perfect cube must end with or 1 or 8. 18G. Find the sum of the products which can be formed by multiplying together any three terms of an infinite G. P. ; and shew that if this sum be one-third of the sum of the cubes of the terms the common ratio is ^ . 187. A vessel is filled with a gallons of w*ine, another with h gallons of water ; c gallons are taken out of each ; that from the first is transferred to the second, and that from the second to the first ; this operation is repeated r times : shew that the quantity of ttO G C ■wine in the second vessel \vill be 7 (1 - ;/) where p = l 7 . a + b^ ^ ^ ah 1 + 2a: 188. By comparing two expansions of ^j — ^, shew that JL X (-1) =l-6n + j2 [3 (3^1 - 3) (3?i - 4) (3?i - 5) (392 - 6) ... +— ip- ... where n is any positive integer, and the series stops at the first term that vanishes. 189. Determine whether the following series is convergent 1 |2 13 |4 or divergent : 1 + -^x+^x'+ -r^x^ + =^x* -{■ .,, 190. If log , ' —3 be expanded in a series of powers a ~~ X "^ X "r X * 3 1 of X. shew that the coefficient of »" is - or - accol'ding as ?i is n n even or odd. MISCELLANEOUS EXAMPLES. 561 191. ^Ive {l+xy = 2ax{l-x'). 192. Shew that if x, y, z are real quantities x^x-y) (x-z) -^if {y -z){^j-x)^z^{z-x) {z-y) cannot be negative. 193. Solve cc* + 2/' + 1 = m^xy — aj'y^, xy {n^x -y) =x- n'y. 19-1. Shew that the equations ax+by-\-cz=0 and ax'+by''+cz^=0 will be satisfied by takinoj ^ — .^ = :r^ — = ^— — -y—. : where X y a + 6 + c + ahcv' = 0. 195. In Art. 458 we arrive at an a. p. of which the first term is — 7— + 7^— , and the common difference is —„ : shew that if this be arranged in groups of q terms each, the m}^ group is equal to the m'^ term of the A. P. of which the first term is a and the common difference is 6. 196. The first term of a certain series is a, the second term is h, and each subsequent term is an arithmetic mean between the two preceding terms : shew that the n^^ term is |(6-a){l-(-J)" 197. If all the permutations of n things a, h, c, ... I taken all together be formed, and from any permutation as abc ... I he formed the fraction —. ^r-? i ^ 7 1 ^ > shew that a {a + b) {a + b + c) . . . {a -\- b + . .. I) the sum of all these fractions is n-l + a. abc ... r ] 9S. Shew that (n ( n-l) (71 -2) ) a 3 x-' + , n'x (n{n-l)Y , ( n..Yf^^''^''-^^ ^ , n{n-l)(n-2)in-3) x\ | = (i4-a:)|i + -j-^^j^4- 12T2 {i-^xy^'-)' 199. Determine whether the following seiies is convergent ^ /3^ 3\-^ /i' 4\-' /5V 5\-\^ . or divergent: [j,- ^) + (^3, - 3 j + [j.- l) + - T. A. '^C' 502 MISCELLANEOUS EXAMPLES. 200. If n is any positive integer, find the value of n""-' - n {ii - 1)"^' + '-^^^ {7i-2y'' - . .. 201. Multiply out (1 - x) (1 - x') (1 - x') (1 - x*) (1 - x') ; and find the fonn of the series up to a;'' when the number of fiictors is infinite. 202. Shew that 203. Shew that money will increase fifty-fold in a century at 4 per cent, per annum compound interest, ha-\dng given log 2 = -301030, logl3-M13913. 204. Shew that Jf + 4j3 =^ + _-^^ .j — ... ^ 1+J9+1 + P+1 + 205. Find the number of ways in which a substance of a ton weight may be weighed by weights of 9 lbs. and 14 lbs. 206. If T^j — o"T7i — 9 ;^ be expanded in ascending powers ^i — AXj \\. — tdX -i" X J of X, find the general term. 207. If n is a positive integer, and x a positive proper fraction, 1 — «""*■* 1 - a" shew that — ^-r— is less than 71+1 n 208. Shew that n* - hi^ + 571"' - 2n is divisible by 12 for all values of n greater than 2. 209. From a bag containing 10 counters, 3 of which are marked, 5 are to be dra\vn ; and the drawer is to receive a shilling if in his drawing the three marked counters come out together : find the value of his expectation. 210. Determine whether the following series is convergent or 1 2^ 3^ divergent : 1 + ^^ +-^3+-^^+ "• 211. If the square of the sum of n real quantities is equal to times the sum of their products taken two and two together, n- 1 the n quantities are all equal to one another. MISCELLANEOUS EXAMPLES. 563 212. Shew that 25 {{b - cy + (c ^ ay + (a - ly] {{h - of 4- (c - ay + {a- by] = 21{(6-cy + (c-«)* + (a-6)T. 213. If a man 48 years old can buy an annuity of £150 a year for £1812. IGs,, interest being reckoned at 5 per cent., deter- mine what is considered the expectation of life at 48. Having given that log 2 = '3010300, log 3 = -4771213, log 7 = '8450980, log 1-1872 = '0745239. /t J. 1 214. If -- denote the r^^ convergent to ^^^ — , shew that qr ^2 215. Find the proper fractions which satisfy the condition tliat the sum of five times the numerator and eleven times the denominator shall be 1031. 216. Shew that if n be a positive integer, and x such that no denominator vanishes, 1 n I n(n-\) I (- 1)» x+\ la; + 2 1.2i«+3 a + w+1 ^ ^ {x + \){x+2) ... {x + n+\)' 217. If 2^ he a positive proper fraction, and a and b positive quantities, shew that {a + by a^~'' is less than a +2^b. 218. If 3, or 5, or 7, or 9 be raised to any power, shew that the digit in the tens' place is always even ; if 6 be raised to any power, shew that the digit in the tens' place is always odd, 219. There are three balls in a bag, and it is not known how many of these are black ; a person draws a ball from the bag and replaces it ; this is done three times : if every drawing gave a black ball find the chance that all the balls are black. 220. If cc = 7 + 7,^ - — . . . shew that y = x- ■-. — - ^— ... •^ 2?/+ 2y+ ^ 2x'- 2a;- 221. If ^, + — ^ + TT-T — ^ 1, shew that 26c 2ca 2ao two of the three fractions on the left-hand side must be equal to 1, and the other to — 1. OtZ .) 224. Convert / ia^ + ,-) into a continued fraction. 564< MISCELLANEOUS EXAMPLES. 222. Solve yz+zx + xy = 0* -x"^ ^h"^ -y^ = c* - z*. 223. If jO yeai-s' purchase must be j)aid for an annuity to con- tinue a certain number of years, and q years' jDurchase for an annuity to continue twice as long, find the rate per cent. ye*?)' 225. Resolve 2x^ - ^Ixy- \\y^ -x ^ 3-iy - 3 into rational factors of the first degree. 226. Shew that a recurring series whose scale of relation is 1—px — qx^ is convergent or divergent according as x is numeri- cally less or greater than the numerically least root of the equation 1—px- qx^ = ; the roots being supposed real. 227. Shew that if all the letters denote positive quantities and p , 2h) 2h •'■ ^^^ ^u ^^25 ^31 •-• ^^'^ both in ascending or both , „ ., 1 2h^h^ -^ P^f^i^ + ■ ' ■ + P ci>^ . in descendino: order of maojnitude, " " is 2h+P2+ "•+2^n greater than ( -^ '[ ) . _ 228. If a^ 4- 6^ = c^, and a, h, c are integers, shew that one of them is divisible by 5. 229. A number, of n digits, is written down at random: shew that whatever be the value of n, provided it be given, the chance that the number is a multiple of 9 is ^ . 230. If n be any positive integer, shew that the integer next greater than (3 + ^5)" is divisible by 2". 231. If the two expressions x^+2yx^+qx+r and x^+p'x^+q'x+r' r-r p'f-pr' q'r - qr' liave the same quadratic lactor, then 7 = ^ — y . P~P 9.-2 r — r 232. Shew in the preceding Example that the third factors ny /p' fp — ijpf are x + - — ^ r and x + - — —, r' respectively ; and that the quad- o Q — q ratic factor is x + , x + r — r r — r' p-p p-p MISCELLANEOUS EXAMPLES. 565 233. The present value of an annuity of .£100 on the life of a person aged 21 is by the Carlisle Tables of mortality £2150, interest bein^ at 3 jier cent. If out of ev^ery 10 children bom 6 reach the age of 21, find what sum ought to be paid down immedi- ately on the birth of a child in order to secure it an annuity of £100 on its reaching 21, the deposit being forfeited if the child dies previously. Ha\-ing given log 43 = 1*63347, log 2 = '30103, log 103 = 2-01284, log 1155 =3-0628. 234. Convert / (^^ — ) i^^^o a continued fraction, n being greater than unity. 235. There is a number, of two digits, which if its digits be reversed becomes less by unity than its half:, find the number. 236. Shew that if n be a positive integer, and x such that no denominator vanishes, 1 n n(n-\) x+l (a;+l)(a;+2) (a; + 1) (aJ-t-2) (cc + 3) + (- 1)" \n 1 (x+\) {x + 2) ... {x + n + \) x + n+\' n+\ -n—X 237. Shew that ic" — 1 is greater than n{x^ — a; ^ ) if n is any positive integer, and x any positive quantity greater than unity. 238. In the successive powers of 4 shew that the digit in the tens' place is alternately even and odd ; in the successive powei-s of 2 and of 8, shew that there are alternately two even digits and two odd digits. 239. A digit from 2 to 9 inclusive is taken at random, and raised to a high power : shew that the chance that the digit in 5 the tens' place is odd is ;r-^ . lo 240. Determine whether the series whose n^ term is 2n' + 37H-2 -. z— -^ r— -T- is convergent or divergent. (n4- 1) (71 + 2) (n + 3) ° ° 241. A series a^, 6,, a^, ^a? ••• is formed in the following way : a^ is an arithmetical mean between a^ and 6„_,, and b^ is an harmonical mean between b^ and a„_i. Shew that ab^ = aj)y_. 566 MISCELLANEOUS EXAMPLES. 242. Shew that tlie following equations are either incon- sistent or insufficient for determining the values of x, y, and z : X* — a^ = zx + xy — yzy y^ — li^ =xy + yz- zx, z^ — c^ = yz + zx- xy. 243. A person starts with a certain capital which produces him 4 per cent, per annum compound interest. He spends e very- year a sum equal to twice the original interest on his capital. Find in how many years he will be mined. Having given log 2 = '3010300, log 13 = 1-1139434. // „ 4a-f 2\ . 244. Convert . [c^^ — j into a continued fraction. .245. A farmer laid out £25 in buying sheep at £1. IO5. a piece, and bullocks at X5 a jiiece : find how many sheep and bullocks he bought. 246. By comparing the coefficients of the various powers of a;, shew that - 1 ~ xY + —J 1-0.')"^+ —, \.. ^ o^ (1 -a; "-^+ ... m^ ' m {m - 1) m (m - 1) (m - 2) ^ ' 1 n X nin—1) x^ iii — n \' m — n+\ 1.2 m — n + 2 ' " "* n being a positive integer, and m such that no denominator vanishes. 247. If all the n letters a, h, c ...h denote positive quanti- ties, shew that n {a^^^ + ¥^'^ + c^^^ -Y ... ^k^^^) i^ greater than 248. If n be a prime number, and N not divisible by w, shew that N"" — 1 is divisible by rC ; where m stands for rC - n^~\ 249. A box contains three bank notes, and it is kno^vn that there is no note which is not either a £5, a XI 0, or a X20 note ; one is drawn, found to be a X5 note, and replaced : determine the value of another di^aw. 250. Apply the process of Synthetic Division to divide a* + 3a;* — 12x + 4 by a:'' — 4a; -K 12 as far as the term involving a;"*; and give the remainder. 251. Solve x*y -¥x=^xy + x^y^ - 4y + 4, xy+\- Zxy* - x^y^: MISCELLANEOUS EXAMPLES. 567 252. There are two numbers a and h: it is required to find n intermediate numbers csj, a„, .-.ci^,, so that a^ — a, a^— a^, a^— a^, ...b — a^, may be in arithmetical progression with the common difference d. Find also the limits between which d must lie. 253. When the 3 per cents, are at 88, the sum of £100 is given for a pei'petual annuity of £3 per annum, and an annuity terminable in 30 years : supposing the value of money to be fixed by the price of the 3 per cents., find the amount per annum of the terminable annuity. Having given log 1*1 = '04131), logl-3 = -H394, log2 = -30103, log7 = -84510, log 3-658 = -50320. 254. If -^^^ , — , "-^ be three successive converijents to J{a' + 1), shew that 2 (a' + 1) q„=2^n-i + Pn^i, ^IK = Qn~i + Qn^i- 255. A boy laid out a shilling in buying apples, pears, and peaches ; the apples were five for a penny, the pears were one penny each, and the peaches were twopence each, and he got a dozen in all : find how many of each kind of fruit he bought. ci + hoc 256. If '" be expanded in powers of x, shew that (l-c..)(l-f) a + bc— {ac + h) c' the coeflicient of £c" is c''(l -C-) 257. ehew that [nf is less than i^'Lt'^^^'l±^^^T ^nd that {nf is less than fli^^LtlTj". 258. If ?i be a prime number, and 2i not divisible by «, shew that N"* + 1 or ^'^'" -lis divisible by n* ; where m stands J, n{n-\) for — ^-^— . 259. A number taken at random is squared. Shew that it is an even chance that the digit in the units' place of the result iB an even number, that it is 4 to 1 that the digit in the tens* place is an even number, and that it is 59 to 41 that the next higher digit is an even number. 5G8 MISCELLANEOUS EXAMPLES. T ^1 • ^(l + cx) {l+c"x){l+c%) ... ., 260. In the expansion of ;.- f-p, i-^-7^ o— — , tne num- ^ (l-cx)(l-c^x){l-c^x) ...^ ber of factoi-s being infinite, and c less than unity, shew that the coefficient of iC* is c** -/.- — ^,^.- — 4;-.i {,'" , , pr'. 261. If a and /3 are the roots of the equation aaf + hx + c = 0, find the value of a* + a/S' + jS*. 262. If the 9?i*^ term of a series in harmonical progression be n, and the n*^ term be w, then the r*^ term will be — . r 263. The first term of a certain series is «, the second term is h, and each subsequent term is a geometrical mean between the two preceding : shew that as n increases the n^^ term tends to the value U{ah^), 264. If T be a proper fraction shew that it may be expressed thus : 7- = — ' ^ + , . . H , where o'l , q„....q are 5 positive integers. Take for example = . 265. The diameters of two coins are '81 and 'Q^Q inches respectively : find the smallest number of coins which can be placed in a row of 9 feet long. Find also the smallest sum of money which such a row can be made to represent, supposing that the value of the larger coin is twice that of the smaller, 266. Shew that the difierence between any two consecutive odd convergents to J{a^ + 1) is a fraction whose numerator is divisible by 2a. 267. In a geometrical progression of which ^11 the terms are positive the arithmetical mean of the extremes is greater than the arithmetical mean of all the terms. 268. If a' + 6" = c", and a, h, c are integers, shew that ahc is divisible by 60 ; and that if a is a prime number greater than 3, then h is divisible by 12. MISCELLANEOUS EXAMPLES. 5G9 2G9. There arc n tickets in a bag numbered 1, 2, ... ti. A man draws two tickets together at random, and is to receive a number of shillings equal to the product of the numbers he draws : find the value of his expectation. 270. If -.4 be the present value of an annuity of £1 on the life of an individual, shew that in order to receive £P at his death the payment to be made immediately and repeated annually during P AP his life is -^ — , where 7? is the amount of XI in one year. 271 If ^(y + --^) ^ y{^ + ^-y) ^ ^{x + y-z) log a; log 2/ log^ ' shew that ' tfz^ = ccV = x^y*. 272. Solve J{x' + a'){y' + h') + J(x' + b') {y' + a') = {a + h)\ x + y — a + h. 273. Find a series of square numbers which when divided by 7 leave a remainder 4. 274. If — be the ^z^*" converging fraction to J(a^-\-\), shew that g--^V(a'.l) <"^^gZl>'-<--;<^^\ 1 + 1x x^ 275. Expand — ^ — =-77— in a series of ascending ( i 4" ox) \^~~ 1-^Xj powei-s of flj, 276. Find the scale of relation in each of the following series : l+4:x + 1 8a;' + 80a;' + 356a;' + . .'. 1 + 2a; + 3a;' + 8a;' + 1 3a;' + 30a;' + 55a;« + . . . 277. If aS' be the sum of the m*** powers of the n positive quantities a, b, c, ... k ; and P the sum of the products of the quantities m together; shew that |n— 1 S is greater than \n — m \m P. 278. If n be a prime number greater than 2, shew that any number in the scale whose radix is 2/i ends with the same digit as its n^ power. 570 MISCELLANEOUS EXAMPLES. 279. A bag contains 5 coins, and it is known that they can be nothing but shillings or sovereigns ; two shillings are drawn together, and are not replaced : determine the value of another draw of two coins. 280. If n be a positive integer, and m such that no denomi- nator vanishes, shew that -{i+«^r- / , , (1 -^ ^r-' + , '' ^\ '^}^ ON (1 +^)"'''- • > ■ m ^ ' m(7a+l) ^ ' m {m + 1) (7?i + 2) ^ ^ i (1 _ xY- '' a - xY-' + '^(^^-^) a-^xY-'- \ (m + n - 1 {m + n-2>)\d ) x^ — 2x— S 281. Determine the limits between which -^r-r, — r lies 2x + 2x+l for all real values of x. 282. Solve a;-U 2/^ = ft^ {x' + y')K (2x2/)^- = b. 283. If "" be the n^^ converijent to the continued fraction % 1111 — — — — 5 — . . . shew that p and q are respectively the co- a + b+a + b-h ^ " ^'' ^ / . /.I . 1 + bx~ x^ efficients ofx" in the expansions of the expressions ., — v-f — wz—i — ;? J. — [ClO'TjjjX -r X , a + (ab + l)x- x^ and = j—j :r~ r. 1 - (ab + '2)x' + x* 284. Shew in the preceding Example that if X and fx are the values of x^ fpimd from the equation 1 - [ab + 2) x^ + x* = ; ab(k"-ur) X^-"' - fji"*' - X" + fi.'' 285. Find two numbers such that the first may be equal to the product of the digits of the second, and also less by 100 than twice the second. 286. If A^^^ denote the value of an annuity to last during the joint lives of m persons of the same given age, shew that the MISCELLANEOUS EXAMPLES. 571 value of an equal annuity to continue so long as there is a sur- vivor out of n persons of that age may be found by means of tables giving the values of ui^ from the formula 287. 1£ X, y, z be real quantities, shew that a' (x-y) (x-z) + h'{ij- x) {y-z) + c' {z - x) (z-y) cannot be negative ; provided that any two of the three quantities a, b, c are together greater than the third. 288. Shew that any square number is of one of the fonns 5m or 5m ±1. Shew that n^ —n is always divisible by 30; and if?^beoddby 240. 289. A bag contains n balls, but nothing is known about their colours. A ball is drawn out and found to be black ; it is replaced, and then a second draw is made with the same result : supposing the ball drawn the second time to be replaced, shew that it is 3/1 + 3 to n—\ in favour of a third draw giving a black ball. 290. If ic is a proper fraction and p positive, shew that w^as" is indefinitely small when n is indefinitely great. 291. If 1, a?, jc^ and 1, y"^, y^ be each in H.P., shew that — ?/', y, 07, £C* will be in a. p., and that their sum will be x^-\-y^, supposing X + y not to be zero, and x and y not to be unity. 292. Shew that IV + 3V' + 5V' + ... + (27z -4)^ r" r (1 -f 6r -f r') - {(2/^ - 1) ( 1 -r) + 2}^"-^' - 4r"-^' (1-rr 293. Shew that if r be less than unity and the series in the preceding Example be continued to infinity it will be convergent : and find the sum to infinity. 294. Find two solutions in positive integers of ar' - ly"^ = 1. 295. In converting JN into a continued fraction if the first two quotients be each 5, find N. 572 MISCELLANEOUS EXAMPLES. 29 G. Shew that if oj is positive the least value of the fraction x^ 4- 2a' is when x ~a. X 297. The amount of fuel consumed by a steamer varies as the cube of the velocity. She consumes 1 '5 tons of coal per hour at 18 shillings per ton when her speed is 15 miles per hour. She costs forother expenses 16 shillings per hour. Find the least cost for a voyage of 2000 miles. 298. Shew that if any odd number has an even digit in the tens' place, then all its integral powers must have an even digit in the tens' place. 299. There are three tickets in a bag numbered 1, 2, 3 ; a ticket is drawn and put back : if this be done four times, shew that it is 41 to 40 that the sum of the numbers drawn is even. 300. Prove that the continued fraction 1 1 1 1_ ^ ~ ~ 2 2~ ' 12 6 n 1 1 1 (-1)'"'' where S=r. — ::- o^^ +^~7 ""••• + ~7 — rT\' Hence find the value of the .continued fraction when n is infinite. ( 573 ) ANSWERS. I. 11. III. I. 1. 23. 2. 35. 3. 63. 4. 88. 5. 92. 6. 26. 7. 15. 8. 6. 9. 5. 10, 2. 11. 9. 12. 10. 13. 0. 14. 26. 15. 43. 16. 38. 17. 76. 536. ^ 18. 9. II. 1. 9a -76 + 4c. 2. lOx'-ix + U. 3. 12a;^ + 6xy -y' + Sx + 4y. 4. ix^ + a^x. 5. 2ah-\-2x''+2ax+2hx. 6. Za-h + c-M. 7. 2a;' + a;. 8. 2a' - aa;. 9. a-h + c-d, 10. 2hx+2hy. 11. a-h + G-d. 12. a-6 + c + ^. 13. a -76. 14. 5a. 15. 2a-h-d. 16. 12:K-8y. 17. 3a. 18. a. 19. 2a + x-2b + ij=d. 20. 3x-'. III. 1. 3pq + 2p'-2q'. 2. 7a'-hl6a'b-ab'-10h\ 3. a'-a'6- + 2a6'-6\ 4. a' - a'T + 4a6' - 46*. 5. a' + 4a'ic + 4aV-a;'. 6. a'-8aV+16a;\ 7. a-6+(a-6)'a;-2ax--ic'. 8. 60ic^+42a;'a-107£cV+10a:a'+14a^ 9. 6a;*- 96. 10. 4a;'-22a;-2/ + 42a,y-27/. 11. 12a;'-17a;V + 3a;2/- + 2/. 12. x'-xY + xY-y'. 13. x'- iy- + I2yz- 2z\ 14. 6a;* + x^y + 2a;V' - Uxy"" + 4y*. 15. a;* + x^ (j/ +z) + x^ (y^ + yz + z^ + xyz (y +z) + y'z^, 16. a' + 6'-c'+3a6c. 17. x' + y'+dxy-l. 18. x' + 151a; - 264. 19. x' - Alx - 120. 20. 4a;'' - 5a;' + 8a;* -10a;' -8a;' -5a; -4. 21. a;V10a;-33. 22. x' - 7x' + 21a;' - 17a;* - 25a;^ + 6x' -2x-i. 23. a« + 2a' + 3a* + 2a' + 1. 24. a* - a;*. 2o. »*-10a;' + 9. 26. a;« + a;*-fl. 27. a;«-a;V+2a;'a6-(6'+2ac)a;*+2a;'(6c+a£?)-(c'+26J)a;'+2a;ct/-c;'. 30. a6c + (a6 + 6c + ca) a; + (a + 6 + c) a;' + a;l 31. X* -x^ {a + h + c + d) -\- x^ {ah + aG + ad + hc + hd + cd) — X (bed + acd + abd + a6c) + a6cc?. 32. 26V + 2cV + 2aV - a* - 6* - c*. 33. 6' - d\ 574? ANSWERS. IIT. IV. V. VI. 34. 4 {a' + h' + c' + cV), 36. 2 {a' + 6' + c"). 37. ^x\ 38. 2 (a* 4- i' 4- c"). 39. 4 (6V + cV + rt'6'). 43. a" - 22a;' + GOx" - 55a;» + 12a3 + 4. 44. re" - 2a;Vfc + 2xW - 2xW + 2ajV - 2a;a^ + a\ 4o. rt-'-rt'6-2a"Z>-' + 2a'6' + a6'-6\ IV. 1. aj^'-aj + l. 2. 9a;'-6a:y+4/. 3. rt' + c^6-6^ 4. a'-3ab. 5. 32a;* + 1 Ga;^ + SxY + ixY + 2a;^* + y\ 6. a* - a^'b + a'6' -- a6' + h\ 7. a;' + y\ 8. x' + 3aj + 2. 9. 1 6a;* -8a;V + 4a;y- 2a;/ +2/*. 10. x'-xy + y\ 11. a;'-a; + l. 12. a'-2a6 + 36^ 13. a^ - 2a% + 2ah' - h\ 14. 16a' - 24a'6 4- 36a6- - 276'. 15. a;' + 2a;' + 3a;' 4- 2a; + 1. 16. x' - 5a;= 4- 4. 17. a' - 2ah 4- 36^ 18. a;" - 8a;' 4- 16. 19. a;' 4- 3a;- + a; - 2. 20. 2a;'- 8a;' 4- 3a; - 12. 21. a + x. 22. x'-u''. 23. a + h + c. 24. 3a;'-2a6a;-2a'6'. 25. a;' -2a; 4-1. 26. 3a' 4- 4a 6 4- 6'. 27. x'^-xy + y' + x + y^-l. 28. a' + 6' 4- c' 4- 6c 4- ca -^ a6. 29. 6(2a'4-3a'6-a6'4-46'). 30. a6-ac;4-6c. 31. h + c-a. 32. (6 4- c) (c 4- a). 33, a* - 4crbc + 7b'c\ 34. a' 4- aa? 4- a;'. 35. (x 4- 2z) y' 4- (a;' - 2z^) y - a;-;; (a; + z). 36. a6 4- 6c 4- ca. 37. a;' - (a 4- 6) a; 4- a6. 38. a;-6. 39. a6 - ac 4- 6' - c'. 40. a'4-6'4-c'. 41. a4-a;. 42. (a 4- 6 -c- ^) (a- 6 4-c- c^). 43. X- - ax 4- a'. 45. The quotient is 7xy {x 4- y), 46. Eacli is a6c - a// - hq^ - C7*' 4- 22)qr. 47. (a - a;) (a 4- a;) (a' 4- a;') (a* 4- a;*) (a* 4- a;**). 48. (a 4- 6 4- c) (6 4- c - a) (a - 6 4- c) (a 4- 6 - c). 49. (6 4- c 4- dJ - a) (a 4- c 4- c? - 6) (a 4- 6 4- (Z - c) (a 4- 6 4- c - c?). V. 2. 9. 3. 70. 4. 6. 5. 2/V ll2/'4- 47^4- 932/ + 69. YI. 1. a;-2. 2. a? 4- 3. 3. a;' 4- 2a; + 3. 4. a;4-l. 5. 3a;4-4a. 6. a;-?/. 7. 3a;-7. 8. a;-l. 9. x-2. 10. a;' 4- a; 4-1. 11. aj4-2. 12. a;-3. 13. 2a;-L 14. x-y. 15. a;'4-2a;4-3. 16. a(2a-3a;). 17. 2a;-9. 18. ax-hj. 19. x' + {a + y)x + y\ 20. (a;4-l)'. 21. 2a;'-4a;' + a;-l. 22. aj-2a. 23. a;-2. 24. a;'-l. ANSWERS. VII. VIII. 575 . VII. 1. (2x'+3x-2)(3x+l). 2. (a^- 1) («+ 2). 3. (x'-0x'+23x~16)(x-7). 4. {Zx-2) (^x'-ix'-x + l). 5. (a;+l)-"(a;'-l). G. (x' - f) (x' - Af), 7. 16aj'~l. 8. X {x' - 1). 9. {x' - W)\ 10. {x-l){x-2){x-S)(x- 4). 11. {x-2)(x-3){x-i){x^b). 12. (a;^-l)(a;^-9)(a;+7). 13. x*-Ua\ U. (aj-«)(a;-6)(a;-c), 15. (aj + c)(2a;-36)(a;'+«a;-6^. 16. 3Q{a*-h')(a'-hy (a^-h'). VIII. 1. ^. 2. ^. 3. a:-3. 4. a + h. 5. oj+l. £c + 7 ic- 5 . 3a;-l ^ 3aj + 2 . 2a;4-3 . 3aj + 9 10.^? 11. ">!"^? 12.-.-.^.. 13. ^"-^^^ aj + 5 * ' aj^ - 2a; - 3 * * a;^ - 2a; + 2 * ' a;^ - 3aj + 1 ' 11 ^' + ^' in ^ ifi 7 (a;'' + a;y + y') a' + b' o 2a ^. 9 18. i. 19. -— ^:L— . 20. — . 21. . .r^ ^-,. 22. rr-^^^^— ox. 23. ^^\ 24. JL_. 25. 0. (a;^- 1) (2a;+3) x^-h' x + 2 ,2G. -^.. 27. "'r/"^/ . 28.^14^. 29.-^. a* -6* {^-y) {a -by a + x 30. ?1|^. 31. 0. 32. 0. 33. 0. 84 o4. -y — — — — =-1. 3i). -r . 36.0. 37. 0. (6 - c) [c — a) [a — b) aoc ■38. Mi. 39.-^^. 40.?^. 42.1^. x{a + b) y (x + y^) iy x 43. -^ ^. 44. -. 45. 2. 46. -, , — -.. a*-x a a +ab + b 47. -T + -2 + 1. 48. a;- 4- 1 + , . 49. ; , ,-. . oO. -^^ ~ . a a a; (a; - o) x [a + x) 51.41^. 62.-,-^^^. 63. -A. 54. l±y. 36' (a + 6) X ~xy + y^ x' + y^ y 55. 1. 56. 1. 57. a;'-a; + -- -3. 58.-^-. X X X 576 ANSWERS. VIII. IX. X. 59. ^^±^±1. GO. a^-6^+c»+2ac. 61. ^±^. G2. a'-5'+c'-2rtc. 63. p; . 64. ■ 65, — -r-. x+ha ax ac+oci ^^ a' + x* -^ hc + ca + ab _„ a* + a%" + b* 2ax 0C + ca- ao ab {a - b) bc{b-cf xy -, {ar_ + by 4 ^.^ adf+ae 76. -^ a;^-a** 26c * * 3 (a; + 1) * ' bdj + be -t cf IX. 1. 1. 2. 20. 3. 3. 4. IL 5. -. 6. 13. 7. 8. 8. 1. 9. 7. 10. 7. 11. 4. 12. 3. 13. 5. 14. 28. 15. 2. 16. 2. 17. 3. 18. 10. 19. H- 20. 21. 21. 5. 22. i. 23. 13.^ 24. 9. 25.4. 26.4. 27.9. 28.^. 29.13. 30. | . 31. 4. 32. 56. 33. 7.. 34. 8|. 35. 41. 36. 2j%. 37. If. 38. 3. 39. .2. . 40. 12. 41. 12. 42. 2. 43. 3. 44. -2. 45. 1. 46. 1. 47. 5. 48. ~, -^ «o ^r^ 8a ^-i cd-ab ^^ a^{b-a) 49. 3|. 50. ^. 51. -. J. 52. j-{ :• ^ 2d a + b-c-d b (b + a) a (1 - Z>^) p, , a-c + b^a + c^b — a — b — c ^ ac ^^' J^^T^T) • , • ac + &c + a6 - 1 * . T ' ^^ ab {a + b- 2c) ^^ rb - eg ^g _a^ ^^ ^^-^^^ • a'^ + 52_^^_^^' • pQ^^ci' 'a + b' .' m — n' 60. ^ . 61. ^(a + 6 + c). 62. 2. 63. 20. 64. 5. . - X. 1. £1290, X2580. . 2. £120, £30Q. 3. .£5,. 4. £140. 5. 28, 18. 6. 38 cliildren, 76 women, 152 men. 7. £720. 8. £144, £240, £210, 9. £350, £450, £720. ANSWERS. X. XI. XII. 577 10. A £162, B X118, C £104. 11. 3456, 2304. 12. 126 quarts. 13. £2. 155. 14. £3. 10^. 15. £600, £250. 16. 400 inches. 17. 30. 18. 42. 19. 7, 8. 20. 8, 6, 3, 2 ; 24 kings in aU. 21. 3. 22. 6 shillings. 23. £3600. 24. 11 oxen, 24 sheep. 20. 5 shillings taken by each ; there were 20 shillings in the purse. 26. 240. 27. 90 by 180, and 100 by 230. 28. 48 minutes. 29. £8750. 30. 5. 31. 60 oranges and 240 apples, 32. 10 from A, 4 from B. 33. 11, 22, 33. 34. £420. 10^. 35. 6/i past one. 36. ^. 37. 2s. Sd. 38. 40. XL 1. x=n, y = L 2.x = 5,y = 7. 3. x^lG, y=7. 4. x = 2, y=13. 5. x = 8, 2/=l. 6. x = 2, y=G. 7. a; = 3, y-S. 8. x=r3, y=L 9. i>; = 12, 2/ = 3. 10. iu-4, 7/= 3. 11. x=l^, 2/ = 20. 12. a:=60, y=36. 13. x=\2, y=20. 14. x = -G, 2/=l^. 15. x=l8, y = 6. 16. x=7, 2/ = ll. 17. x = 2, y = 7. 18. a; = -4, y = '\, 19. x = i, y=\. 20. ^ = ^, y = -^. 21. a^=12, y=6. 22. x = 2, y = -l. 23. x = 3, y = 2. 24. a^ = 18, y=12. 25. x=5, 2/ = 6. 26. a; -10, 7/ = 5. 27. a; = 2/ = ^^ ^, „^ Qic + hd mc-ad 28. x=y=m^n. 29. 0.-3^, y^-lh. 30. x=— 7 , y=—r— . 2>l. x=h + c, y=^a + c. 32. a; = (a + 6)', y = {^i- &)"• XII. 1. cc = 7, 7/= 5, 2; = 4. 2. 03= 2, 2/ = 3, ;$;= 4. 3. x=\, y = 2, z = 3. 4. x=2, 7/ = 3, c; = 5. 5. a;=2, 7/ = 3, « = 4. 6. a: = 8, 7/ = 4, c = 2. 7. a; = 10, 7/= 2, 2;- 3. 8. a; = 4, 7/ = 3, s^ 5. 4 4 9. a: = 3, 7/ = 4, ;2; = 6. 1 0. a; = „ , t/ = 4, ^ = . . o o 11 7 7 21 ,, 1 1 1 11. 0.= ^,^ = -^, . = -. 12. x = ^^, y = 3, z=~^. ... 20 46 13. a: = 2, y = 3, z^l. 14. re = 6, y=y, -= 3- • T. i\. 37 578 ANSWERS. XII. XIII. XIV. XV. 15. a; = 4, ?/= 9, «=16, w=25. 16. u = 4c, x = 12, y = 5, z = 7. 17. a = 3, i/=l, u = d, z^b 18. a;- 3, 2/ = 2, w = 5, ;2 = -4. 19. rc=2,7/=4,;c-3, 7^=3, 1^=1. 20. rc=2, 7/=l, «=3, w=-l, v=-2. 21.^=^,2/4'"=2- ^^•^ = *-^^- 23.a:=2a,y=26,.=2c. 27. x = b +c-a. 28. a; -- abc, y - ah + hc + ca, z = a + h + c. XIII. 1. ^. 2. 250, 320. 3. ^. 4. 5, 6. 5. 42s., 26^. 8 15 6. 756'. and 35^. 7. 5 and 7. 8 7, 10. 9. 25. Gc/., I5. 8c/. 10. 1, 3, 5. 11. Tea, 55. per pound; sugar 4c/. 12. 50. 13. £3000, .£4000, £4500, at 4, 5, G per cent respectively. 14. 2 persons ; Gsliillings each. 15 8 and 12. 16. £540; 17 pence. 17. 300,140,218. 18. £70. An ox costs £10 and a lamb 1 85. 9c/. 19. ^ wins 21 games, B 13 games. 20. A lis., B 38s., G 33s., D 32s., E 36s. 21. 90 miles. 22. A could do the work alone 11 21 in 80 days, ^ in 48 days; A must receive — of the money, and B „ - of the money. 23. ^ in 5 minutes, B in 6 minutes. 24. 2|, 2 miles per hour; distance 5 miles. 25. 100 miles; original 777)7 rate 25 miles per hour. 26. A26,B1 4, C'8. 27. ^ in -^ days, ^in -^ days. 28. — ^ miles per hour. 29. 4 yards and m-n a-c 5 yards. 30. 27. 31. 63. 32. Coach goes 10 miles an hour; train goes 30 miles. From A to B is 16| miles; from A to C is 20 miles ; from C to ^ is 40 miles. 33. 600 yards. XlY.l.a. 2.'-^'. 3.^. 4 0. 6. ""^^^ X + o ' a' ' 2 8. x = a + b. 9. x-=a, y^b, z = c. 10. (a;+l) (aj+2) (a;+3) (a;+4). XY. 1. ^' + -^ + ^, 3. ^ = a-. 6.^. 7. aj = 6-c, ic'' + a; + 1 2 y = c — a, 2; = a — 6. 8. Clear the given relation of fractions ; thus ANSWERS. XV. XVI. XVII. XVIII. XIX. 579 we find {a + h) {b + c) (c + a) = 0, therefore one of the three factors must vanisli ; hence the required result follows. 9. Each cliild obtains £1920. 12«., and each brother £960. Gs. 10. x = -3a. XVI. 1. 1 -f 4^ + 10.i;' + 12a;' + Ox*. 2. l-2x + 3x"' - -ix^ + 3x' - 2x' + x\ 4. l+(jx+lox'+20x'+l5x'+6x'+x\ 5. 2 (l + lox'+lox^+x'). 9. The numerator will be found to be equal to 5 (1 + x^)* and tho 5 denominator to ( 1 + x'Y, so that tlie fraction = . . ^ ^ 1 + a;^ XYII. 1. x'-x + l. 2. x'-2x-2. 3. 2^:;" + 3a; - 1. 4. 2x" -x+ I. b. 2x^ - 3ax + 4rt-. 6. 5a;' - 3aa; + \a\ ■ 7. (x-ay. 8. a' + b\ 9. {a' + b'){c' + cr). 10. a'~b'+c'-d\ •11. a;-2-^. 12. x^-~+-. 13. 'f +^-•^ X 2 X 2x0, '14. a=' + (26-c)a + c'. 15. {a-2b) x' -ax + 2b-?,. 16. M4. 17. a;' -3a; + 2. 18. 2a;' + 4ca; - 3c'. 19. 2x''-Zcx + ic\ 20. 5-51. 21. 9009. 22. 22-22. 23. 111111111. 24. x-~. 26. The given expression = (o;^ — yz) {[x^ - yzf — {y^ — zx) [z^ — xy)] + two similar expressions = (o;^ — yz) x [x^ + y^ + z^ — oxy:::] + two similar expressions - {^^ + y + ^^- 3xyzY. XVIII. 1. xK 2. crl 3. -^-,-. 4. 1. 5. C' {bxy \^ 6. a^6 '^ + a'^b^ + a -Z/"^. 7. o;-^ + x^y - xy''- -y^-. S. a' - 1. 9. a + rt^-l +«-:'+«-'. 10. -4a-'6-' + 9a-'6. 11. a; + //. 12. a;i-a;-^a-^ + al 13. a" + 1 + a"". 14. 2.*;' - So;?/ + 2/. 15. « + a- ^ - 6. 16. - — , . 1 / . 1. ^ -^ r~i • X + 3a;a + ct' x'^ ly^ 18. 2a' - 36U Ic-^. 19. 16a.-' - 16.c^ + 12 - 4a;-i H- x'l XIX. 1. a" + a^^Ua^-6^ + a^»-'+a'z.Uz.». 2. 2^ + 2^ 3-^ + 2^. 3' + 2.3 + 2^3U3l 580 ANSWERS, xrx. XX. 3. 3^-3.5^ + 3-^5'-5l 4. -2670492. 7.3^/---4 + 3 /^ . 8. a-2Jb-^-h. 9. 1 + ^3. 10, 2-^3. 11. sj^ + J2. 12. VIO + 2 ^/2. 13. 3 V7 - 2 ^3. U. y/^^ + ^J . 15 (-(a+c)(5+c)) V I 2 J V I 17 V3/A_-'3X ^(^_c)(&-c)) 2— /• 1 f 16. ^^3^-^ + 18 4X1-^0 I 1+c 2 + 2/ l-^\) 19. 3^/3 -2. 20. 1. 22. 1 + 21. 1+V2+ V3. V2 \/2- 23. V<3 + ^/3-V5-l. 2L \ + J2. 25. 1 + ^/o. 2(o. J2> - J2. 27, JQ - ^5. 29. a: = 2o, oU. a; =7. 31. x^ r. 32. x= 4o XX. 1. x=\, 3. 2. 1, 4, 5.3,1. 6. 17, 3' 9. 3, 11. 13. 17. 4, 21. \ 1 10' 11 7 5 * 1 9 2' 2* 25 ? A • 3' 10* 29. 3, 24 13* 33. 3, - 5. 37. 3,-*. 10 5 -1 2 ' 2 ■ 14 i i 13' 60' 18. 6,-1. 26. ±^6. 30. 2, 16. 34. 29,-10. 38. 2, J. 3 ^ ^ ^- 3 ' 2 • 7. - 4, - 6. 11 - -- • 3' 2' 15. 4,-1. 19. 5,--. 23. 10,-2. 27. - 1,". 31. -2, -16. 35. 10, -29. 39. 8,-8. 4 4-- • ' 3 8. 5, - 3 12. 2' 16. 3, - 32 7 • 1 "2* 4 3 20. 8, I 24. 2' 50' 28. 7,-^. 32. 5, - 3. 36.1,^. 40.10,-?. ANSWERS. XX, xxr. 581 41. 2, -3. 42. 24, t^ . 43. 3, - ^^-. 44. 3, -|. 45. 3,-1. 46. V, 1. 47. ~, 1. 48. 0,4. 49. 0, 1. 50. 1,-^. 51. 2+J3, and - (2 + ./3) 2. 52. a^b. 53. a^J(a'-b'). 54. — f , ^. 55. J (a + i + c ± ^(ct* + h^ + c'-ah -he - ca)]. 56. a + b + c. 2a6 ^^ ^ 2aS-ac-5c -„ 2a-h 3a + 2b 50. 0, 7 . 00. j 7 . a + b — '2c ac be CI. • ^ \ah +bc + ca =*= J\a'b^ + Vc^ + cV - abc (a + 6 + c)]]. In the following Chapters the iiTational roots ai*d the impos- sible roots have not always been given ; and some of the roots given are not applicable ; see Ai-ts. 329, 330. XXI. 1.1, J. 2.1,-2. 3. (-41)', 9. 4. U^(-ir. 5.2,3. 6. 2", (-1)'. 7. {->±V(^-c)r. 8. ±11. 12.5 / 13 \' J- I 8\i 9. *2,V10. 10. 8, i^. 11. 8, (-'^^2). 12. 2'-,(-3J". 13.4,-1. 14.4,1. 15. ic,(-^y. IM-l)^©'- 17. 'Tl. 18. 2".A. 10.9,-LS. 20..5. 21. ^.«^. 4 4 U o -> 22. 16,0. 23. 18,3. 24. 2' = 8 or - 10 ; so that a;= 3. ± , /3 ^ s n n — 1 25. 5, -8. 26. 0, *^, a. 27. x' = ^ or -— , . ' 2 ;t — ii 7Z 4- 1 28. x' = -ab^\{a'-h')s'^. 582 ANSWERS, xxr. 29. {J{x + 2) + J(x^ + 2x)Y -{a-x- Jxy, a quadratic in Jx, from ,vlnch Jx= ^^^ . 30. 1, ^^.^. 31. Multiply up and arrange x{s^^{a-x)-J{a+x)]=Ja{J{a'-x'')-a], square, etc. a;=:0,±^. 32. ± 2«. 33. 1,-^-. 34. 1, ^ . 35. a=2a, ±2r,V(-l), 3C, a;"=0 or -^".^ 37. ^ , - 1-' . 38. =.„,.!. 39.. I''. .^4?^. 40. V^^ 41.0.^"^^«)''. 42. "(l=.,/5). 43.x^=:-^. 44. «' = 9. 45.^'==^,. 2^ ^ ^ 4(w--I) 7a'-26' 46. ic' = — .-,^-". 47. {c±V(c -1)1"-^'. 48. 0, i!:- . 49. ^2a, ^aJ{-G). 50 ^ ^ 51 4(«+&)(a^ + 7r) 4 50. - , 3 . 51. 0, - ^^^^^^ ^ -^^^^ . 0.. 1, - 3 . 53.8,-^. 54.-. 5.x 1, ^3^. 59. 0, i {a + b + c ^ J{a' + b' + c' - 2hc -2ca -2ah)}. GO. 0,^^. 61. 0, ±^V + 6-). 62. 0,±J{mn+a{m-n)}. G3. 0, a( 1 i 2 /- ) • 64. Transpose and square; we get 2x (2a; + 1) ^(a;' + 2) = 2 (ic" + 1) {2x + 1) ; it will be found from this that the only solution is a;=-l 65. 1. 66. 4,-9. 67. 0,2. 68. 0,-5, 3 , - y . G9. 1, - 4, ^f . < 0. 1, ^- . 71. 2,-5, -J {-3^^/241}. 72. a + 2,-'''^J. 73. 2, - J . 74. 1, — 2. 75. x^ + 5ax = - 5a'' =t J{a^ .+ c*) ; whence x. 1 9 76. a;' + 3a: = -T or — V ; whence x. 4 4 77. ax ^^— ^, drc. ; x = -^{-l^J5). ANSWERS. XXI. XXTI. 5S3 78. « = (^ - 2 "*" 2) "^ (^ ~ 2 ~ 2/ • Q'^^^^^^'"^ ^"^ (^ ■" 2) • 79. {x' - xY - (x' -x) = a. 80. 4, - 3. 81. {Jx + J{x ^1)Y + Jx^ J{x + 7) = 42. a: = 9 or (^1^\ 82. (x-iJxY-^2{x-i^x) + \ = 0. a; = 7=t4^/3. 83. { Jx + J {a + x)Y + Jx + J{a + x) = h + a ; &c. 84. (a;' + rc)'+4(a;* + a:) + 4 = lGx*. a;=lor2. 85. {x'+ay = 2a'{x-ay. 8G. (x + -^- Y ■{- a(x + — ) + 5=— . \ CLXJ \ CIX/ Qi 87. I -2 ( 1 + 1=0. 88. re + - = -^ or - -7^. \a xj \a x) X 6 6 89. ix \-lix J 4-1 = after expunging ^{x - 1). 90. 1 + ^/3 =t ^/(3 + 2 J3), 1 - ^/3 =«= V(3 - 2 ^/S). 91. (a;+l)(a;^-a; + l) = 0. 92. (a;+ 1) {1 +?i (a;'-a; + 1)} = 0. 93. re = 5 is obviously one solution. 94. x=6 is ob\dously one solution. 95. ic = 5 is obviously one solution. 96. £c = is obviously one solution. 97. (x-* - 4) (re + 1) = 0. 98. re = a is obviously one solution. 99. 8x' - 1 + 8 (2re - 1) = ; therefore re = J is one solution. 4 1 / 2\ 2 100. re' - - = - ( re + ^ ) ; therefore .r = - - is one solution. 101. re* = 1 is obviously a solution. 102. re = — m is obviously a solution. 103. x^ a, h, or — (a + I). 104. re + p - 1 is a factor. 105. re (;> - 1 ) + 1 is a fiictor. XXII. 1. 3{x-5)(x + ^Y 2. (x + GO) (re + 13). 3. 2(rc + 2)Cre-^y 4. (re- G 2) (re- 2 G). 5. re'- 14re + 48 = 0. G. a:'-9rc+20=0. 7. rc' + rc-2 = 0. 8. re*-2.c-4 = 0. 9. 42, 3G, 117. 10. m=8. 11. ^-^^ , P (p' - ^q)- 12. ere* + 6re + a = 0. 584 ANSWERS. XXIII. XXIII. I. x = ^3', y = ^L 2. :r = GO, 40; 2/ = 40, 60. 3. a;=2;2/ = 2. 4. a;=4,-^^; y^3, '^ . 5. x=7, 5; ij = -5, -7. G. a;=2, 5; y-6, 3. 7. a; = ± 7, ± 4; y- ± 4, ± 7. 5 3 8. a; = -l, ^', 2/ = -l,^. 9. a;=lj 2/ = l- o o r 333 ^ 370 10. a;-±3, =f8j 2/ = ±5. 11. re = 5, ^^ ; 2/-= 9, -^ . 23 5 1 12. ic=±3;±36;2/=±5,=F — . 13. ir=±3,± _^ ; y = ±2, =t ,^ . V^^^ "' v^ 14. x = ^2,^jI) y = ^\, =^2 /|. 15. a:-±3,±-^; y^^l,^^^ : 16. a; = ±4, ±3 J3; y=^5, ±.^/3. 1-5 3 "" TO o '^3 227 17. a; = ±-^,2/ = ±— ^|. 18. a; = 3, - -^^ ; y = -i,-^^^, 19. ic = ± A^ 2/---=p./2- 20. a; = i6, 2/ = ±3, =f3. 21. a::=±3V2; 2/ = =^n/^,=^n/-'- 22. a; = 0, 4;2/ = 0,5. 23, a;=0,-l j 7/ = 0, --;". 24. a: = 0, 15 ; y = 0, 45. 25. a^=0,2,=tV-;2/=0,2,2=?V2- 26. aj=0, 4, -2; 2/=0, 2, -4. 27. a:=:5,|^; 2/^=3,^. 28. a: = 4, 2; 2/ --2, 4. 29. a;=:2, 0; 2/=0, -2. 30. x=l, 4; ?/ = 4, 1. 31. x=l, 10; 2/ = 10, 1. 32. jc^ 3, 2; 2/ = 2, 3. 33. a;=8, 4; 2/ = 4, 8. 34. a;=17, 1; y^l^H- 35. x = i,2,-l^J^-y=2,i,-l^ ^^^ 3 ^ ^~ ' ' ^V 3 • 36. a: = 4 ; 2/ = 1. 37. cc= 1, 4 ; ?/= 4, 1. 38. a; = 2, 3 ; 3/ = 3, 2. 39. x=^2,y=^2- ovx=^2, y=^2. 40. x = ?,,y^l; x=\,y=?>. 41. x=5, -2; 2/ = 2, - 5. •42. x = ^2,^l', 2/=:±l, =^2. 43. ^-i(9±V'3),2/-i(9=FV^3). 44. a; = =t3, =!=2; 2/-±2, ±3. 45. a; = ±5, ±3; y = ±3, ±5. ANSWEP.S. XXIII. 585 46. a; = ± 3, d= 2 ; 2/ = =^ 2, =t= 3. 47. The first equation may be written tlius, xy (y ■¥x-2>) = ^{ix + y - xy) ; combine this with the second equation x=± J{- 3), ± ^3 ; ?/ = 3 =f J{- 3), ± 2 ^3. 48. a;=8, 2; 2/ = 2, 8. 49. a; = 9, 4 ; 7/ = 4, 9. 50. a; =8, 64; y = ^^, 8. 61. a;=5, 13; y=--i,\2. 52. a; = 4,9; 2/=9, 4. 53. cc= 2, 8; 2/ = 8, 2. 54. V^' = 2 ± ^6, 1 { ^ ^l{ro) - 5} )Jy=-2^ JQ, i {^ V(15) + '^l- 55. x = 5,y=^3. 5G. x=^l,y = 3. 57. a; = ^'2/ = 2' 58. rc^ = 4 {.r ± V(^^+ 46*)} ; y^ = i {- a^ =^ J(cU 460}. 59. icy = i{2a-± ^/(2a* + 26*)} ; whence we may proceed. 3ab-a' , Zal-l' is 26 miles ; from ^ to C 52 miles. « -r^. -, T 2abc , , 6, Either xa = yb = zc = , — r ; or else xa + yb + zc = •^ bc + ca + ab '^ and x^y^-z--\. 11. ic-6, 2/ = 8, 2;=10. 12. x=^a{b^-c^), y = ^b {c^ — a^)^ z = ^c {a' — b') ; also x, y, and z may each = 0. XXVII. 1. 3. 2. G400. 3. 57. 4. ^^-^-^i-l^ . 9. Suppose xy ^ ad = be : then a + <:Z - (6 + c) = « - Z* - ( c - - ) = — — — - . ^ \ aj a 10. In the first the wine is <\ of the whole ; in the second |. 11. A has j£72 and B has £90 ; each stakes f'j of his money. 12. Female criminals four-fifths of the male. 5S8 ANSWERS. XXVIII. XXIX. XXX. XXVIII. 1. 9. 2. a-^bb. 3. 4. 4. 1. G. |. 7. 10. 8. 11x'==Ay\ 9. y=2a;4-?. 10. IG. 13. 10. 14. (r=+/^)l 15, We have y + z — x = A^ {x + 7/ — z) (x + z — y) = Byz ; tlius x^ — {y — zY — Byz, therefore x^ — {y + z)' = {B — i) yz, therefore {x- y - z)[x-it- y +z) = {B - 4:) yz, or —A{x + y + z) = {B—i)yz. IG. 2 (?i - 1) hours. 18. 4 hours. XXIX. 1. 1022G34. 2. 321420111. 3. 3015333. 4. 2084Gi. 5. 209. G. 624. 7. 2223. 8. 15^. 9. 1105^. 10. 22441. 11. 1?-G. 12. 75346-K 13. 134M11. 14. 124-9G. 15. 1099-39. IG. 1589-349609375. 17. 450, 1214; product 613260. IS. 3483. 19. 152. 20. 11111. 21. 44-4; in scale 3 it is 1001-2. 22. 62444261; square root is 7071. 23. IIOIIIL 24. 8ie7. 25. -739. 26. Eight. 27. Six. 28. Eleven. 29. Five. 30. Six. 31. Eive. 35. 2"'+2V 2^ + 2'+ 2V 2^ + 2 + 1. 36. 3' + 3' + 3*- 3^+1. 37. 3''-3'-3-l. 38. 3"-3^- 3^-3 + 1. 39. Three feet eleven inches. 40. Twentj-three inches and a third. 43. r" — 1 and 7-""^ ; r being the radix and n the given number. 45. The number is one hundred and twenty. XXX. 1. 800. 2. 4. 3. -333. 4. -28f. 5. -2. 6. 6U. 7. 5. 8. 425. 9. 0. 10. ?i(8 + n). , , n{\o ~ n) 11. — -p — . 12. Common difference - 3. 13. 9. 14. 4 or — 11. 15. 2/z — 1. 16. Number of terms is 10 or 12 ; last term 3 or - 1. 17. Common difference 7. 18. 5, 9, 13,17,21,25. 20. *J{2+4(?z-l)} that is n{2n-\). jj 21. 1111. 22. 20. 23. J(7^-l)?^(27^-l)7ards. 24. 1,1334. 25. Nine means, 3, 5, 7, ... 19. 2Q. Number of terms 19 or -2. 27. 5 or -10. 28. 4 or 7. 30. The number of terms is m + n — 1 or m + w ; in the former case the last term is 1 ; in the latter case the last term is zero. 31. 4 or 9. '62. p + q + {m-\)^. 37. 17. 38. 100 or -107. ANSWERS. XXX. XXXI. XXXII. 589 39. Number of teiTttS 7; middle tenn 11. 41. n'. 42. -vi(-l)". 43. l{l-(2?i+l)(-l)''}. 4G. 9. 47. ^ n (72+ 1) (n + 2). 62. 25 months. 50. -(19-.). 51.-^-^^, -jy-, j-^, -. 53. 15. 54. , hours. 55. 468. r + \ ^-^'^^- ^-V'©'-!}- 2--3i^'"-l!- ^-^ll-®"} 4.?{l-Q"}. 5.2. 6.1|. 7.1. 8.9. 9.10. 4 50 2 '>7 1 1 10.3. 11. -j«. 12.3. 13. ri. U. 1. 15. J. ic. . . 3 V2. 11. m. I) . 18. iz!!i::i' . q^:^i . ^ 24 \o ov 1-/* (1~'^) 19. 4-(?i+2)2-"^\ 20. 6-(2;i + 3)2-'''^\ 21. l|2 + (-l)'-'^^j. 23,81. 24. £108, £144, £192, £250. n* 1 25. -^ - {a^V-lV-U. 28, £3.45. 32. Common ratio -^tt-^ . a V 1 ^ ^ ^ ^ 10+1 33. -~ — —~ , 38. r = 2, a = 3 : r is found by an easy cubic. (r-1)- r-l ' ' -^ -^ „,?£. 15. 9, 3. 43.05,8. 45. - ■2' 2 ' 2' 2" -"•-,".- '"'• (l_r)(l-6r)" XXXII. 1.1, J. 2.1, J, J ^^. 3. Let;, denote it, then 1 = -U (h - 1) (] - 1) . 4. ^^^-^,?^ . 8. 2 and 4. 11. 2, 3, G. 12. Tlie terms are {-^ and ^ ; then the series can be continued. 14. "We may shew that A = -^ ; and G = -^ j- ; as A and 6^ arc thus known in t^'nns '2a- 2a - 6 590 ANSWERS. XXXII. XXXIII. XXXIV. of a and 6, we can find the two quantities in terms of a and h. 19. a* + abf a^ - b^, a' - ah. 20. The common difference in 2 the arithmetical progression fonned Ly the reciprocals is q- . ^ ^ 71 — i XXXIII. 1. 1341-1323. 9. A £100; i?£80. 7. 36 miles. ^XXIY. 1. 1120. 4. 34650. 195 195 8. 6, ^ 1_M) ' [K) \So ' G. 9. 2. 453600. 110 \2\3\5' [60 7. 8. 64 gallons. 3. 454053600. 20.19 19.18 1.2 ' 10. 2r. 1.2 11. 2* ^2^8- 12. Suppose one person to remain fixed, and all possible permu- tations formed of the other n—1 persons. This gives |n — 1 as the number of ways. But this counts as different ways a pair of cases in which each person has the same neighbours, but the light-hand neighbour of one case becomes the left-hand neigh- bour of the other, and vice versa. If such a pair of cases is counted as only one case, we must divide our former result by 2. For example, if there are three persons, there is only one way of arranging them, in the latter view. 13. [9, 1 10 — [9. ,, 12.11.10 16.15.14.13 14. ^ X i- 15. If there is only one thing, it may be given aAvay in ?i ways ; then as a second thing may be given away in n ways, there are ?i^ ways of giving away two things; and so on. 16. qz -~ 2'r + 1 ; r—S. 17. \m in -^= — X ,— , "~ — X j 5 -i- r. Or if the m things are exactly r m — r \s \ii — s ' \s + r ahke, and also the ?i thmgs, 1^77 . 18. n (n — 1) (n — 2) 20. 4080. 21. 86400. 22. |^ x ^ ; if however the three letters are to retain an invariable order, the answer is |5., 23. 10.9.8.7-9.8.7 with 4 fiags ; 10.9.8-9.8 with 3 flags ; 10.9-0 with 2 flags ; 10 with 1 flag. 5275 signals in ail. ANSWERS. XXXIV. XXXV. 591 152 24. 90. 25. 36. 26. 3x14x14. 27. ^r. 28. nV^^- 29. 120. n(n-l) ;?(;?-!) ^^(^^ - 1 ) (^^- 2) ;j(p- l) 0j-2 ) ^^- ~0 O"""" ^- ^^' ~~~[3 [3 • |24 32. Increase tlie preceding result by unity. 34. , .^ , -,. 5. 35. [7; if however each set may be in order, either from left to right, or from right to left, the answer is 8 x [7. 36. L 8.7.6.5 cases without 7 6 (4 repetition. II. -— ^ x U cases in which a occurs twice : also as 1.2 2 manyin which ^ occurs twice; and as many in which n occurs tv\dce. K . . . III. ,-— — cases in which a and i each occur twice: also as many in [2[2 ^ ! . which i and n each occur twice j and as many in which a and u each occur twice. Total 2454. 37. 53. 39. [4 x 11111 x 15. XXXV. l.H4ia"6'. 2.^^«a'. 3. i^HM^'V. i.'J, I. 'A |4 200*^ 2001 * 4. ^^-^ a''af'\ 5. 625-2000a;+2400a;'-1280a;'+256x'. ct'x^ 10. — ^(aV + aV). 11. 64a'-96a* + 36ci^-2. 12. \0c\ 14. This follows directly; or thus, {\ + xy^\\-x)^- (\ + xy{\-x^. 1 2/1+1 \)i — r\n+r-\-\ 12/1+1 16. From 2nd to 5th terms of (3 + 2)'. 18. ^7 ;— r (-l)''~^ 1^1 (- ir' a;^"-^'-^' 1 2n{^iy-'x"'-"'-' (- l)"[2ri -J |2^i-r + l ' |r-l |2n-r+l ' \n\n ' 20. {x' + a^)" = [x + a ^/(- 1)}" {x-a J{- 1)}" = {.l+i?^/(-l)}{.I-i?^/(-l); = ^» + i?». 592 ANSWERS. XXXVI. XXXVII. (>^-l)(2>iri) (3,^-1) {(r-l)n-l} 7.9.1i:....(2,-. 5)^_ 20_ 1.5.9 .(4.-3) ^ 28. 2nd and 3rd terms t ^ o = q • 2^- ^^'^^ *^^'^ = "t'-^o" ^ = ^ • loo i . Z ^0 n^ ., , . -, 3.4.5. 6 /5\* 9375 30. oth and oth terms = - 2r 4 \7j 2401* 31. 3rd term = ^'-3 f r^,^ . 32. If n = 1 the 2nd and 3rd 3. b \12/ terms are the greatest ; if n = 2 the 2nd term is the greatest ; and for all other values of n the first term is the greatest. 33. ILilll?. 34. Sixth term. 37. "^p^ {2n' + in + 3). i 3 o ~ 13.5... (2?'-l) 38. Coefficient of a;-' is '"!,'' j coefficient of x^'*'^ is 2 a '\/r obtained by dividing this expression hj a, 41.M--J , ,, , . /.. ,9 lA^ ,^ S^^(2v^ + l) (2^ + r-l ) that IS, j2. 42. ■. 4o. ^ ■, -. XXXYII. 1. 6. 2. -16. 3. 2^3'+2^3 + 2^3^ + 3*=1905. 4. 3. 5. - 2^5 + 2^ 3\ 5 - 2'3^5. ,-,J_2^ 2^ 2^ 2 \\ ' '— l[8|4■^l7:^|3"^|6J^'^|TT6"^|4;8^ 7. 2^ 5 . 7' - 2\ 3 . 5M + 2 . 5\ 8. - 64. 9. - 20. ^. 15 35 63 37 _^ 1 _. o r i ^ ^=5 ^^- "8 -T--8-=-T' ^^-"i- ^^' -3 + 6 + 10+^. ANSWERS. XXXVII. 598 13 f3_^-I:iI-l^V 14. 50. 15. ^^^-'-^'-' - ^- /3.7 7.11. 19\ 24: 16. The expression is {(1 +x){l - xy-y. Hence the coefiicient is 7.6.5.4 7.n.5 14 7.6 UA5 7 14.15.16 14.15.16.17 1 - 1 1 Q n(n-l){n~2){n-3) n(7i-l)...(n-4) 1 / . 711 + 1. 1 b. FT «J H jn— o -^ '' , n{n-\). .(71-5) n(n-l)...(n-6 ) n(n-l)...(»-7) , 511 T^ L? 19. 0. 20. 5a.* + 20a,a^a^^+10a^\'. 2^^ ^O^-m^-2) ^.-.^^^. ^ .(n-l)..^....(n-3) ^^,.,^^.^^ ^n(n-l). .(n-4)^ 22. -23. 23. -| +^'. 5 Jo 7)1 (m — 1) (m — 2) s -^ _- 24. ma,+ m{m-l)a^a^+-^ -^ ^a,'. 2o. 20, 26. -210. 27. 1260. 28. 12G00. 1.2 lo ^(n-l)(n-2) liP |U .0. ^^^ cZ (a + 6 + c). ^1- j]3pl- Mi^* o. 1 r /I 36^ , /35 56^ 3 /3 156^ 3.>6^ , ,2 2yV2 2 J \S 4 8. 36. a-'-a-'bx-{a-'c-a-'b')x' + {2a-%c - a-V) x^ + (^'V - Za'^b'c + a-%*) x\ 37. l-.a:4- "^V^) cc-- "("-^»"-^^ .-. 38. May Le proved by Induction. 39. For the first part put x=l. For the second part, let S denote the series, so that S-a + 2a„ + 3a^+ ... +7ira^^^; and as the coefficients of tei-ms ' equidistant from the l^eginning and the end are equal, by Ex. 38, >S'= a„^_j + 2a„,_, + . . . + 7ira^. Then, by addition, 2S= 7ir {a^ + ttj ... + a„J = nr (r + 1)". T. A. 3S 594 ANSWERS. XXXVII. XXXVIII. XXXIX. XL. 40. {1+X + x-f - a^+ a^x + a^ +... + a^„_,x"'-'+ ct^„.,x'"-' + a^x'"; change the sign of x, and, since the coefficients of terms equi- distant from the beginning and the end are equal, we have (I ~x + x^)" - a, — a„ ,a; + a, „x^ - a„ .pc^ + ... \ / £ii 2n — 1 2u — 2 2'i — .5 ]Multiply together, and select the coefficient of a;'" ; this will there- fore be equal to the coefficient of x^" in (1 + a; + xy {\-x + x-y, that is, in (1 + cc^ + xy. Tlien put «p" for a^^^, a^ for «,„_,", ... and divide both sides by 2. XXXYIII. 1. 4. 2. 2. 3. 1. 4. 5. 5. 3; -2. 6. -698970-2; -732393. 7. -778151-3. 10. i [log 10 -3 log 2}. 15. 20. 20. About 125 years. XXXIX. 1. This is an example of equation (1), Art. 545, in vrhich 7ii = (a; + 1) (a; — 1) and n — x^. 2. log{x + 2h)x-lor^ (a; + A)= = log[l -— ^— |. 3. See Ex. 1. 5. log(3 + 3x + a;^)a;-3 log(l+cc) = log| 1 --r -3L 6. We have ^x 2x — 1 to find a series for log (x + 1) — I02: x + ~ , I02; (x — 1 ), that is, for loi:; ( 1 + - ) + ^ ^ I02: (1 — ) , that is, for ' °\xj2x-\-l°\xJ' ' ■ 1 2x . f^ l\ 1 . '^x ^ /", xy J^ x^ x^ X XL. 1. Series = — s 1 &c. [ ; convergent by a {X X + a X + 2a } Art. 558. 2. Divergent if a; >1, convergent if :c - , and the series is divergent. Of + I n 3. Convergent if a>l; divergent ifal, convergent if £c1 +-- + ^ + &c., and therefore divergent. 1+21+31+4 ANSWERS. XL. XLT. XLII. XLIII. XLIV. XLV. 595 7. Divergent if a;>l, convergent if x 1, convergent if a; < 1 ; if a; = 1 it is a series discussed in Ai-t. 562. 10. Convergent if a; < 1 , divergent if ic> 1 ; if a: = 1 the results are the same as in Art. 562. XLI. 2. X900. 3. ^-^. 4. 2^. 5. 40 : 11. 6. Between 48 and 49. 7. Nearly 32. XLII. 1. 7 years. 2. 120 days. X y z the eriven sum o t-. i. i.i ic • i. 4 . — = J— z= — - = -5 8. Equate the coemcients Jf" ir' R-' R-^ + H~' + li'' ' of x^ in (1 +a;)'*=(l+ic)-(l ■\-xY~\ 9. Equate the coefficients of a;"* in (1 + xy = (1 + rr)""'"""' {x + 1)""'. 10. It will be found that the whole coefficient of a vanishes, and also the whole coefficient of /?. XLIII. 1. £24. 105. 2. Cent, per cent, 3. 4 per cent. 4. £6400. 5. 3^. 6. £7297-98. 7. £225iWr. 8. , ^ ; / °, = a little more than 9. 9. p^rr-TT— :» where ^ loH 5 - los: 4 ^ ^*' - ?^^ ^o " •'^cs is the first payment ; 7?i mnst be- less than R. 10. e . 11. p(^J:±^, 12. P{l-r)\ 13. ^x 2-617238. 1111 1 1 i_ J_ L_L XLIY. 1- 1 + 3:; 5:^7:;. 9' ^' r^2Tr;i+i+55* . 1 I_A_±i_L_L 4 1+lA L2__L-Ll_i. ^' 2+4+3+2+1+2+170* 4+1+1+1+2+3+1+3 3 22 355 c \ 1 ^ — r y ll3' 4' 29' 33' 161' 2 3 14 17 ? 1^ ^IT L^^ XLV. 1. y, 1^ -5 » e • "• 1' 6 ' 37 ' 22.S' o 3 4 11 15 4 33 268 2177 ^- 1' I' T' T* *• r 8 ' 65 ' 528 • 4 9 13 48 5 51 515 5201 • I' 2' y n* 1' 10' lUl' 1020* 38—2 596 ANSWERS. XLV. XLYI. 5 26 2G5 1^51 c 5 Z ?I ^ ^- 1 ' "5 ' "sr ' "200 • • 1 ' 1 ' 4 ' 5 • 7 22 29 51 .^ 10 201 4030 80801 ^' 1' y '4 ' T* • T' "20"' lOT ' "8O40r* 1 1 1 a 2a'+l ia^^Za 8ft* + 8a" + 1 ^^' "■^2'^2^ 2«+ '" I' "^2^' 4a^+l ' 8«^ + 4a * 1111 a-\ a 2a:'~a~l 2a?~\ 12. «-"^+y:^2(«-1)+I+ 2(a-l)4-"*~I~' 1' 2«-r' 2(x 1 1 11 a 2ft + 1 4ft' + 3« 8ft- + 8ft + 1 2+ 2ft+ 2+ 2ft '■' 1' 2 ' 4ft + l ' «ft + 4 14. a-1 + -— 2+ 2(ft-l)+ 2+2(ft-l) + ft - 1 2ft - 1 4ft- - 5ft + 1 8ft^ - 8ft + 1 1~ ' 2 ' 4^ 3~^ ' 8ft-4 [13 and 1 4 are connected, because a' - ft = (ft - 1)' + ft - 1 .] _ 25G _ 1520 ,„ 1 ,1 ,^1 ,1 lo. --p. 16.:^^^. 18.7j7-,and— -— ,. 20. and 71* 273* (44)^ 2(49)^* (240)^ 2(2111)^' 1 1 1 3 13 42 ;,. 485 211 (273)^ 2(2885)^' 2' 7' 30' 97' "''396' 80 •" 1549 251 114 17 ^^- -360"^ 360* ^^- Tf ^^- m- ^^- '^^• 33. Positive root of a;* + 2a;- 2 -0. 34. That of 7a;'- 8a;- 3=0. 35. That of 7a;' + 8a; -3=0. 36. That of 59a;'- 319a; + 431 = 0. XLYI. 1. a;=2, 2/=l. 2. fl; = 4, 2/ = 5. 3. a; = l or 6, 2/ = 20 or 1. 4. y=l+7«, a;= 41-10)5. 5. a; = 2o-7^, 2/ = 25 + 3.'. 6. a; = 90-19^, 7/=13i5. 7. a;=8, y = 3. 8. a; = 7, 2/ = 5. 9. a; = 11, 2/= 18. 10. a; = 37, 2/ = 13. H. 4 or 5. 12. 19 or 20. 13. 4, or 5. 14. 2. 15. 16. 16. 5. 17. 3 guineas, 21 half-crovrns. 18. 3 sovereigns, 20 francs. 19. 185, 15; 119, 81; 53, 147. 20. 28, 20. 21. "When n is even, the common difference is 2 ; when n is odd, the common difference may be 1 or 2. 22. 245. 23. 104 + 3.5.7.^. 24. 97. 25. Ascribe to y succes- sively the values 1, 2,... 8; and in each case find the correspond- ANSWERS. XLVI. XLYII. XLVIII. 597 ing values of x and z. 26. a; = 1 + ?it, y = 51 -It, z= Q'^ + IZt. 27. AUowiug a zero, there are 15 solutions ; excluding it, there are 1-i. The solutions are found from 100 -i half-crowns, Qt shillings, and 100-7^ sixpences. 28. Allowing zeros, 4 solutions ; excluding them, 2. The solutions are found from 4 — < guineas, 5t crowns, and 12 — 4^ shillings. 29. 6 crowns, 4 half-crowns, 2 florins. 30. 100. 31. 205, 502. 32. 974. 33. 5567. 34. 80 ducks, 19 oxen, 1 sheep; or 100 sheep. 5 8 17 6' 9' Is* 104"" divisions reckoned from either of the common ends. 38. We must solve 5cc + 4?/ + S;?; = 20 : the accompanying table exhibits the solutions of this equa- tion. Then we can use (1), (4), (5) j or(2),(3), (5);or(3), (4), (4). (1) (2) (3) (4) (5) (6) 39. £2. lis. 6d. 40. £2. los. 35 36. 49, 43, 38. 37. The 107'^ and X y 2 5 1 5 1 3 1 2 1 2 4 z 4 XLYII. 1.0^ = 2, 2/ = 4;a;=3, 2/=l. 2. x = i, 2/=21; x = b, y = 7 3. a; = 18, y = 6. 4. x = \0\ y=\. 5. 360. 6. 1684 square yards. 7. 10 and 7. 9. x = ^, y = ?> ; x = 2, y = \. 10. a; = l, 2/ = 3; a; = 53, y=\o. XLYIII. 1. 1 r2xv 3. (2 2"-' + o \ 3 o •? »r+ 1 2. (3(-ir--^] x". X . 1-p 5. ()i + l)of. 6. {7a + b){3x)\ 7. {n + iyx\ 8. l + x-x'-x* 9. l + 2x-hx'-ix^-Ux* 7x* ,^ 1 X 3x' x' ^^- 2^2-^X^2^ 8 11. _- _ + % a- a^ a^ a^ 12. l+2)x+2)(2)-^)^'+(p'--I^' + ^^' + 2^(p'-^p'+p + 2)x\ 1.. -X fJ 1_V a - 1 \1 -h a; 1 + a"x/ _ _l fj. 1 1^ 1 \ 15. a=l, 6 = 11, c = ll, cZ= 1, e=U. 598 ANSWERS. XLIX. L. 4. a; less than \. 5. 2"'' {6n + 6). 6. 3" - n - 1. l+n' ' '8 (4 2(7^+l)(?^^-2)7^ 32* ^1/111 1 1 1 \ 11 6. 3 VI 2 3 n+l H+'2 11 1 3 ^ 11 96 2{n + 2){7i + 3) 4c{a + 1) {71 + 2) {71+ 3) {n + 4)' 96' H ^ __ 3;i+5 ^5 r^ n(n + 1) (?i + P.) * 6 (;i + 2) (vi + 3) ^ 6 ' 6 • \l ^"^^(^-i)-ir + ^"^^-(.^+i) ., . _^,,„_, ■l-i. 7 7-^ . 12. naria-\-br) . ^ ( ex c^x^ ") 1 3 . Expand and we o-et 7- - 1 + -, — -i h . ^ ^ {l-xf[ (l-o:)- {l-xy ) If ^n(^ n(n + l) 2 7i(7i+l)...(n-Tm-2) ,„ ,) I 1.2 l^/i— 1 J 1'5. (^1 - ^2) = 2" ('1 - I) ". IS. 16.5. 19. 460. 22. Proceed thus ; suppose (1 + xv) (1 4- x'v) (1 + x%) (1 + x''v) = 1 + A^v + Ay + + ApV'', where A^^, A^, A^, do not con- tain V. Kow change v iTito xv ; thus we can infer that {1+A,v + Ay+ + A^v") (1 + x'^-'v) =={1 + A^xv + A.^ x-v"" + + A^^f^v") (1 + xv). 2n'ow equate the coefficients of the same powers of v on the two sides. \ -\- X 1 ^^' IT^^ ^ \^x-\-x' '' *^ei'e^*^i'e (l+a;){l-x^ + a;«-a:^+ } l-rcV "^1-a/ ~l-a; (l-a;f "^ (1 -x)' ~ (1 -re)* "^ Expand each term of the last line by the Binomial Theorem and then equate the coefficients of ic" on the two sides. ANSWERS. LI. Lll. LIII. 599 LT. 8. 2x^ is > or < £c + 1 according as oj is > or <1. 16. This depends on the sign of (a —h){b— c) {c — a). 22 and 24 depend on Art. 681. 23. As many of the fol- lowing inequalities as may be required will be found to hold : 2 (?i- 1) > n, 2> [n — 2) > n, ; then by multiplication the result is obtained, 25. This may be deduced from Ex. 23. 29. See Ex. 3 of Chapter XXV. 31. Multiply up ; then use Art. 681. 32. Put l-a = h, and expand (1 - hf by the Binomial Theorem ; the series will be convergent. We shall then have to shew that 1 — -^ — rj^— + -r;i > 1 ; and this is obvious^ since cc is < 1. LII.' 2. 60. 3. ^.5'AV. 4. 2^3^5^ 5. 2'. (823)^ 12. Suppose n to lie between m* and (m+ 1)'; then n — ah = (7n' + 7/1 — oiy. 19. n" — n+l is gi*eater than (n— 1)" and less than n'^. 20. Suppose, if possible, oi^ + 1 =m^; then n^ =^ (m—l) {m + 1). Now no factor, except 2, can divide both m — 1 and 7?i + 1, and 2 cannot here divide them, for n is odd. Hence m — 1 and m + 1 must both be perfect cubes ; but this is rmpossible ; for the difference of two cubes cannot be so small as 2. 35, 36, 37, 38. These all depend on Eermat's Theorem. 40. 48. 41. 96. 42. 400. 43. 22680. 44. 2"-''5"-\ 45. 12. 46. 12. 47. 160; 1481040. 48. 6. 49. 126. 50. 24; 15. 51/ (n + iy. 53 and 54 must be solved by trial; the answer to 53 is 2*. 3^. 5, and the answer to 54 is 2'. 3'. 5. 7. 57. a; = 2 . 5'. 7'. ^'; 2/ = 2. 5. 7.^. LIII. 1. 27 to 8 against. 2.-;?. 4.''). 5.]. 6. A. ° 4o 4 4 16 7. — - . 8. 7 to 2. 10. yl's chance of losin;? is |, and of neither 36 ° ^ winning nor losing is J ; D's chance of winning is |, and of neither winning nor losing m ^ ; B and C have each the chance ^ of winning, ^ of losing, ^ of neither. Or more simply, ^'s chance of winning is ^, ^'s and 6"s ^, and Z)'s |, if we suppose that one of GOO ANSWERS. LIII. 5 2 3 1 the boats m«s^ win. 11. j. 12. -.Tk • ^^' tt* 1^- 9 71 _ 18586 _ 31031 ^^ 12393 ^, , /5^' 16.2.. 18. ^30^- ''■~^- '°-125mJ- 21-1-V6 „ 1 f^'\' 93 2551 ,4 13'x6 ^•'^ ^^- ^"Vssr ^"*-^'^- -"*• 51x50x-19- -'"• 52.51.50.49 • 32. Tlie chance of the sovereign being in the first purse is to the chance of its being in the second as 10 is to 9. 33. h 10 3 1 r|23 10|13) 34. 4^. 35. ^3. 36. 1. 39. ''— '— ' g3. ^^- Q3' ^^' 2- ^^- io"'[9l|U [4 /• 40. l.(^): 41.-033. 42.4. ^ 1 6 111 6 2 /ly .„ W\n . 44. w + ^.^.Tc.7r+-.7r- . 45. m both cases. 771 + 71 7 7*2*2*3 7*3 \2 ,,_ ^l&iA , ,7. I»^. 48. lltoS. 49. 6; [J. 51. 1«. \n n 6 3o 53. Let -4's chance of winning a single game be x, and ^'s chance . x" (1 — x) 9 \ — X \ then -4's chance of winning the set is ^j — ^^ -. . 54. vt; • ^ \—x-\-x 1 o 55. p,+p,+p^-p^,-p,P^-p^V,+PiV22h'^ PiP,+V2l\-^PzP-'^PJ\P.' Kc^ 64 56 49 .^ , _.7 _ 30 ,31 ^^•169' 169' 169- ^^* ^•^^)- ^^- 6l""^6T* 59. 21 shilKngs. 60. 42 shHlings. 61. £400. 62. 35s. M. 63. XIO. 64. A florin. 65. 3 florins, 1 sovereign. Q(:>. 2r4-l 2 to 1; ^ of what each stakes. 67. — ^ — . 68 3r(r + l) • 2 (2r+ 1) * 69. 33333 shillings. 70. -^ n shHlings. 71. f . 72 1 73. 2 ^, 3 1265 _ 5087 nin+\)' 5* 1286^ 5144* 76. £^;;-. 46 77. 2 en ■^■'" , «6 + ac + 6c 3' 50* (a + c)(6 + c)* 82. 4. ANSWERS. LIV. LV. LVI. 601 LIY. 1. J(l-x')^-l^J3. 2. Substitute for x' from the first equation in the second ; thus we obtain either y" ~ hi or x — - . y-h 4. Square; and jnit the equation in the form (.:c'-4x)" = 24 (a;- 1)^ 5. c = 220. G. Multiply up in the given relation. 9. Equate the coefficient of ic" in the expansion of -^ 2, X ^ »// "n Q%Kj and in the expansion of the partial fractions into which this expression may be decomposed. LY. 1 . {J{m' -f n') J {a' + 6^) - naf. 2. 1 + V| + ^^ • 3. 8. 5. 0. 6. aj = 2G^; y=495-2U. . 7. 1-^1^^^, where 2^ = 2". 8. (1 — x'Y + x" {\ — x)' is never negative. 12 3 n—\ 12. — lo^n = W ^ . K • T • Hence we may reacard the general term of the series as — 1- log (1 — ] ; and by expanding lo£f (1 ) the ojeneral term is found to be numerically less than s. Then see Art. 502. 14. If ho draws ac^ain from n the same bag, his chance of getting a sovereign is f, and his chance of getting a shilling is f ; thus his expectation is -^ shil- lings. If he draws from the other bag, his chance of getting a sovereign is >, and his chance of getting a shilling is 7 ; thus w • H. i-ir ^r (n-l) R -n + R''" his expectation is : shillmgs. 10. —r. — t-z , where R is the amount of one pound in one year. LVI. 6. Convergent if x is less than unity, divergent if x is greater than unity ; if cc is equal to unity, convergent if a is negative, divergent if a is j>ositive. 8. Divergent if x is greater iJaan unity, convergent if x is not greater than unity. 602 ANSWERS. LVI. LVII. LVIII. 9. Divergent. 10. Convergent if g - ^; - 1 is positive, diver- gent if ^-;; - 1 is negative or zero. 13. Convergent if x is less than e~\ divergent if a: is not -less tlian e~\ IG. I. Sup- pose a — A positive : the series is convergent if /? + 1 is greater than a, divergent if ^ + 1 is less than a; if /3 + 1 = a the series is convergent if a — A is greater than unity, divergent if a — A is not greater than unity. II. Suppose a — A negative : the series is divergent. III. Suppose a — A = -^ then apply Art. 767, and discriminate as in Case I. LYII. 4. 2^^ = hcf-' + {n - 2) h'a'-' + ^^--'^K'^-^) ^s^^n-s then q^ can be obtained by Example 3. 1 0. Every component has unity for denominator ; the mimerator of the first component is 1, of the second is ^x, and generally of the (27-f is /'"^ ^^^ , and of the (2r + I)**' is '"^ (2r-l)2r' ^^ ' 2r(2r + l)' LYIII. 2. ah ^hcA-ca + 2abc = 1. 3. {a' + b' + c'f=-8{ah+bc + caf. 4. a' + b' + c' - ahc = L 5. a'b'c' {a' + b' + c' + 2abc) = a'b'c\ 6. {x^ + y'f = z^. 7. 5 (a' - b') {2a' + b') = 9a (a' - c'). ''"'''^*=1. 9. a/5=l + y. ac J \ ac 10. {a-by(a'+b') = a%'. 11. {a + pf + {a- (Sf =^ 2. 1 3. X (f - z') + 2y {z' - x') + iz {x' - if) = 0. 14. {a + bf-{a-bf = {8cf. 16. 399. 17. This problem can be solved by the aid of the princij^les I. and II. of Art. 814. Let 2^^ be the probability of a single event with three cards of a selected suit ; let p^ be the probability of a selected pair of events j let i^^ be the probability of a selected triad of events ; and so on. Then P^ = mp^ ; F^ = — — 2^2 ' I\ =— ^ -^ 2\i We have now to find p„ 2^^, i\, ANSWERS. LYIII. 003 Imagine three cards fastened together, so as to form one card ; we should then have mn - 2 cards instead of mn. The nnmber of favourable cases would be \7nn—2, and the whole number of I mn — 2 cases [mn ; this would give a chance denoted by " : and to ' ynm obtain p^ we must multiply this result by [3, for the cards imagined to be fastened together could be permuted among themselves (5 _ Q^. 77171— 4: in ! 3 wavs. Thus w — . Similarly ;? =: - *— " ' onn [mn — 1 ) and so on. Hence, finally, the required chance is .,m (m - 1) m(m-l)(m-2) 6m ^ 2 ^ [3 mM + mn {mn — 1) m7i . . . {inn — 3) mn [mn — 5) I m + n . X x^ x^ 18. 'r — ~ . 19. The expression ^ — ^ — ^ + , m [ri ^ l-x' \-x' l-x'' becomes by expansion x + x^ -^x^ ■¥ x' + x^ + ... «A/ tA^ vU »0 %J{j "~~ , , « + iC^ 4- X^^ 4- CC"' + iC^^ + X^^ 4- ... _ ^'■_ 'r^' _ ^-^^ 'V'^S 'j-SS Then, by adding the vertical columns, we obtain »A> *0 iA/ r+^^ "^ iT^ ^ rTa7^ "^ 20. Let a = (l-a;)(l-aj-^) [\ - x') ..., /? = (1 + a:) (1 + re') (1 4- a;*) ..., y - (1 - x')[l-x') [l-x") ..., 8 = (1 4-a:^)(l 4-ic') (1 4- a;*^) ... ; then a/? = (1 - a;')(l - a)'')(l - cc'")..., yS = (1 - re") (1 - x') (1 - x'^) thus a/?yS = y ; therefore a/58 = 1 , and therefore - = (So. 21. 4 {r" =. V(2^''-" -pV)} = {q' ^ J{^r''-P")Y' G04 ANSWERS TO MISCELLANEOUS EXAMPLES. AJSSWERS TO MISCELLANEOUS EXAMPLES. L Ix-^u - (^z. 2. ax*+bjc' + c. „ ox +2 20.^- + oSax 3a^ + Gax 4. 23. x + 2. 24. I. 25. 3. 26. x =^ -^ , y = ,^ ' 7x- 4:' ' {DX+ 3a) {7x + 9a) ' {2a + x) {a' - x') ' 5 7 5. L 6. a;= - , y- -. 7. i? travels 6 i hours before ho ovei-takes J. 8. 80, 128. 9. a' + a--.. 10. 2, ^. 11. 10. 12. a;*-(-ia' + 96-)a;- + 3Gtr^)-; 7ic' + 5a;V - Sx/ - 3/. ^^ , ^^ / ,v .1/ 15a;" — 4a;;c + 2rt-^ -^ 3 13. a; + 2) a:4-3)(x + 4). U. .-^ -— ^.. lo. - ^ ' ^ ^ ^ ' (3x + 4a) (4^ + •oa) 1 16. ic = ll, 2/ =6. 17. 49 1\ minutes past 9. 18. Each in 50 days. 19. 2x-?>y-vz. 20. 2, 4. 21. 17. 22. a;'V(a + 6)x'-(Ga--«6 + 66')a;'-6a5(a + 6)a; + 36a'6'j a;'+4a; + 15. 3 1 2 27. Di miles from Ely. 28. 90 benches; 10 persons on each. 29. x''-2x' + x-2, 30. ^, -^. 31. 7, 1, 3. 30 A_3^,?^_,3..o3i. 33.™L:!:li. 34. ^-il±^-5. 100 10 lU ' 2\x'^t>x^^\ \-x^ 35. 15. 36. a;=ll. 2/=7. 37. 48 of each kind. 38. A man receives iC 1 II c£4. 45., a woman £3, a child £1. I65. 39. tt — -^r- • ' y 2 2x g 40. 6, . 41. 4,2, 4. 43. The second expression M'ill divide the first; so the second is the g. c. m., and the first is the l. c. m. 44. ('^-±2^ 1^^^^. 45. H 46. a;=3, 2/=5, ^=7. 47.30. x^ 5 48. 10. 49. J{a-b) + J{b-c). 50. 1,3. 51. 4:(ax+by+cz). 53. x' + T/'. 54. 2. 55. i 56. a; = ^ , 2/ = - ^ • 57. .1 in 36 days, B in 60 days, C in 15 days. 58. 4| miles. / 2 2\3 59. 2a;^^-a;' - 3. 60. 0, ± ^/(ai). 61. 2 (x + 4). 62. ^^^^ . Q ANSWERS TO MISCELLANEOUS EXAMPLES. G05 ^X x-2>y. 61 '^.l"'^'/! -. Go. 2. m.x = a,y = h, iU T it + i C7. £1000. Q>S. 84 for the resolution, and G 3 against it. G9. b. 70. I^, ^. 71. 9. 73. x-D. 7-i. x\ 75. 2rt. / 7 / 7 76. x=y=z^a^+h' A-c'-ah-hc-ca. 77. In 1 more minutes. 78. Twopence on tlie first day, § of a penny on the second day. 80.-4,-7. S?>. x'-2ax + a\ 84.-^3-,. 85.2. 86. ic = 1 (6 + c), 2/ = i (^ + «)' ^ = 2 (^^ + ^)- 8"- £600000 of ordinary stock. 88. 3, 4, 5 miles an hour respectively. §9. -05772. 90. c,c--J-. 91.20. 93. a;V(2m-3)cc-6??z. 95. a;=l. 96. re - a = ?/ - 6 = ;:; - c = - 1 (a + ^> + c). 97. 60, 30, 12. 98. 14 miles from A to B, 16 from B to C. 100. Ill, 112. 101. ±1, ~ ^ ^ ^^ . 102. 3, 4, 5. 103. a; = 3, 6; 2/ = 6, 3. 104. 3, - 12. 105. - 71 106. ^- (l - ^ ; f . 107. 6. 110. Between 90 and 119, both inclusive. 111. ±(a + 6), ± (a-6). '^rtc - h' 112. a;^ + a; + 1 = 0. 113. a; = ± 2, ± 4 : v= ±4, =t 2 114. 7. 115. 162. 116. A |i_(^/2 _!)»}. 118. 91 - m + 1 if ?• is not greater than m ; n-r +\ if r lies between m+1 and n+1 both inclusive; if r is greater than n + \. 119. Divergent. 120.3-06864. 121.1,-4,^^^. 122. 2Z;^=9ac. 123. 3,4,-6^2^6; 4,3,-6=^2^6. 124. 30,36,45. 125. ^=^ must be a positive integer, and I- 1 j + ~ must be a perfect square and a positive integer : these two integers must be both even or both odd, and the former integer greater than the square GOG ANSWERS TO MISCELLANEOUS EXAMPLES. root of the latter. 126. ,\~f. ^. 129. 3. 131. 3, ^. ar" (1 - r) '3 Ic (b-c) • k {c~ a) k {a - 6) , 1 jj. x-a = , y - 6 = ^ , z — c = — - -j — ^ , where be ' "^ cct ' ab ' 7 A ^ y a^b-c) + h- (c-a) + c'' (a-b) a oi ^ ir k = or = - lohc -^, {:, — 5-77 -i — i^T 7^- • 134. ^ 31 half- a {b- c)--\-b-{c -ay-¥c-{a — b)- 'crowns, 16 shillmgs, 13 sixpences; ^29 half-crowns, 24 shillings, 7 sixpences. 137. (o\^. 139. Divergent. 141. 6,-3. 142. 6400. 3 /2 13/2 143. a;=± ^ , ±^; ?/ = ± . =^-^- 144. 30 miles an hour. 4 1 '2i o U9. 18. 151. 5, -g. 153. . = 21(1)*; y = 2l,gy. 154. 75 per cent. 157. [7 -2 [6. 159> 0. 160. 1666 nearly. 161. a; + - = -4± ^6, whence re may be found. 162. ^10 miles o 1 q an hour, B 12 miles an hour. 163. x = -2, -— ; v= — , 0. ' 11 ' ^ 7 ' 164. a; = 0, y = 0, ;^ = 0; or :c = ^ , y = |t ;^ = _^ . 166. Either 2 a=^b=^c; ovb=-2a, andc^ia. 169. - ^^^ . 170. 1-21534 nearly. o 171. =^=8, =tg. 172. a'x'-2a'{b'-2ac)x + b'{b'-iac) = 0. ,^^ ^ 23 25± j'19968 ^ 69 21 2o±Jl9968 ^'^•^=^'-5' 19 ^2/ = 3, --,.-. f^ . 174. 9 (lays. 176. 1, 2, 4, 8. 177. ^""^"^'"^ Ink n — p\n — q 181. 1, -, --. 182. .£1045. 183. a:-81, 16; 2/=16, 81. 184. l=a'-hb'+c'+2abc. 186. "- /—l^, + 6 l(l-r)' 1-r^ (1-r) (l-r^')j * 189. Convergent if x is less than e; otherwise divergent, 191. X- -=-a± J(^a^ - 4), whence a; may be found. X ANSWERS TO MISCELLANEOUS EXAMPLES. 607 TYL 193. X and y may be found from a;^+l = =t— a;, y'^ + \=^mny. 11/ ^^^c^ n i. or^r\ I o ( '^ n(n-l) n(n-l)(n-2)) 199. Convergent. 200. \ n + 3 {^^ + \ ^^ ^ + — -r~ -\, 201. l-a;-fl;=+a;'+x''+a:'-a;"-x''-a;"'W+a;'*-a;"; l-x-x'+x'+x'-x'\ 205. The solutions are found from 14^ weights of 9 lbs. and 160-9Mveights of 14 lbs. 20G. (2"+--n - 3)a'". 209. — of a shilling. 210. Divergent. 213. 19 years. 215. The solutions are found from taking 20-4 -lU for the nu- merator, and I + 5t for the denominator j so that t may have any 3 inte<2jral value between 13 and 18 both inclusive. 219. 7 . I'I'l. x' — a =y —b'^z' — c^ T-^.n, . 223. rp+^~ = 2. 224. The quotients are a, b, 2a, b, 2a, ... 225. (aj-ll3/+l)(2.K+2/-8). 233. £693. 234. The quotients are a-a,l,2(^-l),l,2(a-l),... 235.52. 240. Divergent. 243. ISnearly. 244. The first quotient is a; then we have 1, 2, a, 2, 1, 2a, which recur. 245. Either 10 sheep, and 2 bullocks; or 5 bullocks. 448ic-'- 2496a;-" 249. £84. 250. a;'+ 4a; + 7- 32a;-^-208a;- '3 ic" — 4a; + 1 2 4 3 r (r— 1) 251. x=^l, ^; y=l, -- . 252. a,-a = r(a^-a) + -^ — ^ d; put ?i + 1 for ?• and b for a^+j ; thus a^ becomes known : d must lie between - ^i^-^| and ?^^^ . 253. £-645 nearly. 255. Either n{7i+l) w(>i + l) 5 apples, 3 pears, and 4 peaches; or 12 pears. 261. ^ ^ + —3- . O/ CC tt 265. The smallest number of coins consists of 121 of the larger and 15 of the smaller; the smallest sum of money consists of 10 of the larger and 150 of the smaller. 269. ^ ^-j^ '^ shillings. r;()8 ANSWERS TO MISCELLANEOUS EXAMPLES. 272. x=a, h; y-^h, a. 273. (7i±2)-. 275. The coefficient of a;" is 10'' + n(-3)"-'. 276. l-ix-lx'; l-?^x'-2x\ 279. 11 i sliilliiigs. 2 7 2 281. Between land -4. 282. (xy)-^ =- ^ ^ "" — -: from this and the fii-st given equation we can find o;^ and y^. 285. 56, 78. 293. ^3 \ 294. x=8, y = 3; ^=127, y=48. 295. 27. 297. £240. 300. 2 log 2-1. cxy. r. -a in-.F.: i-aisn::' et c. j. clay. m.a. at Tiiw ltni\ tusirv puess RETURN TO the circulation clesl< of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY BIdg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS • 2-month loans may be renewed by calling (510)642-6753 • 1-year loans may be recharged by bringing books to NRLF • Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW APR 2 2001 12,000(11/95) f GENERAL LIBRARY • U.C. BERKELEY BD00SSS132 *' 984604 'T6 THE UNIVERSITY OF CALIFORNIA LIBRARY I