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Les diagrammes suivants illustrent la mAthode. 32X 1 2 3 4 5 6 "'wrniiimiiiioiiMifiiiiW'rniiTiil^ «Ma«a«WWWMIfi i lJffiitg i Ti rMaiaMlWW .wi B wina mtmm^'' L L NOTES ON ALGEBRA. FOR THE USE OF THE CADETS OF THB |lopI Pilitarg College of Canaba. T^ EDGAR KENSINGTON, MAJOH ROYAIi AUTILLERY, PROFESSOR OF MATHEMATICS AND ARTILLERY. KINGSTON : WILLIAM HAILIE, PRINTEH. 1879. ■ "-^ ' W. " %' ■ ' t ' l^i'i" ! . - "'!)!- ; k NOTES ON ALGEBRA. (1.) The earlier letters a, b, c, &c., are generally used Commonest ^ , •' . . uBesofU'ttera- for known or constant quantities. The later letters a?, y, z, &c., for unknown or variable quantities. Thf middle letters k, I, ?»-, «, for numerical coefficients or abstract numbers ; also m, n are generally used for in- dices, which are always abstract numbers. The small letters a, b, c, I, a, a, y, z, &c., for lengths of lines. The capital letters A, ♦fee, for puints, angles or areas or surfaces. Be. The triangle ABC\\a6 its angles denoted hyA,B, C, its sides by a, b, c, viz., a opposite to A, &c. The semi- perimeter or ' i.) is denoted by s. The area of the triangle by S. The Greek letters a, ^9, >-, &c,, are very frequently used to denote angles. The Greek alphabet is as follows: a b g d e z e th i k 1 m n X o p r s t u ph ch pb o (2.) Terms are connected by the signs of + or — . Termi. Factore arc connected by the sign of multiplication. Factors. Letters in Algebra merely stand for Arithmetical num- bers, abstract or concrete, more frequently tl>e latter. Hence all Arithmetical rules apply equally to Algebra. Coneider particularly the rules for dealing with fractions. (3.) " Greater than " > ; " Less than " < ; the point is Symbols and always towards the lesser quantity. *°form8."" " = " must never be used except when the whole ex- pression on the one side is in every respect equal to the whole expression on the other side. ,-- -' k 8 InHnitu 00 is that which is greater than can bo described, Anomaloua or which is greater than any assignable quantity, or wincli cannot be appreciably altered by adding or subtracting finite quantities. Nothing or zero is that which is less than any assignable quantity, or which is so small as to bo inappreciable, or which is the intinitesiinal part of a finite quantity. .-, • = .-. ' = 00 Also 00 X may be finite and a|)preciable ; for if an inch be divided into an infinite number of parts, each part is infinitesimal, but when multiplied by the infinity we have the inch again. Thus a square inch nitiy be regarded as n linear inches each of thickness ^; placed side by side ; n being infinite and therefore - infinitesimal or nothing, n Nothing is so small that its addition to, or subtraction from, a finite quantity does not appreciably alter the value of the latter. (4.) It is frequently advisable to simplify before substi- Substitution, tuting. Be careful to put its value for each letter, and to perform all the algebraical operations correctly and distinctly with- out confusion. JEc. Ifa=l;J = 2;c = 3;rf = 4 ^a -f 2a» + ft^ _ h^ - 2&C + c^ j4c^_- ^cd + d^ a -\-h b — c c A. 2 = a_|_c_^2(2o-(f) = 4 + 2(6-4) = 4 + 4=:8 Ana. Ex. Same values. l/3a + 2c3+4= 1^3 + 18 + 4= 4/ 25 = 5 I ( ■ -• ••■" i 'l 1 nrF ii rnr I ff - itr'V- fMtkmi'>mm»'it* ■ < .V,t^w.-.a..» -^-i-'-fti-.^r^ (5.) Any imiuljer of qimntiticd incliulcd within Imickuts Urucket*. aro to bu truiitod as oiio (iiiuntity, every jmrt of which is iMlually aftected by wliatever aft'ects the whole. Ex, 3 X (» - ^> + 26'). Every i)!irt is to be inultipllod by 3, .-. It' the brackets are removed we have •da — 36 + Go. '^ Be particnbirly careful in the use of brackets, always to insert them whenever required, aii'l never to omit them without good cause. Generally remove outside brackets tiist when multiplying. (6.) (i^ means a . a or lX'« • o ; that is I multiplied by indices. "(t" tvs many times as the number or index jtlaced over a. lit definition. This definition is only intelligible wheJi the index is a posi- tive integer. For a more extended detinition see §60. ^ Thus c^ =1 X a . a . a. a or a^ = 1 X a. rt" == 1 because it is not multiplied at all by a. a ,in = 1 X a . « . a m times. (7.) a™ . a^ =(t . a . a . {m times) X a . a . a . n times. Indices. Thus there are m + n of these letters "a" multiplied Multiplicafn. VI ■}- n times together a ,m rt" =20, . a . a . =«•" + " Hence the rule. To muUijply different powers of the same quantity j add the indices. , a™ a . a . a (8.) Similarly v = ^ ' a a . a a m times n times = a m — n (9.) An index is essentially an abstract number, and lias Divlelon. therefore no dimensions. (10.) Every letter used in Algebra is said to be of one Dimensions, dimension, except letters used as indices and letters that are known to be of more or less dimensions. Thus a,b,c 'iw«^:.vSk-3si»"" luMii<; Oiicli of onu tlimciisiou, u^, ah, bo iiro of two dimoii- siuiia, a'S U'b, ubc iiro ot'tlireo (limciisioiiB. Only (juiintitiiw of tlio hiimio (liiin'iirfionrt ciin bo addod or rtulttnicteil, and tlio diineiiHioiu reiimiii the same. Kjo. a -\- b — o\* of one dimension. Whon two exprcdrtionu are multiplied together, the sum (•f their dinionHions is tliftt of the product. i Kx. ub ; a {b + <•) ,* {a — b) {b — v) are each of two dinienBions. Ax". a' {b^cf ; a^ {a-b) {b~o) {,'.-a) ; a {a^-\-b^) o are each of tivu dimensions. For division, suhtract the diinenHions of the denominator from that of the numerator to obtain that of the quotient. ^ (l+l-'^') (/' — « + '•) {a-b^c) c ' ' d^ + 6'^ + c^ each of one dimension. lay 1 Ex. are Ex. a bo ' x'8 — ya ' a:+ y are each of — 1 dimensions. (11.) An expression is said to be homogeneous when all lloiii(«in<"us terms connected by the sign of -|- or — are of the same dimensions. ir^-^a ^e. X -\-y -[-"^ ^ is a homogeneous expression of one dimension. ar9— ya Ex. aa -|- 5 or oj + y + f— arc not liomogeueous. An expression or equation that is homogeneous continues homogeneous to whatever Algebraical processes it may be ordinarily subjected. Every step must be homogeneous. If at any time it is found to be no longer homogeneous there must be an error, unless a reason can be shewn. This is a most important test of error, and aid to memory, throughout all Mathematics and every scientific research. ll •*^^. M Kiirtlior, no matlicuiiitiral (iXpressioii or equation e«u liave any intelligible meaning unless it is (in reality if not in appearance) liomogen':ou8. Ex. (One hour + one dollar) is not homogeneous, and is meaningless. " Homogeneous " means " of like kind. " The principle which is here so strongly urged ia only an extension of the Arithmetical truths that concrete quan- tities of like nature only can be added or subtracted. Non-homogeneous expressions are frequently used in Algebra, for practice in Algebraical oj)eration8. Expressions that are apparently not homogeneous are often really so, it being understood that certain fi«;,ure8 are really concrete numbers, or that certain letters are of no dimensions or perhaps of two, three or more dimensions. j^]p %,, — 3y =: 5 I.e. 5 units of the same nature, as ,/• and y. Ex. y = mx -\- c. Here m stands for an abstract num- ber or ratio. Ex. Ax -\- By -{- G =■ 0. Here C is of two diuien- sions if the other letters are each of one dimension. (12.) Before commencing any Algebraical operation ar- range the expressions to be used as follows : ,^ 1. In alphabetical order. Arrangement 2. In regularly ascending or descending powers. Algebraical 3. With special reference to the principal letter with expressions, which the question is concerned ; x and y are generally the principal letters. Ex. a^ +'2ab + b^ - c^ divided hy a + h - c. Eb. x^ - 2aii^ + {a^ + ah -h^)x — a^h -f- ab^ . Ex. '2xy^ g3 ^ 3a.2 2^ 3 _ 53,8 yz% ^ (a ± bf =a^ ± 2ab + b^ {a ± bf =a^ ±6a^b + 3ab^ ± b^ {a ± 6)* = a* ± ^a^b + 6a^ b^ ± ^ab^ -+- &* Involution of Algebraical expressions. I i M ife The notation means; either the upper sign throughout, involution, or the lower sign tliroughout; thus (,, + hf =a^ + Za^h + 3aJ2 + &3 (a - bf = a^ - ZaH + SoA'^ - 6» Observe the signs either all +, or alternately + and — . The indices are regularly descending and ascending; the whole is homogeneous of the same degree as the exponent. The coefficient of the 2nd term = the exponent. Multiply the coefficient of any term by the power of a, and divide by the number of terms; this gives the co- efficient of the ])ext term. * (rt ± ft)5 =«» ± haH + 10«H2 J. io,3j8 4- &c. (a ± &)«=«« ± '^aH + 15«n2 ± 20aH'* + &c. {a ± h)' r= a' ± 7a«i + 21«H2 ± Zba^b^ + &c. And generally _ ^ ,, w.n-1 __o,<) M.n-l.«.-2 _ 0,3 (,«±i)n=«"±na"-^i4- ^ J .2 a" ^h^± f- 2^^ »"-** +• y^. (3/— 2ai/)^= (3a^)3-3-(3.«2)22,^y + 3-3^(2fly)2-(2ay)3 = 27 ar« —Max^y + SGa'-^ie^yS _8a3y8 (14.) Square every term and add twice the product of Square of a <'oery two, taking care to give the proper dgn. This can Multinomial, be proved by multiplication. Ex. {a + h — c + df =a^ -\-l^ -\-c^ ^- d^ + 'iab-2aG + lad — lie + Ihd — 2cd. (15.) «2 ±2ab+ b^ =(a± bf ; 4^2 ± 12a6 + 9j2 ={2a±Zbf A few « „ « o ,\o common per- a3j^4^j^4j3_(^^2ft)2;4a2± 4a*+ b^ = (2a± bY feet squares. «2±6a6+9j2=(a±36)2;9a2± 6aJ+ b^ ^{Za± bf Observe that the square of the mean always equals four times the product of the extremes. Observe the numerical coefficients; thus 121, 144, 169, 529, 441, 961, all perfect squares. m Jii 8 (16.) No progrods can be made in Algebra until simple Factors, factors of an expression can be readily seen. Mathematical work is often simplified by taking ont a factor. See Ex. § 17, Notes on Arithmetic. Ex. (i? — ab = a {a — b). Ex. fiOa" ¥ h' It must he curefully ohscrvod tlisit when dividing rt" hy (I f A tho tornn arc ultuniatoly I- and — ; when dividing hy II — /' all arc +. a' + b' a + h a—h lu IV •} — ith f- fi^ T— = (H + ab + ¥ a — a" + b" a ± h n lM" is divinible by a — b. and till) quotient in hoth cases has no simple factors if « is a prime nnmher. E.i: a» + &» is divisihle hy a + b, also hy a^ + b^ which contains the factor a-\-b; for rt" + />» --= d'f + />" '' = .<•' + 2/' snppose. It often simplifies ditRcultios if suhstitntion of this kind are made, namely .<■ for a^ and y for ¥ . Then, wlien the calculation is complete, the original quantities must he replaced. «« 4- J9 ,^3 4. yH Ex. = ftO _ ^4:1 j3 ^ y^ a? + ¥ ~ a? + ¥ ~ X -\- y suppose = ar« — xy + y^ = a* — a'^ ¥ + ¥ -?T*?--** . ^ ■j ii u ii . wi gj !f . y J w * i 4«' rTn"'""'':n ii "Tr """ i r"^ — ~ — ~.^-j~ ' — ~.,.- 11 In tliirt iniiiinor n\\ tho simple factors may often bo ob- tained in:^toii(l of p(!rhiips only tlio one wlucb i6 first soon. Kw. a' + b' = {a' + ft" ) («" — «=' ft» + ft" ) = [a + ft ) («» - nh + ft'^ ) {a' - a' b' ^-ft" ) (21.) fa ijeiicr.d, when n ih any Integer (t^" — //" is til- vlmhle hij rt* — h^ , that h hij both a -\- ft and a — h. A few examples will shew tliat tliis nuiftt be bo. Ex. a' — ft" = {il + ft) (« — ft) &. a* - ft* - a^ 1' - PJ = a' + b' ) 1«* -^' ) Ex. «« - ft" = {a' + ft* ) («' + ft' ) («' - />" ) Tii,. a« - ft« = rt^ l' - ft^ :' = («' - ft' ) («* + «' *' +*' ) 7i>;. «"'-ft'«=«^ I -ft'^ ^■=(a'^-ft'^)(««+«"ft'+«*ft*-f«'i''+^') 7^:^-. «''^-ft''^=a« ' - ft* f' = (a* - ft* ) ('f« + «* ft* + ft" ) :=(,,'i _/;i ) („2 + fri ) (a« +«* ft* +ft^ ) Ex. a"" — ft™ = a^f — ftV'^ contains a factor «" —ft" which a"ain contains the factor d^ —U^ . Thus the proposition may bo easily seen to be generally true. (22.) But there are other important factors to be seen by arranging the factors of the index differently. Ex. a'-b'= «» i' - ¥f = («' J^P)(a'-h^) = (a + ft) ia' — aft+ft^ ) {a—b) (a* + aft + ft») from which we sec (comparing former Ex. a* — ft" ) that «* + d^ b^ + ft* = (<*" -«ft+ft' ) («' + aft+ ftM /^'^X^+ft* This is of frequent nse, but is very easily forgotten. Note the difference. a* — a** ft* + ft* has no simple factors. a< + «» i* + ft* = (a^i — aft + ft''* ) {a' + aft + ft» ) ^. a« + a*ft* + ft' = (a* - «''ft* + ft*) (a2— aft + ft") (a» + aft + ft") (See ^x. a"— ft"). ^1b. a^o — ft'" = {a" f ft" (a"^ — ¥ ) Ex. a" —ft'" = (rt« + ft" ) (a' + ft' ) (a" — ft' ) ^*'. a" — ft" = (a^ + ft' ) (a' — ft' ) rt4_a262 + 6« prime. Tlio«o oxiimpliw h1u)\v tho hcMt in«tli.>'l of scniiiriri^r iiU the F«ct..rii. r.ictors without iiiirtHing my. Othcrwis« many umy I'O lout. Alter this Htt'i) it would 1)0 (lirticult to kcc thu nMimiiiin^' tiictori* hIk'WII iiltovo. (23.) u'^" +^''" <» w^W'" (l!>}'»if>f« h ("' + ^) "'' ('^ ~" ''^ ' '«'° + ''''" h„tb,jh,','akin,j the indcr hUof adorn an ahovf, the upper jj^j^^j'^^^y „„,■ h,'ingo,ld^ the expnm'ion may he eanlly broken into (a + b)oiio h) factors. Kr. a" 4- // = («' + i" ) ("* - f>' i' + i* ) a'"+ />'»= (rt'^ + i» ) K — «'•//' + a* // — «» />* + />') ,t'^-f />''J= (rt* -f // ) (a' — a* // 4- // ) TTlJl .jiH-f />»«z= a' " + F" or = a" ' + 'i wlitMic-o WO SCO fiictorn rr'^ + b'* or ^t** + // , the hitter of wliifh cimtiiins tho former. But a' 4- P , a* + b* , ^s** + b^ &c., where tlie index contrtiiis no other fiietor than 2, have no ratioiml factors, tliat is, iriay be considered prime. (24.) The following is a summary : «»" + ' ± i'»" + ' is divisible by « ± J | _^it]i_ . «2n_j2n is divisible by a» — i' ftnd Rl8o=(a" +i" ) («»- J") «'•'" + &*" is never divisible by a ± b, but may have factors if tt contains an odd factor. - Let n be odd or even. n Let/ be an odd factor of n; let .- = y Then a« + fti is a factor of a" + J" , Factors of rt" ±6" ' I ' ) 'l ,„ i . ' l',-].W > W--J"IM ' 'liiilf IP I wm i 13 Examples ^ p^^^^^ „, ^2 -|- ^2 is prune, ^.6. has no rational factor. a" ± 6» ,,2 + i^ r= (rt + 6) {a — h) a^ + h^ = {a + b) {a" - ab + P) a^ — P = {a — h) {(i" + ab + V ) (t* 4- b* is prime. ,(♦ — b* = («' + b' ){a + b){a — b) a^ + b' = (a^ + // ) {"'* -an' + b*) «« _&«=(« + b) {a - &) («* + aH' + b' ) = {a + b) {a-b) {a^-ab f b\{a? + a& + h') (i^ + b^ is prime. ,,8 _ J8 ^ (,,4 + J4 ) («2 _^ J2 ) (,j _|. I) (^a-b) „nj^ W- = {a*- + b* ) («« ~aH* + b') Count up the diiueiisions of all the above examples to test tlieir accuracy. (25.) Factors may often be seen by the aid of arrange- Coefficients iiient. When the coefficients occur in pairs, try arranging ac- cording to the coefficients. Ex. sx^ - 'i-<^ y + ■"/' - v' = 3*' i^-y) + y' {^-y Ex x* + '2,e' + 2.C— I ^ ^ — 1 + 2a! {x" + 1) = (ic^ + 1) (ar" -t- 2a;— 1) Ex. x*—an^ + ba^—ax + b—i=.e*—l—ax{x^ + l)-\-b{x^-{-l) =(^j,^+ I) (x'—ax+b—l) (26.) There is one more very general and most important if ^ {a)=0 method of testing for factors. t^'^" '<'-« '" » If ,« — « is a factor of any expression, then if x is put = a factor of f {x) in that expression it will vanish, i.e. it will equal nothing. And conversely, if a; = a makes an expression vanish, then X — a is a factor of it. ,|i M.l lf l .H . . l/iiij lll * *'! Ex. ^? — a^=.0 if X a;" — a" = if X - a - a .:{x- A^ + a* = if X = — II .'. af + a'^ = if .'■'= - a* .-. s» — Sx — 4: =^ \( X = 4or ■ 4), {x + I) are factors. X — a is a factor. ;/! — (t is a factor. rf! -f- « is a factor. x'^-{- a^ is a factor. — 1 y<_3,,3_4 — if,«! = :^» _3,i/> + 2 = if .c = 1 .•. (./) + 1) is a factor. 1 .•. (.c — 1) is a factor. (27.) Always look for factors to simplify operations, as siu'^ested in Todlmnter's Algebra for beginners § 95. Jix. Find the G.C.M. of 6,r'' -11,/' + 5 and 3.*;^ -2./;* -1. Here x = 1 makes each =i .-. x — 1 is a factor. The otbor factor {Ox — 5) of the first is prime and cannot divide the second . .'. * — 1 is the G.C.M. (28.) The word " prime" has been applied above to ex- pressioiis that have no rational factors. Such expressions can be broken into irrational factors. Ex. «2 + &2 = (al + b%) (ai — «t hi + il) a — h = {Va+ \'h) ( V~a — V b ) «2 + ah +V= (a+ S'ah-\-l) (a— Vab + h). See § 22. (29.) Let A and B be the two expressions. Let i? < J. B) A {p Bp C)B{q D) C{r Dr E)D{8 Ea A—Bp-G A = Bp+ C. Factors. G.C.M. Proof of th*j common rule. - i ^i 'j ^^v ^ ^^" ■ f y 'i* " , ' m Mr m m^ 16 Every common factor of .1 and B divides A — Bp i.e. C and is therefore a factor of l)oth B and C. Every common factor of ^ and C divides Bp + C i.e. A and is therefore a factor of both A and B. :. the pairs A &:, B and B & C have all their com- mon factors in common. .-. they have the same G.C.M. .-. G.C.M. of A and B = G.C.M. of B and C. = G.C.M. of 6'and/>. = &c, =*the Answer. N.B. At all times during the process strike out the simple factors, if desirable, retaining those only which di- vide both expressions, i.e. both A and B or both B and C, vtc, and multiplying the final result by these to obtain the required G.C.M. See Notes on Arithmetic, § 17. Also, when desirable, either expression may be multiplied by any quantity which is not a factor of the other, so as to facilitate the process. The method of division is adopted in order to diminish each number, by adding or subtracting a multiple of the other. This multiple may be any multiple. Thus Bp may be greater than A. In this case it is better to subtract the upper from the lower, in order to keep the sign positive. The following example illustrates the ordinary method. The 2nd divisor and dividend are respectively ^ — x — 1 and x^ — ^ -I- 2j; + 3. The quotients are omitted, not being required. i Ex. *•» — 4«!^-H2rf!-t- 3 2u!* — Qar" -I- 2,«* — 8a!» 4- Vi^ — 7 4ar» -f 6.» .^ X 6,e 2x 7 3 4) 4ir' — 4x x^ — X — 4 -- 1 Sar' — 3a; — 3 | G.C.M. = Q? — X — \. 'I G.C.M. -•^j#fe* 16 It is often convenient to subtract eucli a multij)le that the hist term vanislies instead of the first; /.^'. to divide from the right instead of from the left. I Q,a _|_ 4s.,.2 + 52,,. 4. If, I 48,/'2 -h Sx — KJ Kc. 6«2 + ,/•— 2 6.^'^+ (U./' + 40 21)<;3,/' + 42 3./- +2 3j;)\).i^ -j- 9t)./;'^ -f tip./; 3j!» + 3air+2o"" 3,/'=' + 2.B 30.^ + 20 .-. G.C.Jr. = 3,r + 2. Here addition has i)een nsed instead of subtraction, fix. 35.*,'=' + 47,j'H13./'+l 42.r*+ 41a;''- 9.c^- 9x-l _140.t'*—188*«- 52^-2- i.v 35r3 + 40,r2+ 5,7! _| ' ~ fx^-^ 8«+"l"~ .-. t^ ,w a, .y^.yy^-. :j .,^^ iru' 50. (J. CM. nnil I.C.M. 17 Ee. .'■' -H 5,/- + 10 ; ./-^ — lOr —30 ; ,/ irere .'" + 5,/- + 10 is priiiic, and is tlu'i-otbro tlie only |ios8il)lc G.C.M. It'so, tlie rt'inainin^' tiictors of tlio others ,,,.e ,,. —3 and .r — 5 rospectivciy. Try theso mentally. .-. G.C.M. = .'.^ r r»./'+ 10. . •. /. 6'. i/. = (./•— 3) (./.--SX,/''^ + Tw + 10) = (a'~3)(,/-»— 1 5.r-50) iia. .*•- 4- 3./' + 2 ; u^' 4- *'« + ^^ ; •^•" + •'^•" + «• (,*; + 1) (..> + '2) ; (./' + 1) {.c + 3) ; (,r + 2) (./^ + 3) There is no G.C.M. The L.C.M. = '^c-\- 1) (,/'+2) (.c+S) The Z CM. must eonrain the highest power of every liiftor that is to be tiamd in each of the given expressions. ' If the factors are not evident, they may be fonnd, as in Arithmetic, by G.C.M. (31.) The following elementary simplitications ought to siM.|.lificati(m lie seen through easily without additional steps. They are forms that are constantly met with. {a + hf + (a - hf = 2 {h' -\-V') {a + hf — {a-- hf = 4ah a-b _ 2 {a' + ¥ ) -r a + 6 "^ " <^—¥ a — h " a-\-b 1 1 a a + h a-- h a + h a — ^ 1 a — It 1 a — h _ 1 "" « + l ^ah a' 4- 2a a -^ h 1 1 __2/^ a' — b^ n.n-}-l Thu8i-.\ = |; i-W = A When more than two fractions with Algebraical denomi- nators have to be added, it is generally best to add them in pairs, selecting those that go most easily together. This is specially to be remembered in working equations. ■it*!,,1!'* llifi! P 5 •m^' 1-- 18 a—x _ g"— a ;' _ 2 (a* 4:.'^' a" — «* Siinpllflcntinn _ 2(«« +2a^ ,<•" + .<•* ) — (g* — 2a'^ j- ' +J,'* ) _a* +(W .i^ -{-x* «♦ — J* ' "' ~ a* — .i*'~' ^' x—'ia ,e—a x-[-^a j'+a~ x^—M* a*— a» ~ X*— lOa^x" -f- W r'J2^ ^i rot « -(«'-*")' +(*'-«'')' +(<^»-a^/ _ ^ Simplification Hero D contains a factor a—b •.• it'a=b; D=0. (Sec § 26). '""'«""""«• It has also factors b — a; c — a. .-. D = k (a — b) {b — c) {c — a) where k does not con- tain a, b, c, because tlie reuiaining factors are of tlie given tliree dimensions. .-. by synjinetry N also = k{a^ — J'^ ) {b^ —c^ ) (,^ —a\ the k being the same. .-. u = {a -\-b)(b + f) {c f «). ={x -f- y + 5) (ar* + y^ ^ g* — ./-y — ys — 2.,;) factorization. (3-i.) 2 = 3 — 1 is an identity, absolutely true whether identities, the numbers be abstract or concrete. •c" — a^ = {x -\- a) {x — a) is absolutely true whatever be the values of ic and a. Sin'* d + cos" # = 1 is also art identity, for it is always true whatever be the value oi d. Zx = 6 is only true on condition that x = 2. ax = b " " aj _ _*_ ' a j«a _ 3a;-|-2=0 « » a!=lor2. 2 sin d cos ^=1 " « ^=45° or 225° &c. Equations. ^^ ^-^' i iiry'ii A. ' i u«fll --...-*,.... - -^ 19 All c(miition alwuytj involvcw ono or more coiHlitioiiH, wliicli it is our bu»iut}*» to Uuttxmiiie lij "solving" the ciUHtion. Onndltlonii or ThoHC con.lltiouHl v tlues Bro tlio "ro.itH" of the cqimtioii. "^I'J'/j'J^"" (35.) A simple ('»iimtion i» one tluit t'lm ho rothictMl to the Hlmi>l« form ax 4- b = ^K that is involving only the lirHt power of «»"""""• the unknown. A Hinjplo e(miition «!an have only ono root. For let the equation {t' no equation, .". « — /? = 0, .'. a = fl Therefore there is only ono root. The rules for working simple equations depend only on Kudid's Axioms 1, 2, 3, 6, 8, Book I. and 1, 2, Hook V. Ex. 2x—6 _ 2a!— 5 ar— 8 ~ Sx—7 .'. X = 2 Ana. ^'^' 2x'+l 'x-^rViT ^x" —32.f +42=60;" — 3la! + 40 , X— I 60—14 46 5=23 = 2 Ans. 28- .-. 5a'-|-60=28aj+14 ,, 132ar+l . 8a!+5 _„ 43 13 .-. 43rfr— 43 = 39aj+13 .-. x= ^ = 14 Ans. Ex. -hx + 'U — -8 = •75« + -25 "■ 4-2 .-. \'\x — 3aj = 3-2 4- 1 .-. a? = p^ = 3 Ans. Test the results by substitution to see if they " satisfy " the equations. „ ,*-.-tl. HU.'_t.fc-***«i IK!!!! ;i!:'' iii|vi.|| 20 (36.) If a factor can be observed by means of § 26 or Equations, otherwise, arrange the work so as to divide out that factor as soon as possible. 1 _ L - ^~^ ^- x—a ~~ x—b " .tr' — ah X — h — X -\-n, _ a — h satisfied if a = J {x—a){x —b) 0^ —ah yA _ tj^ = .c2 _ {a -f b) x+ ab Divide out (a — b) 2ab . ,e a X .-. arrange I X X a 1 X _ b b ~ X 1 -^^- ,. "I .7! + J a a-\-b . . satisfied \{ a = b a (b — a)x b — a , , jc ' ' ab X . satisfied if x = a. X I arrange - 1 1 a- X — a a-~a + b X + b" a.v {a + b){x-\-b) b{a + b) It is best to arrange the equation so that each side con- tains the observed factor ; as shewn in the last two ex- amples. If 'lA in Ex J- = f — T 1 a lactor [p—c) It as in E-x. „j,_g_^ nx—b—d ' ^ ' is observed ; then ihis factor must be evident in every line of work until it is oivided out ; otherwise there must be a mistake. Hence, in multiplying out the above Example, all terms such as " mx.nx, " " ad, " &c., which do not contain " ft " or 'V must cancel out ; if not, there mnst be a mistake. (37.) Besides the ordinary method for clearing away roots ^^j^ |/iIT6 by squaring both sides of an equation, the following is some- _ g times convenient, i.e. when the division is simple. ii III f. II '!!!'%'! 21 Ex. V® + ^i»— 16 = 8 but it is seen that x — {x— 16) = 16 .-. dividing (/9) by (a) ^z a? — \/x- 16 = 2 (a) Equations. (r) .-. adding («), (r) 2 4/ a; =10 or subtracting, the same result is obtained. but |/® — a -f- ^a? — b= |/a — b X — h — (JB — a)— a — b ^'x — b — |/aJ — a= y'a — b ^x=5 . («) • 05) . (r) .-. 2i/x — a = and 2 y'~x^^ = 2 ^a — b Either result gives x == a. Ana. The ordinary method is as follows : ^' X + \/ X — 16 = 8 .-. a; + 2 ^/x{x— 16) + a;— 16 = 64 .-. 2(a; — 40) = — 2 v^ x^ — 16a) ... ,x.2 _ soj, _|_ 1600 = a!« — 16a! /. x = 1600 64 =25 This may be shortened by arranging the equation in the form ^'^x^^niQ = 8— f/ar. The equation requires to be squared twice in order to remove both roots. ., Example of simplification used for Conic Sections. Given viM" («« + '»)" ± i'Y +{ae-xf = 2a but {/ + {'le + xf}- {f +{ae-xy } = 4a«a! , .'. ^y^ +{ae + xf T i/f +{ae-xf = 2ex .-. adding, 2 |/y« + (a« + ar)^ = 2 (a + ea;) ... j^a + a* e2 4- 2a^ + aj" = a" + 2a«a! + e* a;" .-. y« f (l-<3«)a^ = aMl— e») («) (^) (r) J .wmH!li!^:ia 22 (38.) Do not put .e for the unknown quantity without (lonsideration. In Ex. 10, chap. XXI. put x for half the back. .-. tail = 9 + « .-. back = 2,e = 18 + a. . a; = 18 .-. the Answers are 36 and 27. In Examples like 64, chap. XXIl. put x for the number of men in the side of a square. In nearly all examples of men working or cisterns being filled, &c.,"&c., put ^V for the whole work to be done, or G for the total gallons, &c., &c. The IF divided by each man's time, gives the work he can do in one day or hour, &c. (See Art. 203 Todhunter's Algebra for begin- ners). Problems. G (Ex. 10, chap. XXII.) 20 + T ^ G^ X W « = 30 G 12 W TF , IT ,'IF 7 IF ,. (Ex. 18, chap. XXII.) 24= -^ + ±e^ ix = i^ ■'' "^^^ Ans. 16S, 84, 42 days. In all problems take care to equate like concrete quan- tities, Le. Work to Work ; Time to Time ; Money to Money ; Quantity to Qn.antity ; &c., «fec. (39.) A " quadratic equation " or an " equation of the Quadratic 2nd degree " contains the square of the unknown with or *^"' '""*■ without the first power. General form, a^ + px + q = 0. It involves two conditions or roots, which may be both equal, both possible, or both impossible. Ex. a!* = 4 is satisfied by a? = 2 or aj = — 3 < ^•. !» = ± 2. Always remember to put the ± sign when extracting a root. The ± sign need only be placed on one side of the equa- tion, for " ic = ± 2 " means a; = 2, or » = — 2, and " ± a; = ± 2 " means the same. ■. 5. , 23 (40.) If the first power of .r is also involved. 1st. Sec that the cootticient of a-' is unity. 2nd. Complete the square hy placing, in the third term, the square of half the coefficient of x (adding the same to the other side.) 3rd, Bring all that remains over to tho other side. 4th. Extract tho squaro root and obtain x. Ex. Fir. — 4x + 3 = : X — 4.C — 2 : + 4 = = ± 1 : 1 .'. .1' : = 3or 1. Ans. x^ .1^ -f px -{-pf "" 4 -q = Pi 4 ,(• 2 -i,j This is the type cxamplo of all ^ < -iq the scpiare root cannot be extracted, and the roots are said to be iinpossihle. 2nd. If jt>'^ > 4^ the roots nra possible. 3rd. If ^'^ = Aq the second term vanishes, and the two roots are equal. Tliis is the case when a perfect square 0(puils nothing. Ex. ic^ -- 4a; + 4 = .-. .r — 2 = .: x = 2. There appears to be only one answer. Compare § 42. Further, T/ie sum of their roots with their signs changed jthat is \{p— f'p^ — 4^) + UP+ /p^ — 'i'j)=p} equals the I'oejficitnt of the second term. The products of their roots with their signs changed {that \^\{p^ — {2)^ — 4tq}) = q) equals the third term. Ex. ii? 4- 4a; + 3 = 0. Here the sum of the roots = — 4 the product = 3 .'. the roots are evidently — 3 and — 1. _ ' Ad foe tod qimdratioH, -\-px-\rq = Theory of Quadratic equations. - ja. * r. wi i ^ i ^'t '^;Bi rt KaM^iw i ii w i ii IWWi'tt fiWTai ^MfMjifV^i .^!- WaJJjWUjgjM' WgMMj on 24 (41.) The same result may be obtained by breaking into SnUition hy factora. Ex. a? + Ax -\- Z = = {x ■\- Z){x -\- 1). This is satisfied either if a; + 3 = or if x-\- 1 = .•. X = — 3 or — 1. This furnishes by far the simplest method of solving easy quadratic equations. That is, Find what factors of the third tcrm^ when Al- (jehraically added, yive the coefficient of the second term ; and change their tigns. Ex. {x — a{x — h) — Q.\x = aorh. Ex. ar" + (a — h) x = ab .'. x = — a or h. Ex. o!* _ 39j, + 200 = 92. .-. j^ - Z'dx -f 108 = .-. X = 36 or 3 or solved by completing the square. .!» — 39« -f- 2" 1521_ 432 _ 1089 '4 "~ 4 ~~4~ 39 ± 33 = 36 or 3 1521 1088(33 03)189 Ex. 4.c« — 8a! = 5 ... .x»-2.T+ 1 = 1+ ^= -^ 3 5 1 .-. .t' = 1 ± 2" = -g- or — Y See Ex. g 18. or by treating 2x as the unknown quantity. 4*2 — 4 X 2* — 5 = = (2« — 5) (2.'b + 1) 5 1 Ex. 9a.'2 + 6® — 35 = = (3a; + 7) (3a! —5) 5 7 X ^or-^ 25 (42.) The following Exiim[)le is very important, l>o'igof ^^ ^^^ ^^. a more general form than x' + jtj; + q = 0. _ ^ h ^_ _ P i» — 4«o -h ± \^^ —iito *•" + a 4rt» a; = 2a Here again observe the siviiio relation between the roots and the coetticienta u/ter the f/iiH«ion by "a," the cocjfioient Also, observe that \( b'^ < 4«c' the roots are impossible, if b'^ > Aac the roots are possible, if 6* = iac the roots are equal. N.B. Every line is homogeneous, if a; is a ratio. £x. 13»»— 47« — 90 = 90 X 13 78 X 15 65 X 18 The factors of 90 X 13 whose Algebraic sum is 47 are 65 and — 18 65 18 ^ 6 ... ^= _^^or-^3 = 5or-l^ This short method of solution is applicable to all ex- amples with rational roots. It is the same process as that explained in ,^ 18. The / ^>5v/ 18v tactors are 13^ic — ..j) yx + "ro)- (43.) By observing (a ± bf = «* ± 2ab + J* we see that 4 X «*i^ = {2ab)K See § 15. That is, 4 times the product of the extremes equals the square of the mean {or middle term.) This is the test for a perfect square. • .'. asc* -\- bx -{- c\%^ perfect square if J' = 4a<;. .i^mi y not am i wli 1 bci (liv ati w Be ot ■SHSW9 M (44.) Ill ftolv'm}? equations, ill vide out uny olworvud laetor iiti Boot) an poHrtiblo. Seo ^ 30. ff th'iH factor cont'ti na the unknown quantity, it mmt not he neijli'ctid ; iquafe it to nothing, and one part of the answer is ohtalned. Ex. aj5 — ir* = .-, J' ^ or a. A'/.. (./■+ 10)'^= 144 (100 — «») .-. ,r -f- 10 = 144 ( 10 — u!) or J' + 10 = 2S> ,4' = or 10. (45.) Every cubic equation has tlireo roots, one or all of ^^^^^^ which are iMmsible. (for Iloroor's F2very (!ubic ('(^uation may bo solved either by their "j^'os.) bcitij? re(lu(!ible to (juadratics, or by observing a root and dividiiiLC out the factor thus shewn. 2j + 13 a; + 16 18 The three roots arc 5 ; — "13 » '^^ The ecjuation reduces to the form (.1; _ a) (a? _ h) {ex — 1) = where c = 0. The infinite root must not be neglected without consider- ation in tlie applications to higher Mathematics. (Seo Williamson's Ditf. Cal. Chapter on Asymptotes.) Ex. ar* — 4** + 2* + 3 = 0. Try substituting »= 1 ; 2 ; 3. The last gives 9 X (3 — 4) + 6 + 3 = .-. a?— 3 is a factor. 1 ± vT aj = 3 or .r» — aj — 1 =0 jj = 3 or 2 ' (46.) Biquadratics may be solved if a root can be oh- BiquadraticB. served. ' They may sometimes by solved by completing the square of the first two terms. 1 1 ,t; i |ij i i i .. i i. i « ii ii Qaadratic Problems. m If then it can be put into the form of a qnadratic, the Bi(iuadratic8. sohition is easy. Fx. 4«* — 4ar'' + Tar* — 3a! = 18 .-. {ix" — xf + S {ix^ ~x) — 13 = .-. 23!" — a; = 3 or — 6 3 aj = -„- or — 1 or a pair of impossible roots. In general, biquadratics are insoluble, except by methods of approximation, such as Horner's. (See § 99.) (47.) When problems are solved by quadratics, it gen- erally happens that two answers are obtained where only one is applicable to the question. Then the reasonable answer must be selected. 1st cnae. Sometimes the two roots concur in giving the same Answer. Ex. Divide 15 into two parts so that their product is 54. Let X & 15 — X be the parts. We get a! = 9 or 6 .*. 15 — a! =: 6 or 9. 'inil case. Sometimes the second root answers tlie problem in a different manner to what was intended. Ex. Find the number which, added to its square, will make 210. Here ar» + « = 210 .'. a? = 14 or — 15. The intended Answer is 14; for 14 + 196 = 210. But (—15) is a correct Answer, for —15+ (—15)'' = 210. Zrd case. Sometimes the solution of a quadratic answers not only the given problem, but also another problem of a similar nature but generally inverted order. (See example in Todhunter's .Algebra for Beginners, p. 178.) Ex. A stone is thrown upwards with a velocity of 160 feet per second. Find how long it will take to reach a height of 336 feet. The equation reduces to x^ — lOx + 21 = .". a!=3 or 7. The true Answer is 3 seconds. 7 seconds is the time till the stone arrives again at the given spot, in its descent. tit th (« I'O ' *f!f ^H/ i . w ^o>^ -' ^^KL' >Vi ,PyMi ' >!llV-i^ ' * 28 Ex. Find sin 18" . Let 18° = ^ The equation is sin 2 A = cos S A .-. 2 sin ^ cos u4 = 4 cos' ^ — 3 cos ^ .'. cos il or 4 sin* A + 2 sin ^ = 1. (See last Ex. § 44.) — 1 ± »/ 5" Quadratic Problems. o sin A or cos A = O. The whole of these roots satisfy the equation sin 2 A = cos 3 A which was formed for the purpose of finding only sin 18° , the angle 90° being divided by Jive. The required Answer is necessarily positive, VT—\ / .'. sin 18° = 1 But it may be seen that ( — 54° ) , obtained by dividing ( — 270° ) by Jive, also satisfies the equation, for sin 2 (—54° ) = cos 3 (—54° ) .•. sin ( — 54" ) = the other root, viz., -r sin 54° = ♦/5 + 1 (48.) TJiere must be as many equations as unki -own quan- simultaneous tities, no more, and no less. equations. Eliminate one of the unknowns. Two J5aj. 6a! — 7y = 42 . . . . (1) .-. 36a! — 42y = 252 unJcnowns. 7a; — 6y = 75 . . . . (2) .-. 49a! — 42y = 525 6x21—42 ^ .-. VSx ^ 273 .-. X = 21 .-. (1) y = ^ =12 (49.) Any equation lawfully derived from one or more of Derived ti»o given ecjuations may be used instead of an original '^i"**"'"''- equation. Jiie. (the same). Subtracting, we get a? + y = 33 («) Adding and dividing by 13, we get a! — y= 9 ...(^) (a), (^) are derived equations, and are satisfied by the same roots as the given equations. Any two out of (1), (2), (a), (y9) will give the Answere. ail ' ' «jjn'i"! ? '") ' i'"t ' - ' ' ■■. y.jiW ' >^^ wwy^^ W"-''^J ' r'"*-" (50.) Eliminate one of the unknowns from, Ist one pair ^^^^ of equations ; 2n»l, another pair. Thus two - 4J« ) •*• ( 2y = a =F i^a' - 4J» JEc. .»'•' + y" = <»" (1) xy = h^ .-. 2afy = 26* (2) Add and subtract. Ex. 3^ -{• t^ = a* x -\- y ■= h .'. x^ -{- 2xy -\- y^ = P Subtract, obtaining 2xy ; then as above. Ex. X* +a^y^ + y* = 481 . . (I) ,«»+ ajy +y»= 37 .. (2) Divide : as* — xy + y^ = 13 . . («) Add and subtract. These are examples of the most numerous class of simul- taneous quadratics. (54.) Whenever all the terms of pne equation are horao- geneous with regard to the unknowns ; the ratio— can be y determined ; then substitute in the other equation. Ex. x{x+y)-\-y{x-^)-US .'. se^ +2xy—f =\6S. .(1) 7x{x+y)=72y{x—y) .'. 7ar» -65a;y+72/=0.(2) X 9 7 X 72 {lx-9y){x-Sy)=0 ^-Y ov 8 8 X 63 9 X 56 Substitute each value in (1) viz., y^ y—^ + 2— — 1^= 158. (55.) When the terms (in both equations) which are not of the 2nd degree with regard to the unknowns, are either both constant or both containing the same unknown ; elimi- nate these terms by cross-multiplication and subtraction. J Ex. ^ 4-a?y + 2y'=44 .• 2a)»— a!y+ y» =10 .• .*. y=^x or 3.r. Ex. 1j»—xy-^y'*-^'2y . ^a? +4xy =6y . .: 6a!»— 13ajy + 5y»=0.-. 8S 4^+4^y+ 82/«=17fl .. (1) ^J^^"" 22a!"-lla!y-|-lly' = l 1X10.(2) unknowni. •Oa)» - 6a!y+ y' =0 Substitute in (l)or(2). , 1()»» -5a!y-f Sy =10y , 4»» +8u;y =10y »_15 2 _£ y -T""" ~ (B - 2 iT = TT or — (I) (2) l_ 3 (60.) Equations involving x, y or asy in every term may be solved either by dividing by xy and solving in terms of — , — , &rhy eliminating xy as in § 54. In either case the root ]'"_,)[ tnust not be lost sight of. 3a! + 2y = 6xy . £x. 6 15aj — 4y = 4u;y 21 y = 14 y = y y 2 + 4_ X 4^ X = 10 = 4 (1) (2) 2_ 3 (57.) When both equations are quadratics in a general form, they cannot be solved ; as substitution (§51) results in a biquadratic. (See § 40.) (68.) A reciprocal equation has the coefficients reading Reciprocal the same from both ends, when arranged according to pow- equations ers of », no power of x being omitted ; if one coefficient is stj, degree, zero the corresponding one must be also zero. Ec. aj? + hx^ -If bx -\- a = ^ Arrange the coefficients in pairs, .-. tf («» + 1) + fta? (» + 1) = .-. a; = — 1 Divide out a; + 1 ; and a quadratic remains. Ex. aa^ + ha^ — ? — car' + &B-|-a = Proceed as above. After dividing ont (a? + 1), a reciprocal l»{»inadrntic re- nmins. Solve m tbllowB : Ke. aa^ — *«' + f** — fe* -f <* = <> * Arrnngo the (toefflciout* in piiir« and divide hy x-" 111 ... a(x -\- ~) —h(x -^ -) +o — 'la = (59.) The tlieory of qimdriitic oqimtions (§40) can be ox- Tlntory of tended to eqtiations of all degrees. XSSrens^ Tiiero are aa many roots as the degree of the eqnation. ^«* ^ '^ '''"■ This inclndes possible and impossible roots. The impossible roots can only occur in pairs. The Hum of the I'ootH with their signs changed is equal to the coefficitnt of the second te/w- (that of the first term being unity). The sum. of the products of every two with their signs changed is equal to the coefficient of the third tcjin. The sum of the products of every three xoith their signs changed is equal to the coefficient of the fourth term ; and Bo on. The product of all the roots with their signs changed is equal to the last term. Mb. If the roots are a, b, c, c.) x°-» —{ahc-\-&<) «"-'+ .... aled x= 0. {^^0.) The definition in g 6 has no meaning for fractional Indices, or negative indices. A definition must bo found for them which is consistent with that. Let them, therefore, have such a moaning that a" X <»° shall always equal a" + ° as proved in § 7. -. y.».,i i (ijjj;! m a^ X a^ = a .: a* = v^a Indices. a' u X a^ X (^ factors) = a" = a a|,X a^ X «^ X a^ X J =a^ =^ « J". n Similarly a " = i/a" To find the meaning of a negative index 4 - -n X 0" + ' = »-°+°-i-^ = a a a-" = a 11-1-1 a» + a" Since multiplication is effected by addition of the indices, division is effected by their subtraction. (See § 7). It also follows that (a")" == a" ■mn a" Since -^ = a"-" 1 = a". (61.) Ifa-=m; a; is called the logarithm of m to the ^^^^.^^^^^ base a; thus » = loga". It follows from the rule for indices that, if the logarithms of all numbers are tabulated, multiplication or division can be effected by the addition or subtraction of the log- arithms. Thus if 't* = »i .'. X = log m. o? = n '•' y = log"- flx-t-y = wift .-. x + y = log (mn). a^-^ = m m, n x—y = log — (a»)» = a" .-. nx = log (a"^" The last line shews that numbers may be raised to pow- ers {i.e. involution) by multiplying the logarithm by the power. im<' .'^^"ty'i^^v^?' 86 The usual bnsc; is 10 for arithmetical work .'.log 10=1 ; UflrarithmB. log 100 = 2 ; log 1000 = 3 ; «fec., &c. Loc 1 = whatever be the base .-. «** = 1 1 1 Log vo = -i; log 100 = -2; i«g o 10 - * ' '""^ 100 - "" " ' '"^ 1000 - The logarithms of any quantity leas than 1 is negative. The usual base, in Mathematics generally, is e where ^^=l + l+^ + T.^-^TX3l+ &c.=2-718281828 (62.) A surd is a term containing a root that cannot be 3 extracted. Ex. V3; 2 Vb ; V2 Similar surds have the same irrational factor. Ex. ^/Sluid ^^la" are similar, for -^/12 = 2v'3 3 3__ 3 8 Ejb. 5. ^/l6 and 7 ^5i are similar; i.e. 10 ^2 and 21 ^2 Surds involving diflferent roots can be compared as follows : S 2 8 \/n = 11^ = V^121 4/5 = 5° = 1^125 Fractions with irrational denominators should be ex- pressed with the denominators rationalised. 1 V2~ -t|ii = ..-.. V3-± _ 3-2 V^3+l ^ 2-i/3" i/3+1 3—1 If 35 ± v'y = a ± Vb where the roots are surds, the rational parts must be equal, and the irrational parts must be equal. Thus x = a; ^y = V^ ' Surds, ■ v. i . i^w ^jt i jUDj ji . i __.n!M.-?J!,im ' SA ' J-.V 86 (63.) Rvtios are best expressed as fractions. It simplifies many operations if k is put for the value of the ratio, as 3 is the value in the Example. Thus -^ = ^ = y-=A; .'. a=kb', c=kd ', e=kf. v Tlio numerators are called antecedents. The denoininatord are called consequents. ft. i* A (64.) If X ~ ^ ~ f~ ^" ^"^ fraction, whose numerator is formed, homogeneously, of the antecedents ; and the de- nominator si/»*7a/'/y formed of the consequents, is also equal to /fc, that is any one of the given ratios. a—2G+3e_ kb—2kd+ Skf k{b—2 d+Sf ) _ ^<^' b-2d + 3f~ b-2d+Sf -^b-2d+Sf -'^' The most important example is as follows : a+c+e+&fi. '■b + d+f+&c. See Euclid V. 12. a a If i-=-3 =-:,-= «&c. each d / (65.) rT- is more nearly equal to 1 than j whether j is greater or less than 1. This may be seen by means of the last example, or by putting 64-c or b—o for a, and observing that jt:^ (*'•«• the G diflference from 1) is less than — a a (04.) If T- = -7- , then 7 is the duplicate ratio of y b a ja\ a a a Similarly, if J" =~ = "J t^en -^ is the triplicate b a a a a' of the ratio -^; for ^ = -j-x — x-j-jr Ratio. J '■ y y »ip >i W |iw a¥Wl ' iBrW Ki^*§wmvnv II 87 a If a, b, c, d, e are any quantities, then the ratio — is a b G d compounded of the ratios -J- > "7 > rf » g ' a b cd^ T ^ T ^ d ^ e a e ^QP ._ _ -- X — X T X — • See Euclid V.Dels. 10, 11, A. . a C (67.) a, b, o^dare proportionals it -j = -^ a b 0^ a, J, 0, d are continued proportionals ^'^—"^ — d Proportion. (68.) If r = T Add or subtract 1 from both sides. a+b c+d a — b _ c — d a+b o+d ••• ^y <^'^'«'«" ct:::h = ^^d so±y _ a 2x Ex. Iff- = 3 ^. If x—y See Euclid V. Defa. 15, 16. 3-1 - a ^ ~" a— 6' (69.) The distance a man walks varies as his rate and variation, also at the time he is walking. The time of mowing a field varies as the size, but in- versely as the numl)er of men employed. These variations may be respectively expressed thus : Distance oc rate X time or Distance = A: X rate X time. ^. number of acres ^„ m- „^ — h \/ numberofacres T"ne X „„„b^, ^f ^en ""' ^'™® - * ^ number of men. These are called the equations of Variation, where h is an unknown constant, to be determined by the conditions of the problem. 88 Problenia on Variation. Observe that if A = kBC wliere A^ B, C are all vari- able, it means that ^4 varies as Z? if 6' constant, and as C if B is constant. Also if ^ = kyi ; this means that A varies as ^ if C is constant, but inversely as (J if B is fonstant. If both vary then in the one case A oo B X C; in the 1 other case A cx) a X -jr (70.) Ist. State the equation of Variation. 2nd. Apply the given conditions to determine the un- known constants. 3rd. Substitute these values and obtain the required Answer. £x. The cost of a College is partly constant, and partly varies as the number of students. If it costs $35,000 for 20 and $50,000 for 40 students, how much will it cost for 100 students? Ist, the equation is a? = o -|- kn. 2nd, the conditions give 35,000 = c-\-20k. .(1) ) j 20;i' = 15,000 .-. k = 750 50,000 = c-j-^Ok. .(2) f • • ( .'. « = 35,000— 15000=20,000 3rd, substitute these values. .-. X = 20,000 + 100 X 750 = $95,000. Ans. (71.) In Arithmetic Progression the difference between ProgresBionB. any two successive terms is constant. In Geometrical Progression the ratio between any two successive terms is constant. In Ilarmonical Progression the reciprocals are in Arith- metic Progression. (72.) To sum Arith. Prog, let I'n = sum of n terms. Aritlimetical .-. l'n= a+i'i + h)+{a+2b)+...-\-l where l^a-\-r^l6 P'"«""«'«"- .-. 2'„ = l+{l—b) + {l—-ib) +...+a n n v„ = - -- (a+Z) = -g- (2a-f n-1*) in=-^(a+0 39 The Ar. mean ot «, 6 >9 2" The Ar. incim of a, b, c, ^ + • .... + «'•"-' ... r X ^'n = «r + ar'' + _^ . . . ■ + a/-"-' + q^° SubtVacdngT. . . (r— 1) ^'n = ai^ — a . y _ «'!L- « = ''^-f if; = hist term. This can be verified by dividing out. (See § 19-21. Oeometrical ProgreHsion. i' — afn—a r-1 a to infinity, v _ _« Similarly, y^^ =^ a ^- ar -\- ar'^ + r being less than 1. This follows from the former result, since ar <» = if /• < 1. The Geometric mean of a, h is v'oJ 3 The Geometric mean of a, h, c is ^abo CO 1— r The Geometrical mean of/,/, b, c, d, e is Va, b, r,d, e The Arithmetic mean of any number of positive quan- titicH is greater than the Geometric mean, unless they are all equal. Proof for two quantities a, b, a» + ft» > 2ab because (a" — 2rtft + ft» ) is + " a + b a" + 2ab + i* > 4a6 > ^ab (74.) By (letlnitlon ^— — = -— ^^ if a, i, euro in II. P. a a — b c ~ b — Ilarnionlc Progrvmion. il — h b — I' ah ~ A" That in, tlio Ist is to tho Unl ii8 tlio flifforciKio between the I»t imd 2inl is to tlie ditferencu between the 2ncl luul 3nl. Thus in Kucliil VI. 3 & /I, if the anj^lo /' of the A A /'O and its exterior angle be bisected hy Pli and /'/> cutting the base and base produced in Ji, D ; then Ali r a AD AC Ali A I) AC— Ali liC-^ a ~ AC • ■ AD - AD- AC The straiglit line AD\% said to bo Ilarinonically divided, 1 1 2 2rtft Since ^^ + y - ^ .-. c=—i^. nionic mean. This is tho Ilar- (75.) Draw /*/), PE tangents to a circle at tho points D, E. Any straight line PAliC drawn cutting the circle in vl, C and the chord of contact DE in B shall be Har- monically divided. Let be tho centre. Join PO bisecting DE at right angles in F. Draw 00 perpendicular to PC :. PG= g {PA -t- PC) PB PO PO.PFJi.TD^ _^PA.PC PF=P ■'• ^^- PO — 2Pa ~PA + PC ^"*- Harmonic mean. Harmonic property of the circle and otlier conic sections. (76.) It has been seen that if a, b are two quantities a-\-b Their Arithmetic mean is -„— Their Geometric mean is y'oJ Their Harmonic mean is a+h Means. .rr. \ ': roved by nduction. I title fibc, qii IS 8U ' m i f ti mx 'm mn i M.m m yyj ^ W. . ' ^H^Um B* " 44 (82.) /* means tlie number of perinntations of n quan- PomiutationB ■" , Combination^ tities taken r together. C 18 the nninber of combinations. ohc] hue, cha, &c., are diflTerent Perm" but the same Comb. The order of the quantities is not regarded in Combinations. The order is changed for every Permutation. (83. P = u.n^Ui^ .... n—r + i = iV' suppose. + 1 + 1 If this is true, prove it true for P T ■ Place the (n + 1)* quantity at the beginning of each of the above permutations. Thus we have JV permutations with this quantity stand- ing first. Again we have iV'more when it stands second, and so on. .-. we have altogether N X (« + l) permutations of{r+\) (quantities. Proof by Hatbematical Induction. n + i P = n + 1 w.n— 1 . n — 2 r + l n — r+\ z=n+ln.n—i .... {n + 1 — r+l+l} This is of the same form as the above, g^^j. /» = 71 ,-. /> = n.n—T and so on, .-. the formula is universally true. P z= n.n — 1 . n — 2 n — r + 1 3 . 2 . 1 = 1« Let r = n .: P = n.n—\ n {Si.) If of the n quantities, p are the same, each = a ; q are the same, each = b ; r are the same, each = c. Let JV be the number of permutations taken all together. Now suppose the^ letters a to be different, viz , a,^ a,^ a,^ &c. There are now iV^ X | /^permutations. Similarly for 6, c,ifec P = I'i_ II =: factorial n ^ • l£- • l?_j l!L — !-i- n ^ ~ \PJ\1-- l!! 1 urn but Th Ofi titi tog ( pai HIK ma th( vo tip 46 (85.) Every Combination of r letters will form, by re- ComblnatJonB r arrangement, /* Permutations, i.e. \r__ .'. No. of Permutations = [r x No. of Combinations. n p n.n — 1 . n — 2 n — r + \ \r but n.n — 1 n — r-\-\ . n — r . n — r — 1 3.2,1 = k .'. C = ~ \n n n \ G = C r n— r by symmetry. , — \r\ n — r 0= O r n— r This is evident, since (out of the n quantities) for every set of r quantities there is a corresponding set of (n — r) quan- tities. Ex. Find No. of Combinations of 50 letters taken 48 together. "V = "c' = ^^>^ = 1225. 48 2 2 Ana. (86.) When an assigned pei-son is to form one of every party of 5 selected out of 20 (for example), set him aside and find how many parties of 4 can bo made out of the re- Problema. 19 maining 19, i.e. C\ for he can take his place with any of these parties. When words have to be formed of m consonants and n vowels, find first the Combinations, i.e. Cm X Cn and mul- tiply this by |»M-» to get the Permutations of these. m mi ■■■ipBiM w pii 46 Hlnoiiiinl riiiMirfiii. (h7.) It 1ms l»con prove*! in «$ 79 tlmt wlioro tlioro hi'c h lin-torfl ; l'{(t) ineiiiirt ii -\- h -\- c-f- Hi"^l contnlns n teriiirt. n IXah) means «& -f ac -|- ft<'-|- . . . . and contains T terniH. II l\ab<:) means ube + flftf/-|- a"*^ contains ^' terms. Let a = h — c = d = &c. •I " n.n — 1 , „ n.n — 1 . n — 2 =,!.•» —« ^ \t v^ IMAGE EVALUATION TEST TARGET (MT-3) Photographic Sciences Corporation 23 WEST MAIN STREIT WIBSTER.N.Y. 14SM (71«)I72-4S03 ) I CIHM/ICMH Microfiche Series. CIHIVI/ICIVIH Collection de microfiches. Canadian Institute for Historical Microreproductions / Institut Canadian da microraproductions historiquas 47 (88.) The square or cube roots, &c., of numbers can be extracted by means of the Binomial Theorem. Ex. m = (32 + l)i = 2 + i- X yg = &c. The roots of ratios very nearly equal to unity can be readily extracted. Fx. |/M2_ (1A.-L \* _ 1 _L-1 y 1 inapp.eciable ^1000-V^^^lOOO/ -^^^ 2 ^1000 [terms 1001 1000 „ , ^1001 10002 So also V 1000 = 10000 ~ 1"0002 (multiplying the diflfer- ence by the index, -r) Ex. How far can a man see from a height of'-'-h ^^ feet ? Radius of Earth = 4000 miles. Let X = the distance. «' =(^B+hf - i?* =-• ^ {^W')^ -B?=B? j ( R+U R \- UR X = (in miles) ^ 8000A 5280 (approximate) v'-q- (89.) See Notes on Arithmetic. Let P = Principal or Present Worth. / = Interest for whole time. M = P -\- I = Amount. D = M — P = Discount on M. r = rate per cent. r R =1 + jQQ= amount ofone dollar and its interest for one year. n = time in years. '. for Simple Interest /= j^ Interest. Simple. 200 100 X (107)" . log 2 -30103 loglOt' •0293838 48 • For CompoHiid Iiitt'i-est M = Pi^ Compound Af. In how many years will $100 doulde itself at 7 per Interest. eoiit Conipomid Interest ? . .-. log 2 = //. log 1-07 log 30 103 = 1 . 47860'J8 log 2-93838 = 1 . 4680961 lis 1079 .-. log « = log 30103 - log 2-93838 = 1-0104919 .'. n = 10-2445 years. Am. (90.) It'un Annuity of $4 is to continue for n years, tlie Annuities, value at the end of that time will be M= A + AR + AR^ + .... + AR"-' = A ^~^ P = M R" A fl°-l fl-1 R^ R-i 1— 9' •• A = Pr = Annual interest on P, as must evidently be the ease. (91.) The Prohabilitij or Chance of one event out of probabiiitieB many possible ones is the following ratio: f,, _Total number of ways in which it can occur. that is, Total number of combinations. The number of favourable combinations The total number of combinations. 1 Ex. Chance of throwing a " 3 " with one die is -jr- . (92.) The Chanee of two independent results is the pro- duct of the chances of each result. Ex. Chance of throwing aces is -«- X -^ . 40 (93.) Tlie cliiiiice of a result huppening, added to the ProbHbilities. I'liauee of its not happening is certainty, tliat is, xmitif. 1 '\*\ Ex. Chance of not throwing aces is 1 — „a- = ;r-: ^ 36 36 Ex. Chance of not tlirowing an ace with two dice is 5^ 6 _ 25 ^ 6 ^ 36 11 .•. Chance ' *' throwing an ace is -Vw? It may be seen that there are 11 ways in which an ace may be thrown, viz. : aces and each ace with 2, 3, 4, 5, 6. (94.) The odds upon an event are the ratio Chajice of its happening. Chance of its not happening. - Ex. The odds against throwing aces arc 35 to 1. (95.) Divide iB* — Sar" + 12a!« - 5aj + 4 by » - 2 — »♦ + 2ar»— 12 —10 — 6 Answer, x^ — ^ms^ — 5 — —5 6 6 X- Commence the ordinary process of division, first changing the sign of the divisor, so as to employ addition instead of subtraction. Add the coefficients of*-' and multiply by "2" ?'.«. the root of the divisor. This gives the coefficient of ol^ . Proceed in the same way, omitting all but the coefficients. Add the two rows thus obtained. 1, — 6, 0, — 5 are the coefficients of the quotient, 6 being the remainder. This process is only convenient when the divisor is of the form X ± a. Synthetic mode of division. eqiiationB. 60 (96.) If, in any equation, x -\- a he snbstitutod for x a Transforma- iiew equation will be obtained, all of the roots being "a" *'"" "' loss than in the former equation. For let r be the root of if in the lat equatio!i /. x -\- a =■ r in the new equation .-. /• — a U a root ; similarly for all the other root''. Tho transformation may be effected either by direct sub" stitution or by Taylor's Theorem, viz , if{x -\-a) =. ifx + a f •' X + .-^ f ";r -f- AC" or by the synthetic process, as follows : (1)7.) Required to transform ip{x) into an expression whoso roots are less by " a " Divide by oj — a ; let ^ be the quotient, li the remainder, .-. f{x) =z Qx{x — a) -\- lii Let x — a = y .: „y° 4- «ny"-' + . . . . + Rs y» + flj y + Rx This is the required expression. Ex. Transform aj« — 8.c» + 12.»» — 5a! + 4 = into another equation, whose roots shall be less by 2 than the roots of the given equation. Divide successively by ic — 2 by the synthetic process. 1 — 8 + 12 — 5 + 4 2 — 12 0—10 — 6 2 — 0- 8 - - 5— ♦] -16 -4 — 2 — 8- 4 -21 _2 — 2 12 The last quotient is 1 ; the successive remainders are — 6, — 21, — 12, .•. the transformed equation is a!* — 12a!' — 21aJ — 6 = N.B. — Of coarse sines the sum of the four original roots is 8, the sum of the new roots (t.«. reduced by 2 each) must be zero, .*. the coefficient of z9 vanishes in the new equation. ^.t , 11 | i i j»^in » j i ; i . 61 (98.) Horner's method for extracting any root is let. Ascertain, by trial, the first figure of the root. 2nd. Diminish the roots of the equation by this figure. 3rd. Ascertain, bv trial, the first figure of the root of this equation, which is the second figure of the original root. 4th. Again diminish the roots by this figure, and so on. Ex. Innd a root of ir" 4- 9a) — 16 = 0. Here x lies between 1 and 2 .•. 1 is the first figure. — x—\) I -+- +9 —16 (1 I 1 10 Extraction of C^ube Koota. Horner'a Method. 1 10 1 2 2 l2 1 3 The roots ol a? -\- ^x^ ■\- 12j) — 6 lies between '4 and "5 for the root of 12jj — 6 = is -5. The roots of.*" + 30a!» + 1200a)— 6000 lies between 4 and 5. — a> — 4) 30 1200 —6000 (4 4 136 6344 34 4 38 _4_ 42" 1336 162 1488 — 666 656 41 Hero X is less than ivgg = gg — x- — 4) = nXi .'.a) lies between "4 and '5 ar -3) — a) — 5) 420 4 148800 1696 — 666000 (4-354526 601984 424 4 150496 1712 — 64016000 467013 428 4 15220800 1297 1523377 1298 — 83147 76244 4320 3 — 6903 6100 4323 3 4326 3 1524675 216 1524881 216 — 803 768 -40 30 43290 1525097 — 10 Am. 1-44364526 69 The figures under the douhle lines are tlie coefflclenta of the 8U(!COfl«ivo tniusforincd i-qutttions. As \n the onlinnry method of extnicting cube roots, the let and 2nd cnhimns bonoino iniippreciiihlo and may then ho neglected. Thus the figures 4520 hnve been obtained merely by dividing 0903 by 1525. Any error in asi'ertaining the trial roots will reveal itself in the working, Ity the next root appearing either negative or greater than 1. The numbers in the 2nd column are (it may bo observed) trial divisors, whereby to obtain the next figure. The successive derived equations are .if' + 42ar» + 1488a! — 650 =0 ; root between -4 and -6. .«•" + 432X-" + 152208a> —54016=0; root between -S and -4. .1* + 4329a'' + 15240750a) - 8314700 = approximately, .z'' + 433()0a!» + 1525097000aj— 090300000=0 approx., &c. (99.) Let the three roots be /• and a ± i^^ a-a _^2« + /•)«*+ (a* + 2ra — ,3) x— r («» — /?) Diminish all the roots by /'. 2/') ,c^ + («* — 2/-« + /•' — /9) a! = 0. 0. (2a- Write these equations | ^j. + ^, .^^ 4?^, ^ = 0. It will now be seen that To find all the rontd of a niibic, npproxlinately 2 '-2'^ 2« + 2r) — r = — a and ''■ + \- " - -/9. In the last example 9 ; r = 1-44354526 j}i - 43305 y,= 15-25097 ■^* = 2- 16525 3y = 27 — 1-44354 .-. — a = •7217 — i)»= .-. — /? = 42-'25097 the additional roots are —-7217 ± |/— 42-25097 il 1; ! I ii 1 ' TSff ' l ' A'M",- ' !!' 58 Fj). Find all the roots, to 6 figures, of a!» + 10ar» + 5(b— 2600 = -11)1+10 +5 —2600(1 1-0068 nearly — ifi- —X- 6) 21 82 43006 43012 430188 2)430204 21^5102 —11-0068 236 2596 58800,00,00 — 588258 588516 15 603-5^ —100 4000,000,000 3529548 "470452 4)503-5 — « = 10-5034 125-9 = — ^ .-. the roots are approx. 110068 and— 10-5034 ± \^^—\Mr^ Equation of higher 4 (102.) The expression Oy. bi — (n h is denoted \ai , Jj l)i>t.Tiiiiimiit« by the Determinant, which is of tlie second order. \b\ , ^2 If the columns are written as rows in the same order, the value is unaltered ; thus \ai , b\ (1% , bi If any two columns or any two rows are interchanged, the sign is changed ; «i 5 fla ! __ _ ;aa «i I _ _ 7>»i , is j _ Jj ii I thus bi , bi bi bi Kit , fla fli Ui Ex,. The sum of two determinants differing by the quantities in one row only may be obtained by adding these respect- ive quantities. ai , «a , as ! «' , aj , as j \a,\ + «*,««, rts i c' , OS , c'3 I I i /5i , ii /?a I , j«i «i , ij /^a I (12 «i , «2 «2 f>2 ,ii , />a /^a i (l-i «i » ^2 /9a | '1 , /'i :^0 fO + «, ^ii «2 , f>2 -«a i'l ; , 1 3 «a , Oi 1 aa > /Ja ^a ^1 , «a aa : aa ,ia Tlie above rulrs can be verified, in each case, by develop- ing the determinants. Tlicy are universally true for de- terminants of any order. (103.) and denotes ai X '«2 , fta , Ci ia-s 5 />3 , Ca is a determinant of the 3rd order. *» » ^'a I I ^ s. i *s , Cs 1 _i_^. ^ 1*1 , ci ■bs . Cs +aa X +as X |ai , bi , ci , rfi ifla , bi , e-i , (it ia4 , bi , t'4 ,. di ilibi , c'a , di ' — Oa 1*3 ,03, ds -f-fls ! *i , c, I -r"» " lAa , Ca is of the 4th order, and denotes bs ,cs , di bi , Oi , di bi ,Ci,di 1*1 ,oi,di bi ,ei , di - b\ ,C\ ,d\ bi,Ci,di -tti *a , Ca , di bs ^Ca ids Similarly for higher orders ; but determinants of the 3rd, 5th and odd orders have each determinants of the lower order positive /the 2nd, 4th and even orders have the alter- native signs negative. (104.) Solve the equations I *' ? 1" 5* ?( ~ '"' ,e = Ci , *i I jca , bi ai , b\ ioa , bi I rta « + *2 y = <'a (0 (2) y Solve the equations lCa_, jOa ' 1*1 J ai i l*a , aa Sax X + bi y ■\- oi 2 + di = . .(1) aa iC + 6a y -f- Ca 3 + di = . . (2) Oa « + Js y -I- C3 z-tda = . .(3) .#- / \a h y ii II ,1 Tl orde fli , , (1 IB n Ad( eacl (isei 5fi 1i ? Ai , Oi Determhiniits The denominator of 05 is as , fta , ca ^aa , hi , Ct The numemtor is a similar determinant, writing d for a. Examples. = abe - af' - hif — eh" + 2fyh y , / » " il ,iCi , yi '1 ,i»i», ya ,1 ,»s,y3; Z' > «2 > ^2 W,y , ch ,b3 — x^ ya —Xi yi +Xi ys — ^s yu -j-ara yi — »! ys <'2 (105.) JS». Prove, without developing, that a -{- h -{• is a factor of a, h, c\ c, a,h\ = x h, 0, a\ \b, h, e\ Add the determinants la, a, b\ and c, ft, c J, a, i rt, <;, a each of which is evidently zero. ia -f- J + c , J , «! .*. 35= \a \-h -\- c ,a ,h — {a-\-h +• t") 'a -j- ^ + c , c , a' 1 ,i,c 1 , a , J 1 , c , a Q. E. D. This develops as follows : aS _|_j3 +^8 -3a5c=(«+J-|-c) («« -Jo+ft^ - ac+^a (See § 33.) ah) \ k «n 8UI1 sue tW( fori y / »!W?WBBP!"ef«M