A treatise of algebra, in three parts. Containing

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16 A RE Aetis of PARTII. se iis coefficient the quantit stat arises by multiplying the term of the last quantit hythe indices of e in acti term an dividingili produxis That the messicient of thelast term ut tw-6, --ape 4his deduced in ille same manner fio the term immediatet solioWing that is by multiplying veryter of e' ope in aqe in by the index of in that term an dividing the whol is multiplied into the index os ron in termsought that is by And the nexi term

. The demonstration os this may rasilibe madegenera by the Theorem os finding the wwersos a binomiat, tace the transforme equation consist os thelowers of the binomial thature marhed by the indices os e in the last tum, multiplied each by thei coefficient i, - Φ o. δε ε 3, c. respectively. aue Fro the last in articles e came sit sind the term of the trans med equation Without an involution. The a term is adb substitutini instea of x in the proposed equatio, the nexi term is multiplying very par of that last term is in index se ineach pari, an dividing the whole and the following term in the anne describe in the oregoing article, in respective divisors being

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cΗAp. . ALGEBRA. 16sMinime quantur e multiplied by the index os

one os the smpla quations uia produce

os the values os e canis equa to nothingsor is ver termis no multiplied by x, then: - o cannot be a divisor of the whole equation, and consequently o cannot be one os the values of x. Isi' does no enter into at the terms of the quation the two os the values os cannot be equat in nosting. Is e Ges no enter into ali the term of the quation, thenthree of the values of x cannot be eques to thim, M. 6 6. Suppos no that Wo value os, areequat to ne anotlier, an to ea then it is plainibat two values olla in the tr/nstorme equation

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An consequently by the la artiori the twola term of the transforme equation inust Suppos icis the cubic equation ossos stat

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In generat, When ino values o care quallo each other, an to the two a term of the transforme equatio vanisti an consequently is o multi y the remis of the proposed equatio by the indices os, in minterm the quantity that wil arise,ill e mi, and wil give an equationis a lower dimensi cinthan the proposed that stat have ne of iis more eques to one os the modi of theimpostaequation 'That the last two termet os the e*ration vanish en the values of mare suppostd equa minachmirer, mi OA Win alio appea by considerion

stat since two values of then hecome eques tonoming the produc of the value os, must Vanim, Whichris equa to the la ter of the equation; and because tw of tho Mur values ofare eques to nothire, it sollam alis th M oneo any three that can besta n out of these urmumbe mi and the resore, the procluet made by multiplying any three must vanisti and conseque y the coemient of the last term bubone, whic is qua to the sum os these produm, must vanisti. 37. Aster the fame manner, i there arethree equa root in the biquadratie,' - ΡΤ ε-- and sis beseques mine of them, three values of ill vanissi, an consequently ill enter at the term of

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ead scio his eas demonstratio of this Rules; and wi ille applied in the ex chapter o demonstrate illa, Rules sor Ming the limit osequations. It is obvious however, that though, maheu se of equations whos sigia change alternately,

ille sanie re senilis exaehds o MLother equations.

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Whicli is the produc that arises h multiplying 'the term of the proposed equation by the terms of the series, et M. h. α; histima represent an arithmetica progression.

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be proposita an transform it, sis o into the equation,

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It is lassicient there re in orde to find the limit, o enquire What quantit substitutes sor in each os thest expression - ' in x - γ'- ρο --3 - p, Williive them allio-stive,' so that quantu Uilli the limit required. How these expression are forme stom oneanother, Was explaine in the eginninios illelast chapter.

y o Isthe equationi 2P iox ao es 63 veto, o is proposest ansit is require lio determine the limit that is greater than any of the oolsa ou are to enquare What integer number substitute sor, in the proposed equation, and the sollaminiequations deducta stom

ithyrios, Wil give, in acti, a positive qua

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ira A TREA Tis of PARTII. Is the limit of the negative mota is required, D m y by 6 23, change the negative intopositive mois, and then procred as besere to indthei limits Thus, in the example ou illfind that raris the limit os the negative roota. So that the sive mois of the propostd eqvationare bet ix -- Lan in . D i. Havinisound the limit stat surpasses the reate positive oot calicit Andri youassumebit ni an soris substitute, Mine equatio that wil a rise,ill have ali ita mota positive 'aecause, is supposta in surpasi anthe values os and consequently -- xima must alwayche assirmative. And by this means. av quatio ma be Ganred into Meuhat Isauhave ali iis rom a malive. 'OHis --n represent in limit os the negative '

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i'. et vi suppost that the messicients are eques to e Moster , and is mitis suppose

positive remaining undiminissita a Driori, allthe coefficients of the equationiae Piecome positive. An the sanae is obvious i candis have positive signs, an no negative signs, Moesupposta It appora heresere, 'inat, is, in any cubic equation, the greate negative coefficient, then must surpas the greatest Valu of x. 'ν a. q. y the fame realisniniit appears,thacisis e the greatest negative coefficient of the