Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/142

 no A L G E BRA. It is ufeful to obferve, that, in general, “ when, by which is the limit lefs than any of the roots) four lifubftituting any two numbers for x in any equation, mits for the threeroots of the propofed cubic. “ the refults have contrary figns, one or more of the 2It was demonftrated in chap 17. §3. how the quadratic “ roots of the equation mull1 be betwixt thofe numbers.” 36' —2pe--q is deduced from the propofed cubic <?J—Thus, in the equation, x —2x2—5=o, if you fub(ti- pe1--qe—r—o, viz. by multiplying each term by the tute 2 and 3 for x, the refults are —5, +4; whence it index of e in it, and then dividing the whole by e; and follows, that the roots are betwixt 2 and 3 : for when what we have d.emonftrated of cubic equations is ealily thefe refults have different figns, one or other of the extended to all others ; fo that we conclude “ that the factors which produce the equations muft have changed “ laft term but one of the transformed equation is the its fign ; fuppofe it is x—<?, then it is plain that ^ muft “ equation for determining the limits of the prdpofed be betwixt the numbers fuppofed equal to x. “ equation.” Or, that the equation arifing by multiLet the cubic equation x}—px7'-)-qx—r=o be, pro- plying each term by the index of the unknown quantity pofed, and let it be transformed, by affuraing —£-, in it, is the equation whofe roots give the'limits of the propofed equation ; if you add to them the two mentioninto the equation ed in p. 109. col. 2. par. 4. For the fame reafon, it is plain that the root of the 1 2 —Py 2pey —pe f n > Ample equation 3?—p—o, (/. e. p) is 1die limit3 be+ W +F:C tween the 3two roots of the quadratic 3e —2pe- rq—o. And, as qe4 —3pfl1+2?v—r—o gives three limits of the x equation t f —pe --qs —re-f-jzzo, fo the quadratic x = Let us fuppofe e equal fucceffively to. the three values 6e —3/.'H-y o gives two limits that are betwixt the 3 % of x, beginning with the leaft value; and becaufe the of the cubic t.e —%pe .+ 2qe—r — o and Jaft term —pez--qe—■> will vaniflt in all thefe fuppofi- roots, 41?—p—o gives one limit that is.betwixt the two roots 2 tions, the equation will have this form, of the quadratic 6e —^pe+q’—o. So that we have a complete feries of thefe equations arifing from a fimple equation to the propofed, each of which determines the —py —^P6 c =°» limits of the following equation. + ?2 3 If two roots in the propofed equation are e?u;d, where the laft term 3<? —2pe-~q is, from the nature of then “ the limit that ought to be betwixt them muft, equations, produced of the remaining values of y, or of " in this cafe, become equal to one of the equal roots “ themfelves.” Which perfectly agfees with what was the excefles of two other values of x above what is fup- demonftrated in the laft chapter, concerning the rule for pofed equal to f; fince always jc=x—e. Now, i°. If ^ be equal to the leaf! value of x, then thofe finding the equal roots of equations. Apd, the fame equation that gives the limits, giving two exceffes being both pofitive, 1 they will give a pofitive product, and confequently 3s —2pe--q will be, in this alfo one of the equal roots, when two or more are equal, it appears, that “ if you fubftitute a limit in place of cafe, pofitive. unknown quantity in an equation,” and, inftead 2°. If e be equal to the fecond value of x, then, of of“ the a pofitive or negative refult, it be found =0, then you thofe two exceffes, one being negative and one pofitive, may conclude, that “ not only the limit itfelf is a root their0 produft 3?*—2pe--q, will be negative. but that there are two roots in that 3 . If e be equal to the third and greateft value of x, “ of the equation, equal to it and to one another.” then the two excefles being both negative, their produft “ Itequation having beenx demonftrated, that the roots of the 3^—2pt--q is pofitive. Whence, x' —px --qx—r—o are the limits of the roots If in the equation qe*—2/^+9’=o, you fubftitute fuc- equation 1 ceffively} in the place of e, the three roots of the equa- of the equation qx —2px4-qz=o, the three roots of the to be a, b. c, fubftituted tion e —pez-~qe—r=o, the quantities refulting will cubic equation, which fuppofe 2 for x in the quadratic 3.x —rapx-f-q, muft give the refucceffively have the figns -}-, —, +; and confequently fults and negative alternately. Suppofe thefe the three roots of the_ cubic equation are the limits of three pofitive 2 refults to2 be -f-N, •—M> +L; the foots of the equation 3^—2pe--q~o. That 2 that is, 3' -— is, - the leaft of the roots of the cubic is lefs than the zpa+q—N, 3 3^ —zpb-f-q——M, 3c 3 —2pc--q=-L 1 leaft of the roots of the other ; the fecond root of fince a —pa*--qa—r—o, and 3// —2pa --qa—N'><.at the cubic is a limit between the two roots of the other; fubtrafting the former1 multiplied into 3 from the latter, remainder is pa -—2qa-^r=N'><.a. In the fame and the greateft root of the cubic is the limit that ex- the manner pb1—2qb--or=—MY>b, and pc1—2pc-(-3> —+ ceeds both the roots of the other. We have demonftrated, that the roots of the cubic LXr. Therefore px1) —2q - Jr2)r is fuch a quantity, that equation e*—pe^-f-qe—r~o are limits of the quadratic if, for x, you fubftitute in it fucceffively a, b, c, the re3^1—2pe--q", whence it follows (converfeiy) txt the fults will be -f/VX, —MXb, ft-iXc. Whence a, b, c, roots of the quadratic 3^—2peJrq—o are the limits be- are limits of the equation p 2—2qx--q)r—o, by p. 109. tween the firft i.and fecond, and between the fecond and cpl. 2. par. 8. and, converfely, the roots of the equation third roots of the cubic el—pe'+qe—r=o. So that if ybu px 2—2qx-~^r~6 are limits between the firft and fecond, find the limit that exceeds the greateft root of the cubic, and1Jbetween the fecond and third roots 2of the cubic x3— by die beginning of this chapter you will have (with o. p.K rqx—r=o. Now the equation /x —2qx--y=o a-
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