Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/141

 E B R A. 169 A L G Now if e be fuppofed ftill equal to £-}-f, and of the propofed'equation; of a number that exceeds the ppofitive. and r to be lefs than q, then, a fortiori, all thefe greateft pofitive root. will be pofitive, the negative part, which involves If the limit of the negative roots is required, you may terms and r being diminiffied, while the pofitive part and the (by chap. 16.) change the negative into pofitive roots, and pnegative involving q remain as before. then proceed as before to find their limits. Thus, in the 40. After the fame manner it is demonftrated, that If example, you will find, that —3 is the limit of the nega- r is the greateft ne'gative coefficient in the equation, and tive roots. So that the five roots of the propofed equa- e is fuppofed =7-4-1, then all the terms of the equation tion are betwixt —3 and -f-2. (y^) of7 will be pofitive; and confequently r+i will be Having found the limit that furpafies the greatefl: po- greater any of the values of x. fitive root, call it And if you afiiime y—m—x, and Whatthan we have faid of the cubic equation x't—p-ff for x fubftitute m—y, the equation that will arife will is eafily applicable to others. have all its roots pofitive; becaufe m is fuppofed to fur- qx-~r—o, general, we conclude, that “ the greateft negative pafs all the values of x, and confequently m—x (==y) muft “ Incoefficient any equation increafed by unit, is always always be affirmative. And, by this mekns, any equa- “ a limit thatinexceeds all the roots of that equation.” tisn may be changed into one that Jhall have all its roots But it is to be obfetved at the fame time, that the affirmative. negative coefficient increafed by unit, is very Or, if —n reprefent the limit of the negative roots, greateft then by afiuming the propyofed equation ffiall be feldom the neareJl linM-. that is beft discovered by the transformed into one that ffiall have all its roots affirma- rule in the beginning of this chapter. tive ; for +« being greater than any negative value of x, Having (hewn how to change any propofed equation into one that ffiall have all its roots affirmative; we ffiall it follows, that/=x-f-» mud be al ways pofitive. The greateft negative coefficient of any equation in- only treat of fuch as have all their roots pofitive, in what creafed by unit, always exceeds the greatejl root of the remains relating to the limits of equations. equation. Any fuch equation may be reprefented by x—a Xx—6 To demonftrate this, let the cubic x3—/>x*—yx—x — d, &c. =0, whofe roots are a, b, c, d. See. rrso be propofed; where all the terms are negative ex- Xx—cX And of all fuch equations two limits are eafily difeocept the firft. Affuming^=x—e, it will be transformed vered from what precedes, viz. o, which is lefs than the into the following equation; lead, and e, found as directed in the beginning of this chapter, which furpaffes the greateft root of the equation. But, befides thefe, we ffiall now (hew how to find cfher limits bethuixt the roots thimfeldes. And, for this purpofe, will fuppofe a to be the leaft root, b the fecond c the third, and'fo on'; it.being arbitrary. i°. Let us fuppofe that the coefficients p, q, r, are root, If you fubftitute o in place of the unknown quantity, equal to each other; and if you alio fuppofe e=/>d-i, putting x=o, the quantity that will arife from that fupthen the lad equation tbecomes pofition is the laft . term of the equation, all the others z that involve x vaniffiing. (B)y}-¥2py,Z +p y+l') If you fubftitute for x a quantity lefs than the leaft +3J +3^ >=Q; root a, the quantity refulting will have the fame fign as the laft: term; that is, will be pofitive or' negative acwhere all the terms being pofitive, ft follows that the cording as the equation is of an even or odd number of values of y are all negative, and that confequently e, or dimenfions. For all the faftors x—a, x—b, x—c. See. p-jrl, is greater than the greateft value of x in the pro- will be negative, and their product will pofitive or nepofed equation. gative according as their number is even or odd. 2°. If q and r be not —p, but lefs than It, and for e If you fubftitute for x a quantity greater than the leaft you ftiil fubftitute/H-i (fince the negative part (—qy—qe root a, but lefs than all the other roots, then the fign of the quantity refulting will be contrary to what it was bebecomes. lefs, the pofitive remaining undiminiffied,) a—H for- fore ; becaufe one fadtor (x — a) becomes now pofitive, tiori, all the coefficients of the equation (y?) become all the others remaining negative as before. pofitive. And the fame is obvious if q and r have pofi- If you fubftitute for x a quantity greater than the two tive figns, and not negative figns, as we fuppofed. It leaft roots, but lefs than all the reft, both the fadtors appears therefore, “ that if, in any cubic equation, p x—a, x—b, become pofitive, and the reft remain as “ be the greateft negative coefficient, the© p--i muftfur- they were. So that the whole produdt will have the fame fign as the laft term of the equation. Thus fuccef“ pafs the greateft value of x.” 30. By the fame reafoning it appears, that if q be fively placing inftead of x quantities that are limits betire greateft negative coefficient 'of the equation, and twixt the roots of the equation, the quantities that ree—q--, then there will be no variation of the figns in fult will have alternately the figns + aijd —. And, the equation ofy for it appears from the laft article, converfely, “ if you find quantities which, fubftituted'in that if all the three {p, q, r,) were equal to one another, “ place of x in the propofed equation, do give alterand e equal to any one of them increafed by unit, as to “ nately pofitive and negative refults, thofe quantities f+i, then all the terms of the equation (/f) would be - are the limits of that equation.” Vol. I. No. J. 3 Ec It