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 E B R A. i o8 A L G It is obvidus, however, that though we make ufe of thus, “ the pro'pofed cubic is depreffed to a quadratic equations whofe figns change alternately, the fame rea‘t( that has one of its roots equal to one> of the roots of foning extends to all other equations. f that cubic.” If it 1,is the biquadratic that is propofed, viz. x*— It is a confequence alfo of what has been demonftraftxi+qx —rx-^-s—o, and two of its roots be equal; ted, that “3 if two roots of any equation, as, “ x —/’x’+j'x—r=o, are equal, then then fuppofmg e=x, two of the values of y muff vanifh, the terms by any arithmetical feries, as, and the equation of § 2.l, willibe1 reduced to this form. “ “multiplying «+3^, a-“2b, a--b, a, the produdl will be =0.” y+40 +6e ej> l 3 For, lfince ——‘hP y*_ + s’ >’1 y 3X —2px--qX.bx—o, it follows that tfX 3ff-3^x i~—apx'1—~2bpx'1 --aqx--bqx—ar~o. 3 i -t-zge—r~o; or, becaufe x = Which is the product that arifes by multiplying the 4.V — 3/’x -f2?x — r—c. In general, when two values of x are equal to each terms of the propofed equation by the terms of the feother, and to e, the two laft terms of the transformed ries, a+3#, a--2b, a--b, a which may reprefent any ’equation vanifli: and confequently, “ if you multiply arithmetical progreffion. “ the terms of the propofed equation by the indices of “ x in each term, the quantity tljat will arile will be =0, “ and will give an equation of a lower dimenfion than Chap. XVIII. Of the Limits of Equations, “ the propofed, that (hall have one of its roots equal “ to one of the roots of the propofed equation.” We now proceed to (hew how to difcover the limits That the lall two terms of the equation vanifh when of the roots of equations, by which their folutiou is the values of x are fuppofed equal to each other, and to- much facilitated, f, will alfo appear by confidering, that fince two values. Let any equation, as x3—/x^+yx—r=o be propofed;, of ji then become equal to nothing, the product of the as above, into the equation values ofy niuft, vanilh, which is equal to the laft. term and transform it,yt+w'+P'y+e of the equation; and becaufe two of the four values of 1 —Pj —2pey—pe y are equal to nothing, it follows alfo that one of any + qy + q three that can be taken out of thefe four muft be =0 ; and therefore, the produfts made by multiplying" any three muft vanifh; and confequently the coefficient of Where the values of y are lefs than the refpective values the laft term but one, which is equal to the fum of thefe of x by the difference e. If you fuppofe <? to be taken products, muft vanifh. make all1 the coefficients of1 the equation of y §5. After the fame manner* if there are three equal roots' fuch as toviz. e —pe'+qe—r, 3? —2pefq, —p ; in the biquadratic x4—•y>x3+ j'xl—rx-h-t^o, and if <r be pofitive, then there being no.variation of the figns in the equaequal to one of them, three values of y (=x—s) will all the values of y muft be negative; and confevanifh, and confequently y5 will enter all the terms of tion, quently, the quantity e, by which the values of x are the transformed equation; whkh will have this form, diminifhed, muft be greater than the greateft 'pofitive value ancf confequently muft be the limit of the roots of y +40’ ? # * * =0# go that here ofthex;equation x3—px:L--qx—r=o. — ’5 It is fufficient therefore, in order to find the limit, to Se1—B/’f+S—O; or, fince e=x, therefore, what quantity fubftituted for x in each of thefe 6x*—3/>xff-g=o: and one of the roots of this qua- “ inquire expreflions x3—/>x*+yx—3x*—2px--q, 3X—p, dratic will be equal to one of the roots of the propofed ““ will give them all pofitive;” for that, quantity will be biquadratic. required. In this cafe, two of the roots of the cubic equation the limitthefe expreflions are formed from one another, 4X3—o)pxx--2qx—r=o are roots of the propofed biqua- wasHow explained in the beginning!of the 4laft chapter. dratic, becaufe 3 i the, quantity 6x*—^px--q is deduced Examp. If the equation x — 2x — iox3+30xl+ from 4X —3/x -|-2§ x—r, by multiplying the terms by 63x+I20=0 is propofed; and- it is required to deterthe indexes of x in each term. limit that is greater than any of the roots; you In general, “ whatever is the number of equal roots minetotheinquire what integer number fubftituted for x in the' “ in the propofed equation, they will all remain but one are equation, and following equations deduced from “ in the equation that is deduced from it, by multiplying propofed “ all the terms by the indexes of x in them ; and they it by § 3. chap.4 17.3will give, in each, a pofitive quantity, “ will all remain but two in the equation deduced in the yx3—8x1—3ox*+6ox+63 “ fame manner from that;” and fo of the reft. yx1—6x — i^x-j-iy § 6. What we obferved of the coefficients of equations yx — x—y transformed by fuppofing y—X—e, leads to this eafy deyx —2.4 monftration of this rule; and will be applied in the next chapter to demonftrate the rules for finding the limits of The lead integer number which gives each of thefe equations. pofitive, is 2 ; which therefore is the limit of the rootsof
 * * — o. So that ax1—-apx' ‘-aqx~arz=o ;, and