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 107 B R A L G E that arifes by multiplying the terms of the laft quantity eft root in the one it trdnsfortned into the leajl root in by the indices of e in each term,' and dividing the prothe other. For fince x = —, by e: That the coefficient of the laft term but two, y andv=—, {viz. 6e*—Ipe+q) is deduced in the fame manner from that when the value of x is greafeft, the value of y is the term immediately following, that is, by multiplying leaft, and converfely. term of 41?}—o)pez--2qe—r by the index of e in How an equation is transformed fo as to have all its everyterm, and dividing the whole by e multiplied into roots affirmative, ffiall be explained in the following' that the index ofy in the term fought, that is, by eX2 : And chapter. the next terjn is 4?—.X2. Chap. XVII. Offinding the Roots of Equa- The demonftration of this may eafily be made general the theorem for finding the powers of a binomial, tions when two or more of the Roots are e- by fince the transformed equation confifts of the powers of qual to each other. the binomial y-{-e that are marked by the indices of e in multiplied each by their coefficients 1, . Befo r e we proceed to explain how to refolve equa- the laft--q,term, —r, +/, &c. refpedHvely. tions of all forts, we ffiall firft demonftrate hoiu ah equa- •—p, §3. From the laft articles we can e^fily find the terms tion that has tiuo or more roots equal, is deprejfed to a of the transformed two without any involution. The looser dimenfion; and its refolution made, confequently, laft term is had by equation fubftituting e inftead of x in the promore eafy. And ffiall endeavour to explain the grounds pofed equation; the next' term, every of this and many other rule's we ffiall give in the remain- part of that daft term by the index ofby emultiplying in each part, and ing part of this treatife, in a.'more fimple and concife dividing the whole by <?; and the following terms in the manner than has. hitherto been done. defcribed in the foregoing article ; the refpe<5live In order to this, we mull look back 3to the laft: chapter, manner divifors being the quantity e multiplied by the index of where we find, that if any equation, asx —px^-^qx—r~o, term. is propofed, and you are to transform it into another that y in eachdemonftration for finding when two or more roots ftall have its roots Lfsthanthe values of x by any given areThe will.be eafy, if we add to this, that “ when difference, as e, you are to affume y = x—e, and fublti-/ “ -theequal quantity enters all the terms of any eqqa-. tudiig for'x its value you find the transformed “ tion,unknown then one of its values is equal to nothing.” As equation, in the equation x3—px1--qx—o, where x—0=0 beingone pf the fimple equations that produce x3—/>xi -jy*d-iey'z + o,e'y + g.x=o, it follows that one of the values of x<is o. In —fy —2pey—pe* (' like manner, two of the values of x are equal to nothing; + qy -^qe in this equation x43 —px* =o;i and three of them vaniffi in the equation x —pxi=o. Where we are to obfenm,3 t 0 It is alfo obvious {converfely' ) that “ if v does not en-" i ". That the-lafl term (e —pe --qe—r) is/the very all the terms of the equation, i. e. if the laft term equation that uras propofed, having e in place of x. ^ ““ ter be not wanting, then*none the values of x can be 2°. The coefficient of the lalt term but one is •3le — “ equal to nothing,” for if everyof term be not multiplied 2pe--q, which is the quantity that% arifes,, by multiplying by x, then.x—o cannot be a divifor equaevery term of the lad coefficient e —pe --qe—r by 3the tion, and lconfequently o cannot be ofonetheofwhole indexL of e in each term, and dividing the produdt pc — -of x. If x does not enter into all the terms ofthethe values equao.pe' --qe by the quantity e that is common to all the tion, then two of the values of x cannot be equal to noterms. thing. If x3 does not enter into all the terips of the30. The coefficient of the lafl: term but two is, y—p, equation, then three of the values of x cannot be equal to which is the quantity that arifes by multiplying every nothing, &c. term 6f the coefficient laft found (31?*—2/’e+?) by the § 4. Suppofe now that two values of x are equal to one. index of e in each term, and dividing the whole by 7e. another, and to e; then it is plain that two values-of § 2. Thefe fame obfervations extend to equations of all transformed equation will be equal to nothing: dimenfions. If it is the biquadratic x4—px^+qx1 — rx-f- infincethey=x—<?. by the laft article, s—o that is propofed, then by fuppofing y—x—e, it will the two laft termsAndof confiequently, the transformed equation muft vabe transformed into this other, niffi. Suppofe it is the cubic equation of § 1. that is proy*+Aey3s+6el 1yx z + /ie*y e + 3 pofed, viz. x3—px2-j-qx — r — o ; and becaufe jve 1y —3P 'y—peiy/ —fy —M fuppofe x=e, therefore the laft term of the transformed +£>' iqey + qe >—0 equation, viz. e3—pe2+ge—r will vanift. And fince two values of y vaniffi, the laft term but one, viz. will vaniffi at the fame time. So that Where again it is obvious, That the laft term is the equa- 3?*y—opey-rqy 3e*—2pe-rg~o. But, by fuppofition, e=x3 • therefore, 2 tion that was propofed, having e in place of x : That when two values of x, in the equation, x .—px g-qx— the laft term but one has for its coefficient the quantity r=o, are equal, it follows, that —2px-j-g=o. And thus.
 * it is plain, dudl