Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/138

 106 A L G E B R 3 A. i If the propofed equatiorr is of n dimenfions, the value formed 3 into zthe equation/ —I3/ -{-I4X3Xx-}-i6X9=0»' of e, by which the 3d, term may be taken away, is had or/ —i3/ +42x-i-i44=o. Then finding the roots of this equation, it will eafily by refolving the quadratic equation <?’ be difeovered what are the roots of the propofed equation, 3X=/, or x=4/. And therefore, fince one of the =0, fuppofing —p and to be the coefficients of the fince values of / is —2, it follows, that one of the values of 2d and 3d terms of the propofed equation. y. The 4th term of any equation may be taken away by x isBy the “ an equation is eafily cleared of folving a cubic equation, which is the coefficient of the “ fraftions.”laft rule, Suppofe the equation propofed is x3— 4th term in the equation when transformed, as in the former-part of this chapter. The fifth term may be taMultiply all the terms by the ken away by folving a biquadt atic; and after the fame produdt of the denominators, you find manner, the other terms can be exterminated if there 3 MneXx —nspXx1--meqXx—mnr=o. are any. (as above) transforming the equation into one that There are other tranfmutations of equations that, on Then ffiall have unit for the coefficient of the higheft terra, fome occafions, are ufeful. At>equation, as x3—pxz+qx—r=o, “maybetranf- you find/3—nepXy‘i--m1eznqXy—tnin'ie'lr=o. “ formed into another that ffiall .have its roots equal to “ the roots of this equation multiplied by a given quan- Or, neglefting the denominator of the laft term— “ tity,” as by fuppofing and confequently' you need only multiply all the equation by mn, wffiich and fubftituting this value for x in the propofed will give 3 mnXx —tipXx1-JtmqX.x——-=0. And equation, there will arifej^ l ? —r=o, and mul3 tiplying all by. . y*—-ft>y' --f 'qy—y r=o, where then /3—npXy1 +?«1 «?X/——r=o. the coefficient of the 2d term of the propofed equation multiplied into f, makes the coefficient of the 2d term Now after the values of y are found, it will be eafy of the transformed equation; and the following coeffi- to difeover the values of x; fir.ee, in the firft cafe, cients are produced by the following coefficients of the X—-2— ; in the fecond, x~—. propofed equation, 3(as q, r, &c.) multiplied into the powers of / (/*,/, &c.). For example, equation 3 the Therefore to transform any equation into another whofe roots ffiall be equal to the roots of the propofed. to this form 3Xx3*—4X—*p“=o, and then transformed equation multiplied by a given quantity (/),” you need into / ’ *— 12/— 146=0. only multiply the terms of the prbpofed begin- Sometimes, by thefe transformations, “ furds are tal J equation, 4 ning at the 2d term, by /, /, / , / , &c. and put- “ ken away.”, 3As for example, ting y inftead of x, there will arife an equation having The equation x —-p*/aXx^+qx—nja—o, by putting its roots equal to the roots of the propofed equation mul- y^z^/axx, orx~.A.. , is transformed into this equation, tiplied by (/) as required. yV The transformation mentioned above is of ufe when the high ell term of the equation has a coefficient dif- a*/ a pA/aX^~-rqx^ }- —r»/a—o. Which, by multiferent from unity; for, by it, the equation may be ttanf- plying all the aterms /a becomes/3—pay*+qay formed into one that ffiall have the coefficient of the high- —ra^—o, an equationbyfreean/a, of fords.mu But in order to eft term unit. 1 the ford (yfo) ft enter, the alterIf the equation propofed is ax —-/x^-fyx—•r—o, then make this focceed, beginning with the fecond. transform the equation into one whofe roots are equal to nate“ terms, An equation, asx3—/ix'+yx-—r=ro, may be tranfthe roots of the propofed equation multiplied by (a). “ formed one whofe roots ffiall be the quantities reThat is, fuppofe y=ax or x-l- and there will arife “ ciprocalinto of x;” by foppefing/=—, and /= —, or. 3 1 fo that (by one foppofition), x=—, z becomes z —qz -{-prz— y l—py*+qiy—ra*=o. r*=o. From which we eafily draw this In the equation of/, it is manifeft, that the order of Rut.E. Change the unknown quantity x into another /, the coefficients is inverted; fo that, if the fecond term prefix no coefficient to the higheft term, pafs the 2d, had been wanting in the propofed equation, the laft but multiply1 the3 following terms, beginning with the 3d, ffiould have'been wanting in the equations of/ and z. by a, a, a , a4, &c. the powers of the coefficient one wanting in the equation propofed, of the higheft term of the prppofed equation, refpec- Ifthethelaft3dbuthadtwobeen had been wanting in the equations of / tively. and z. 3 Thus the equation 3X —^x^iqx+id^o, is tranf- Another ufe of this transformation is, that the great-
 * —4x—‘/t =0, is firft reduced