Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/137

 B R A. 105 A L G E =0, then, according to the rule, fuppofe"y-}-]p—x, and Rule. Add to the unkntnvn quantity of ’the given e- fubflituting this value for x, you will find, quation the third part of the coefficient of the fecond term with its proper fign, viz. and fuppofe this aggregate equal to a new unknown quantity (/). From —py—lp2 > =0, this value of y find a value of x by tranfpofition, and +y3 fubllitute this value of x and its powers in the given equation, and there will arife a new equation that ffiall J>% *—ipl+q=0. want the fecond term. And from this example the ufe of exterminating the 2d appears: for commonly the folution of the equaExamp. Let it be required to exterminate the fe- termthat wants the sd term is more eafy. And, if you cond term out of this equation, x3—qx'+sfix—34=0, tion can find value of y from this new equation, it is eafy fuppofe x—3==>', or y+3—x ; and fubfiituting according to find thethevalue of x, by means of the equation _>>+4/=rx. to the rule, you will find For example, >5+9y’+27y+27') Since y'l--q—|^*=o, it follows, that “9/—54T-8I y'^p'—q, and y=tz*/^p y, fo that ,+26^+78 C-o r-° x=y+ip=ip=s=*/T[pl —q; —34A which agrees with what we demonftrated, chap. 12.4 } yi * —-y 10=0. Ifz the propofed equation is a biquadratic, as x —px then, by fuppofing x—-^p—y or x—y In which there is no term where y- is of two dimeofions, --qx —rx+/=o, ffiall arife having no fecond term. and an afterilk is placed in the room of the ad term, to And if antheequation propofed equation is of 5 dimenfions, then ffiew it is wanting. mud fuppofe x—yr±=.±p. Andfoon. Let the equation propofed be of any number of di- youWhen the fecond term any equation is wanting, it menfions reprefented by («); and let the coefficient of follows, that “ the equationin has affirmative andnethe fecond term with its fign prefixed be —p, then fup- “ gative roots,” and that “ the both of the affirmative pofing x—and confequently x=j/+—, and fubfti- “ roots is equal to the fum of thefumnegative roots:” by means the coefficient of the 2d term, which is the tuting this value for jc in the given equation, there will which fum of all the roots of both forts, vaniffies, and makes arife a new equation that ffiall want the fecond tern). the fecond term vaniffi. It is plain from what was demonflrated in chap. 15. generaly “ The coefficient of the 2d term is the that the fum of the roots of the propofed equation is “ Indifference between the fum of the affirmative roots +p; and fince we fuppofe y—x—^—, it follows, that, in “ roots and the fum of the negative rootsand the operations we given ferve only, to diminiffi all the the new equation, each value of y will be lefs than the roots when thehave fum of the affirmative is greateft, or inrefpe&ive value of x by -A; and, fince the number of creafethe roots when the fum of the negative is greatell, balance them, and reduce them to an equality. the roots is n, it follows, that the fum of the values of fo as isto obvious, that in a quadratic equation that wants y will be lefs than + p, the fum of the values of x, by theItfecond term, there muft be one root affirmative andand thefe muft be equal to one another.. «X—, the difference of any two roots, that is, by oneInnegative; a cubic equation that wants the fecond term, there •" therefore the fum of the values of y will be + muft be either two affirmative roots equal, taken togep—p=o. ther to a third root that muft be negative; or, two neBut the coefficient of the fecond term of the equation gative equal to a third that muft be pofitive. of y is the fum of the values of y, viz. +/—p, and “ Let an equation x'—pxl+qx—,~o be propofed, therefore that coefficient is equal to nothing; and confe- “ and let it be now required to exterminate the third quently, in the equation of y, the fecond term vaniffies. “ term.” It follows then, that the fecond term may be extermi- By fuppofing f=zx—e, the coefficient of the 3d term nated out of any given equation by the following in the equation of y is found (fee equation A) to be Suppofe that coefficient equal to nothing, Rule. Divide the coefficient of the fecond term of —2ps+q. and by refolving the quadratic equation ^e1—2pe--q—o, the propofed equation by the number of dimenfions of you will find value of which fubftituted for it in the equation ; and affuming a new unknown quantity the equation the yz=.x—e> will ffiew how to transform the y, add to it the quotient having its fign changed; propofed equation one that /hall want the third term. 1 Then fuppofe this aggregate equal to x the unknown The quadratic 3?into —2/H-£=o, mves v/7,‘—'3?. quantity in the propofed equation; and for x and its powers, fubftitute the aggregate and its powers, fo So that the propofed cubic will be transformed into an ffiall the new equation that arifes want its fecond equation wanting the third term by fuppofing y—x— term. If the propofed equation is a quadratic, as .v*—px^-q A-VA*—37> oxyzzx Vol. I. No. 5. 3 If