T R A 



T R A 



Thus, the roots of the equation *♦ — x- — igx' + 49 x 

 — 30 = o, are + I, + 2, + 3. — 5 ; whereas the roots 

 «f the fame equation having only the figns of the fecond 

 and fourth terms changed, vt% x^ -r x' — 19 x" — 49* 

 _ 30 = c, are — I, — 2, — 3, + 5. 



If it be required to transform aa equation into another 

 that (hall have its roots greater or lefs than the roots of 

 the propofed equation by foine given difference, the 

 method is as follows. Let the propofed equation be the 

 cubic x^ — px- + qx — r — o; and let it be required 

 to transform it into another, whofe roots (hall be lefs than 

 the roots of this equation by fome given difference {e), i.e. 

 fuppofe ji = X — f, and confequently x =■ y -\- e; then, 

 inftead of x and its powers, fubftitute y -\- e and its powers, 

 there will arife this new equation. 



+ qy + 1' 



— r 



zpey -pe-l ^^. 



whofe roots are lefs than the roots of the preceding equa- 

 tion by the difference (?). 



To find an equation whofe roots (hall be greater than 

 thofe of the propofed equation by the quantity [e), fup- 

 pofe jr = X + f, and X — y — e, and the equaiion will 

 have this form. 



(B)>'- 30* + l^'y-'' ") 



-py- + Zpey-pe^l ^^ 

 + qy - q( f 



If the propofed equation be in this form, x^ + px' + qx 

 -f. r = o, then by fuppofing x -f e ^ y, there will arife 

 ao equation agreeing in all refpefts with the equation (A), 

 but that the fecond and fourth terms will have contrary 

 figns. 



(C)>»- $'y' + 3''y-'' 



■^ py' — zpey -t- />«' 

 + qy - qe 



+ r 



By fuppofing x — e =z y, there will arife an equation 

 agreeing with ( B ) in all refpeSs, but that the fecond and 

 fourth terms will have contrary figns to what they have in 

 (B): as 





+ le'y + r' 

 + zepy + pe"- 



-r qy + ?' 



+ r 



Hence we fee how the fecond or other intermediate term 

 may be tajfen away out of an equation ; for, in the equation 



(A), whofe fecond term is 3 ^ — ^ x _y% if we fuppofe 

 e =; yp, and confequently 3 « — ^ = o, the fecond term 

 will vanifh. In the equation (C), whofe fecond term is 



— i' + P ^ y^' fuppofing e = i p, the fecond term 

 alfo vanifhes. But the equation (A) was deduced from 

 x^ — px' + qx — r = o, by fuppofing ji = x — e, and 

 the equation (C) was deduced from x* + px' + qx 4- 

 r = o, by fuppofing y = x + e ; whence may be deduced 

 the following rule for exterminating the fecond term out 

 of any cubic equation ; viz. add to the unknown quantity 

 of the given equation the third part of the co-efficient of 

 the fecond term with its proper iign, -u/s. + jp, and fup- 

 pofe this aggregate equal to a new unknown quantity y. 

 From this value of jr find a value of x by tranfpofition, and 

 fubftitute the value of x and its powers in the given equa- 

 8 



tion, and there will anfe a new equation wanting the fecond ) 

 term. £.g. Let the equation be x ^ — px' -f 26 x — 34 J 

 — o ; fuppofe X — 3 = J, or _y 4. 3 = x, and fubftitutir.g \ 

 according to the rule, we (hall have, 



y^ + 9>' + ^ly + 27" 



- 9J' - 54> - 81 



+ 2(>y + 7S 



- 34- 



y^ * — y — 10 = o, an equation 

 wanting the fecond term. 



If the equation propofed be of any number of diinei - 

 fions (n), and the co-efficient of the fecond term with its 



fign prefixed be — p, then fuppofing x — — — y, and 



n 



X =. y -{■ —, and fubftituting this value for x in the givsn 

 n 



equation, there will arife a new equation that (hall want the 

 fecond term ; whence the fecond term may be exterminated 

 out of any given equation by the follov/ing rule. Divide 

 the co-efficient of the fecond term of the propofed equation 

 by the number of dimenfions of the equation ; and alfuming 

 a new unknown quantity _)r, add to it the quotient having its 

 fign changed : then fuppofe this aggregate equal to x, the 

 unknown quantity in the propofed equation ; and for x and 

 its powers, fubftitute the aggregate and its powers, and the 

 new equation will want its fecond term. 



Let the propofed equation be a quadratic, as x' — p x 

 -)- J =: o, then fuppofe y + i p ^^ x according to the 

 rule, and fubftituting this value for x, we (hall have, 



-py-iP'\=° 

 + 1 J 



f* - kf+ q = o 



Having found the value of y, that of x may be Lad by 

 means oi y -t \p =^ x : e. gr. fince y' + q ~ ^ p' — o, 



j»' = !/>= — q, and^ = + V ^p' — q, and therefore x = 



y + hP = hP+ ^'i/-.-.?- ,. , . 



If the propofed equation is a biquadratic, as x^ — p x^ 

 4-jx' — rx + J = o, then by fuppofing x — ^ p ■= y, 

 or X = X -I- \p, an equation (hall arife having no fecond 

 term. And if the propofed equation is of five dimenfions, 

 we muft fuppofe x = y + -^ p, &c. It is plain, that in a 

 quadratic equation wanting the fecond term, there muft be 

 one root affirmative and one negative, and thefe muft be 

 equal to one another. In a cubic equation wanting the fe- 

 cond term, there muft be either two affirmative roots equal, 

 taken together, to a third root that muft be pofitive. 



Let an equation x^ — p x' -i- q x — r=:obe propofed, 

 and let it be required to exterminate the third term. By 

 fuppofing y :=. X — e, the co-efficient of the third term in 

 the equation of ^ is found (fee equation A fupra) to be 

 3 «^ — 2 p e -\- q. Suppofe that co-efficient equal to no- 

 thing, and by refolving the quadratic equation 3 «' — i p e 

 -\- q = o, you will find the value of e, which fubftituted 

 for it in the equation ^ = x — ^, will (hew how to trans- 

 form the propofed equation into one that fhall want the third 

 term. 



The quadratic 3«'— zpe -i- q = c, gives 



e= p±->^p--Sq _ 

 5 



3 



fo that the propofed cubic will be tracsforir-ed ir.to an 



equation 



