ALGEBRA. 
297 
equation x B —px 2 f-qx — r—o, be a, b, c ; to transform it in¬ 
to one whofe roots are a-fe, b\e, c-{-e. 
Affume x 4 “ e —/> or x—y—e; then 
X 3 —y 3 - yy 2 g-j/jj/- e 3 ^ 
— px z — —py 2 +2 pey—pe 2 \ __ 0i 
-\-qx = + qy — q e f 
—r —r j 
In this laft equation, fince/rrx-fe, the values of y are 
c-J-c, c-\-e. 
If /-f-e be fubftituted for x, the values of y in the re¬ 
flating equation will be a — e, b — e, c — e. 
In general, let the roots of the equation at”— px n ' 4 ~ 
qx n 2 — &c. —o, be a, b, c, Sec. Affume y—x~ f, or 
x=zy-\-e, and, by fubftitution, 
— px >’— 1 2= 
+q * n ~ 2 - 
&C. =Z 
4- ney n ~^ i -)- n. - e 2 y n 2 
— py n 1 — n —i . pcy n 2 
+ qy n ~~ 2 
Sec. 
.4- ne n i y 4- <" 
. . . —n —1 . pe n 2 y •— pe n 1 
. . . 4 -n —2 . qe n 3 y 4" 2 
and, fine cy—x — e, the values of y are a — e, b — e, c — e, 
&c. 
We may obferve, that the laft term of the transformed 
equation, e n — pe v '4 -qe n 2 —&c. is the original quanti¬ 
ty, with £ in the place of x; the coefficient of the lad term 
but one is obtained by multiplying every term of e n — 
pe n ‘4 -qe n *—&c. by the index of c in that term, and 
diminifliing the index by unity ; the coefficient of the lafi: 
, it —1 _ n — 2 „ , . - n —3 
term but two, n. - e n ■— n —1.- pe n J 4~ n —-■ 
2 2 2 
qe 11 * Sec. is obtained in the fame manner from the coeffi¬ 
cient of the laft term but one, and dividing every term by 
2, &c. 
One life of this transformation is, to take away any 
term out of an equation. Thus, to transform an equa¬ 
tion into one which ftiall want the fecond term, e mu ft be 
P 
fo affirmed that ne — p— o, or e—-, (where p is the coeffi- 
TL 
cient of the fecond term with its fign changed, and n the 
index of the higheft power of the unknown quantity;) 
and, if the roots of the transformed equation can be found, 
the roots of the original equation may alfo be found ; be- 
P 
caufe, xr=v 4 " "• 
n 
Ex. To transform the equation x 3 — <)x 2 ~\~q x-f-12=20 in¬ 
to one which ffiall want the fecond term. 
A flume then 
X s —/ 4 - 9 jv 2 4 - 2774 - 2'5 
—9* 2 = —9/ 2 —54/—81 I q 
4-7X — 4- 774-21 r— * 
4-12 — +12.J 
That is, y 3 —20/—21=0; and, if the values of / be a, b, 
c, the values of xare <z-{"3> ^4"3> andc4*3> 
To transform an equation into one whofe roots are the 
reciprocals of the roots of the given equation. Let the 
roots of the equation, x n — px n ‘4 -qx n 2 . . — Px-\-Q 
—o, be a, b, c, Sec. to transform it into one whofe roots are 
Affirme y=~, ' or x— -4 then, by fubftitution, - 
x y y n 
L _ p 
4-yF=7- — —+Q=o, or multiplying by /”, 
1— py+qy 2 • • • — Py n ‘fQy’ 1 —^ that is, Qy" — Py n 1 
.... 4 ’qy 2 — py+1—O ; and fince /=-, the values of/ 
111. 
sre, —, -r, —, Sec. 
a b c 
Cor. i. If any term in the given equation be wanting, 
the correfponding term will be wanting in the transformed 
.equation ; thus, if the original equation want the fecond 
term, the transformed equation will want the laft term but 
one, Sec. becaufe the coefficients in the transformed equa¬ 
tion are the coefficients in the original equation in an in¬ 
verted order. 
Cor. 2. If the coefficients of the terms, taken from 
the beginning of an equation, be the fame with the coeffi- 
Voi,. I. No. 19. 
cients of the correfponding terms, taken from the end, 
with the fame figns, the transformed equation will coin¬ 
cide with the original, and their roots will therefore be the 
fame. Let a, b, c, be roots of the equation x n — px n “-f- 
qx n 2 .-f ? x 2 — px 4-1=20; the transformed equa¬ 
tion will be/” — py n '-\-qy n ~ 2 . ~\~qy 2 — py-fi — o, 
and, a, b, c, muft alfo be roots of this equation, but the 
roots of this equation are the reciprocals of the roots of 
the original equation; therefore, -, -, are alfo roots 
a b c 
of the original equation. 
Ex. The roots of the equations, x 4 — px 3 -\-qx 2 — pxf-x 
— o; * 4 4-fAr*4-i=o; and x 4 -J-i—o, are of the form a, b t 
1 1 
a' b 
Cor. 3. If the equation be of an odd number of di- 
menfions, or if the middle term of an equation of an even 
number of dimenfions be wanting, the fame thing will 
hold when the figns of the correfponding terms, taken 
from the beginning and end, are different. 
Ex. The roots of the equation x 3 — px 2 -\-px —1=0, are 
I 
of the form 1, a, For, in this cafe, if the figns of all 
a 
the terms of the transformed equation be changed, it will 
coincide with the original equation; and, by changing the 
figns of all the terms, we do not alter the roots. 
The equations deferibed in the two laft corollaries are 
called recurring equations. 
Cor. 4. One root of a recurring equation of an odd 
number of dimenfions, willbe-j-ij or —1, according as 
the fign of the laft term is ■— or 4-; and the reft will be of 
the form a, -, b, Sec. For, if 4-1 in the former 
a o 
cafe, and —1 in the latter, be fubftituted for the unknown 
quantity, the whole vanifhes j and, if c, b, c, Sec. be roots 
_ , 111 , _ 
of the equation, -, -, -, are alio roots. 
a b c 
To transform an equation into one whofe roots are the 
fquares of the rdots of the given equation. Let x n — 
px n ‘4 -qx 11 2 — rx n 3 -\-sx u *—&c. =20; by tranfpo- 
fition, x n -\-qx n 2 4 -^ n 4 4 " & c - ~px n '-fox” 3 -[- &c. 
and, by fquaring both fides, x 2 "4-2 qx zn 2 4-^ 2 -f-2.(. 
x 2n ■*4- Sec. —p 2 x 2U 2 4-2 prx 2n ‘4- Sec. and again, 
by tranfpofition, x 2n — p 2 —2 q.x 2 ’ 1 2 4~‘? 2 — 2.pr-\-zs, x 2n ~* 
— &c. 22:0; alTume y—x s , then/”— p 2 —2 q.y n ‘4. 
q 2 — 2pr-\-2s.y n 2 — &c. =0, in which equation the va¬ 
lues of / are the fquares of the values of x. 
Cor. If the roots of the original equation be a, b, c, 
8ec. then p 2 —2y=a 2 4-^ |2 4" <:2 4' q 2 - 2 pr-\-2s~a 2 b 2 
4-<2 2 r 2 4-£ 2 <- 2 4- Sec. 
1'hofe who wifti to fee other transformations, may con- 
fult Dr. Waring’s Meditationes Algebraicie. 
On the Limits of the Roots of Equations. 
If a, b, c, — d, &c. be the roots of an equation taken in 
order, that is, a greater than b, b greater than c, Arc. the 
equation is x— a.x — b.x — c.x-\-d } Sec. ~o; and, if a 
3 G quantity 
