ALGEBRA. 
a —-5 a 3 —36=0, £=~=-~f; 3 a ' b - 
3 3 3 
——, &c. therefore 
3 
V III 
y—~+~i ■+■ -p + &c * and wheB *=--*> J= 3 + p + -p 
4&c. =.347 &c. which is one root of the equation 
y 3 —37+1=0* 
If for_y, the feries 22.v-|+* ! +c*v 5 + < ^* i + Sc, c. had been 
ajjfumed, the quantities b , d, See. would have been found 
—o, therefore the even terms are unneceflary. 
Cor. The lefs ,v is aflumed, the fafter will this feries 
converge, and the more accurately will y be obtained. 
This method of approximation is fimilar to the former, 
in this refpeCt, that the feries will have a How degree of 
convergency; except one value of y is much lefs than any 
other. If this be not the cafe, find in, an approximate va¬ 
lue of y, by trial, and atfume mdzv—y’, then when one 
value of v is much lefs than any other it may be found by 
this reverfiori, and confequently that value of y known, 
which is neareft to in. 
On the. Sums of the Powers of the Roots of an Equation. 
Let a, b, c, See. be the roots of the equation x” — px n ' 
•\-qx n 2 . fwx n m — See. —o, and A, B, C, . . . . 
P, Q, R, S, the fum of the roots, fum of their fquares, 
cubes, .... m— 3, m —2, m —i, m, powers reflectively ; 
then will A—p, B—pA —2<7, C-=pB — qAJj-^r, See. and S— 
pR — qQj^-rP .— into ;. where -\-zu is the coefficient of 
the 77741th term. Then, 
The phopofjtiort alfo admits of th£ following proof. 
3 f =°> and c—ab Tue fame notation being retained, let in and n be equal', 
and fince a , b, c, Sec. are roots of the equation* 
a ’ 1 — pa n 1 4 qa* 2 . . . . 4 w—o 
b v — ■po' '-f qb % 2 .... -pw—O 
1 — n — i.px n ~-\-n —2 .qx r 
. -j -n — m.rox n n 1 — &c. 
x“ — px 11 f-qx" 
1 J 
-lux 
ice. 
x—a ' x—b ' x —c 
and, by aftual divifion, 
1 —1 4. — 4 — 
x ■ a x a * x 2 
b b 2 
, &c. whatever be the value of x ; 
- \r Sc c. 
f— V + + 
X ~V 
b m 
X - C X . 
■ + “T 
x m + 
c 
- 4 See. 
4 -1- Sc c. 
r x• ra •^- , ^ 
and if x be fuppofed greater than any of the magnitudes 
a , b, c, Sec. no quantity is loft in the divifion ; therefore, by 
addition, 
71-3 
4 n — m.wx r ‘ 
nx n ~'— n — i.px n *4 n —2 .qx 
X n — px n ~ t -\-qx n ~ i . 
_ n A B P ^ 
—; + X 2 + 'r x m —2 + x m —1 + x m + 
4 Sec. and multiplying by x n — px n ,J rqx‘ 
wx v m — See. 
Sec. 
-\-wx n m - 
0 
Sec. 
R 
x m +' 
. ., 4 
ji¬ 
nx’ — n —1 .px 
Sec. = 
nx n ~‘4 Ax ’ 1 —- ! + Bx 71 ' 
— npx ri ~ % — pAx n 
4 nqx H ~ 
4 n— 2. qy 
,77- 
4 n — m.rux n 
. . 4 Sx n — *■—‘4 See. ^ 
. .. ;— pRx 7 ‘ m 1 — Sec. I 
. . J r qOx n —‘’4 Sec. t 
.... ~-rPx n m '— Sec. f 
Sec. J 
J r nwx n — m —'-J r Sec. J 
and by equating the coefficients, A — np=i — n — z.p, and 
A=zp-, B — pA-\-nq—n—2.q, or B—pA —2 q. Sec. S—pR 
A-qQ — rP . . . 477777=77— m.w, or S=pR — qQfrP . . . . 
- 77707 . 
Ex. Let -v 3 45* 2 .— 6 x —8=0; then, by comparing the 
terms of this w ith the terms of the equation x n — px n —’4 
qx n *•— Sec. —o, we have p— — 5, q ~— 6, r=8. 
Hence the fum of the roots —p=~^=A. 
Sum of the fquares =zpA —25=25 4 12—372=!?. 
Sum of the cubes —pB—qA-^^r— —185—30424= 
—*191=0, Sec. 
cV—pc" 
Sec. 
-tf c 
4 07—0 
By add. 
5 — pR-j-qO .-47707=0 
Or S—pR — qQ, .—7to7. 
If m be greater than n, multiply the propofed equation 
by x m ", then* 1 "— px m ' y -\-qx m 2 .4 wx m ”=o; 
which equation has the roots <2+, c, Sec. and in —77 roots:, 
each equal to o ; therefore the fum of the 777th powers of 
the roots of this equation is equal to the fum of the 777th 
powers of the roots of the former; that is, S—pR —y 2 + 
&c. to n terms. 
When 777 is lefs than ? 7 , the fum of the 777th powers of 
the roots may be expreticd in terms of p, q, r, — w, where 
777 is the coefficient of the /.v4 1 th term of the equation. 
For p 2 contains a 2 -j- b 2 -fr 2 -!- Sec. with other combina¬ 
tions of the roots, as ab, ac, be , &c. which combinations 
are contained in a multiple of 7; alfo p 3 contains t2 3 4^ 1 + 
c 3 4 Sec. with other combinations, luch ms a*b, a?c, b 2 a. 
Sec. abc, acd , bed, Sec. and thefe combinations may be made 
up of p, q, and r ; for pXq contains the quantities a 2 b, a 2 e, 
b~a, Sec. and r is the fum of the quantities abc, acd , bed, 
Sec. In the fame manner it appears, that a* 4- b* -f c* + 
Sec. may be found in terms of p, q, r, and s ; and, in ge¬ 
neral, a Pt f b m c m A- Sec. may be exprelfed in terms of 
p, q, r, . . . . 7 ( 7 . Alfo, the number of combinations of 
any particular form, as cfb, cannot be altered by the intro¬ 
duction of the root c ; confequently, the coefficient of the 
product pq, by which the combinations of that form are 
taken away, is the fame, w hatever be the number of roots. 
Hence the expredion for a m b a c m Sec. in the equa- 
tjon x n — px n '-\-qx n *. -\-wx n m — Sec. —o, is,the 
lame with the expreffion for the fum of the 7»th powers of 
the roots of the equation x v — px m ‘4 nx m 2 . ~w 
=q; that is S—pR—qO .... —777777. 
This rule was given by Sir If. Newton for the purpofe of 
approximating to the greateft root of an equation. S.up- 
pofe the roots all poffible, and one greater than the reif, 
the powers of this root increafe in a higher ratio than thofe 
of any other, and the 2777th power of this root will ap¬ 
proach nearer and nearer to a ratio of equality with the 
fum of the 2777th powers of the roots, as m increafes; 
therefore, by extracting the 2777th root of this fum, an ap¬ 
proximation is made to the greateft pofitive or leaft nega¬ 
tive root. 
On the ImpoJJible Roots of an Equation. 
It has before been (hewn, that there are as many pofitive 
roots in an equation as it has changes of figns, and as many 
negative roots as continuations of the fame fign, when the 
roots are all poffible. But this rule cannot be applied to 
impoflible roots ; as appears by the conlideration, that an' 
impoffible quantity cannot be termed either pofitive or ne¬ 
gative. 
If then it appear from the terms of an equation that 
fome roots may either be called pofitive or negative, ac¬ 
cording to the above rule, they muft be impoffible. Thus 
two roots of the equation x'-\-qx-\-r—o, or x 3 dzo-\-qxfr 
=0, are impoffible ; becaufe it has two changes of figns 
or none, according as the fecond term is fuppofed to be 
—0 or 4 °* In the fame manner, if’any term of an equa¬ 
tion be wanting, and the figns of the adjacent terms.be 
both pofitive, or both negative, the equation has, at leaft, 
two impoflible roots; and, if two fucceeding terms be 
wanting, it muft always have, at leaft, two impoffible roots. 
Impoflible roots enter equations by pairs ; they alfo lie 
under the torm of two pofitive or two negative roots. 
Let ±724 V— b~, and ±22— f — b 2 , be the roots; then 
xepa—f -r-b 2 X xzpaff — 6 , =x*:qT 222 x 4 '+l"+= 6 ,which 
equation (hews either two pofitive or two negative roots. 
Cor. 
