3°+ 
Cor. Hence, if the lad term of an equation of an even 
number of dimenfions be negative, it will have at leafl: two 
poflible roots, one pofitive and the other negative. 
Let an equation be transformed into one whofe roots are 
tire fquaresof the roots of the former; then as many ne¬ 
gative roots as the transformed equation contains, fo many 
impofiible roots, at lead, are in the original equation, be- 
caufe the fquare of a poflible quantity is always pofitive. 
If any feries of magnitudes be fubdituted for the un¬ 
known quantity in an equation, there can only be as many 
changes of figns in the refults, as the equation contains 
poflible roots. Let* 1 — 2ax-\-d‘-\-b 2 x*— c ■ x — d. Sec. — 
o, be an equation whofe roots are a 44— b z , a —— b z , 
c, d y Sec. whatever magnitude is fubllituted for x, the 
quantity x 2 — zax~\-a 2 -\-b 2 y which is the fum of two fquares 
x—a )*4^% is pofitive ; therefore, the changes of figns can 
only arife from the fubflitution of quantities for x in the 
reftangle .v —c .x — d. Sec. which changes are as many as 
tliere are poflible roots c, d, Sec. 
The limiting equation has at lead as many poflible roots 
as the original, wanting one. 
Let x 2 — 2ax-\-a 2 -\-b 2 y,x — c.x — d. Sec. —o be the pro- 
poled equation, the limiting equation is 
-c. Sec. 
^d. 
ALGEBRA. 
original equation, we know from the figns of the refults, 
what poflible roots it contains. For roots of the limiting- 
equation lie between the poflible roo,ts of the propofed 
equation. 
Ex. Let x 1 } + 3 —a n +'x' l +.p*d n +'—o. 
Its limit is 7 i 43 -*”‘ ! " !1 —; whofe poflible roots 
when n is an odd number are 
” 4*3 
"+ 1 
X a y 
and 
-2ax-\-d‘-\-b > X x- 
4 * 2 — 2ax-\-a 1 -\-b‘y_ x — d. Sc c. 
4 *—a+\/— b 2 X x — c.x — d. See. 
' 1 X”; which fubdituted in the original equation 
give the refults either —, 4, —, or 4, 4, 4; therefore 
it has either four poflible roots or none. When n is even, 
the poflible roots of the limiting equation are 0 and 
—L_ " + ‘ Xo; therefore the equation itfelf will have one 
” 43 ; 
poflible root, or three, according as-n +1 X a > when 
« 4 3 ! 
fubdituted for x, gives a pofitive or a negative refult. 
The roots of a quadratic equation are impoflible, if the 
fquare of the middle term be lei's than four times the pro¬ 
duct of the extremes. 
Let tfx® 4 ^* 4 f;=0 5 then x — 
-bdz yj b‘'■ —4<zc 
>— 0 , 
4 * — a — \J — b 2 x x — c.x — d. See. 
Or by adding the two lad terms together, it is 
x 2 — lax^-cd-^-b'xx — c. Sec. 1 
4-v e - — 2 ax-\-a 2 4 r b 2 y k x — d. Sec. V=o, 
4 2.x— a. x — c. x — d. Sec. J 
In which if c, d, Sec. be fucceflively fubdituted for x, the 
refults are 4> —> therefore there are poflible roots in 
this latter equation which lie between c, d. Sec. or it con¬ 
tains, at lead, as many poflible roots, wanting one, as 
the original equation. It may contain more. 
Cor. 1. Hence it follows that there are, at lead, as 
many impoflible roots in the original equation as in the 
equation of limits. There may be more j therefore, from 
the number of impoflible roots in the limiting equation we 
cannot determine exaftly the number in the original 
equation. 
Cor. 2. Hence alfo it appears, that if the poflible roots 
of the limiting equation be fubdituted fucceflively in the 
which 
X U 
expreflion becomes impoflible when l z is lefs than 4 etc. 
There are impoflible roots in an equation, whenever 
there are impoflible roots in its limiting equation. In the 
fame manner, if the next limit be taken, there are im¬ 
poflible roots in the original equation, whenever there are 
impoflible roots in this limit; and, if the limit be thus 
brought down to a quadratic, when the roots of the quad¬ 
ratic are impoflible, there are impoflible roots in the ori¬ 
ginal equation correfponding to them. On this principle 
the following rule of Sir If. Newton’s is founded, for dif. 
covering impoflible roots in any equation. 
Let the propofed equation be x n — px n 1 .... 4 
Dx”“ r *h»— Ex h ~~ r ^\-Fx n ; — Sec. — o. To obtain a 
limiting equation, which (hall be a quadratic correfpond¬ 
ing to the terms Dx n ' ,f +‘— Ex 11 r -\-Fx" ’ *, let the 
fucceeding terms be taken away, by multiplying by the 
terms of the arithmetical progrellions n, n —1, n— 2, .... 
2, 1,0; n —1, n —2, .... 2, 1, o ; n —2, n —3, .... 2, 1, 
o, &c. and, let the preceding terms be taken away, by 
multiplying by the terms of the progreflions o, 1 , 2, .. , , 
r 4 i 5 o, 1, 2, ..... r, Sec. as follows: 
x n 
— px n 
‘4 qx n ~* . -\-Dx n 
—Ex n ~ 
r 4 Fx n ~ r ~' .... —0 
”, 
n —i f 
n —2,. n — r 41, 
n—r, 
n — r — if .... 
71 —l, 
n — if 
»—3, • • • • » n—r. 
n —r— 
-1, n —r•—2, .... 
n —2, 
n— 3 , 
n —4, ..... n — r— 1, 
n —r~ 
-2, n—r—2, .... 
&c. 
&c. 
4 , 
3, 
2, 
3 » 
2> 
1, 
0, 
ri 
2. r— 1, 
r, 
r4i, 
0, 
1, ..... r—2, 
r —i. 
r, 
* 
a, . r— 3 , - 
r— 2, 
r—1» 
. Sec. 
. 2, 
3 , 
4 , 
..... 1, 
2 , 
3 , 
* 
# 
* 4«—7-41. n — r . Dx 2 
- 2 . 72 — 
-r.rEx-\-r-\-i.rF *—o 
The refpeiftive products being taken, and thofe quantities are impoflible roots in the propofed equation correfpond- 
left put which are found in every produft, we obtain a ing to them. 
limiting quadratic n- r41.11—r . Dx* 2 .n r.rEx~\-r-\-i. Write down, therefore, a feries of fractions-,--, 
rF— o, correfponding to the three terms Dx‘ '+*— / 12 
Ex n r 4 -Fx n ~~ r *; and, if two roots of this quadratic be n — 2 n —r4r n — r .... , 
‘ , . ,-- -, .... --L—,-5 divide each fraction by that 
impoflible, that is, if n — r| X r E be lefs than n —r4i • 3 r 2-41 
n-T.r-\-i.rDF, or ■ 
n—r 41 . r4i 
: E 2 - lefs than DF, there which precedes it, and place the refults 
2 n. 
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