Ax' — 2 
i i i i 
( 5 1 ^ ) 
T> x l — B x ~b i = 0 will be 
> , 5 5 |9 
abed 
impoflible. 
~ , and therefore two of them to wit - , - 
Lemma III. 
In every ./Equation x n — B x n ~ l -f- C a”*” 2 — 
5 D a 1 ” - 3 + £ #”*-4 — . &c. + e at 4 + } + c x 2 Ip 
b x -b_A—0> all whofe Roots are real, if each Term 
be multiply ’d by the Index of x in that Term, and 
each Produd be divided by x , the refulting /Equation 
71 x n ~~ l — // — i B x n ~ 2 -f- n — z C x n ~~ i — 
7 i — 3 © — 4 E x n ~~* — + 4 e # * 
4I3 d x~ c x~^ b = 0 lhall have all its Roots 
real. Thus if all the Roots of the Equation x 4 — 
B x' -b- C x* — © x + A = 0 be real, then all the 
Roots of the .Equation 4 x } — 3 Z? # * + 2 C" * — 
— 0 will alfo be real. This Lemma doth not hold 
converlly, for there are an Infinity of Cafes where all 
the Roots of the .Equation n x n ~ l — n— 1 B x n ~ 2 + 
n iCx*-* — 0 — 3 c Dx n ~*-b &c. + 3 d x 2 -±. 
2, c 'x -bb = 0 are real, at the fame Time fome or 
perhaps all the Roots of the Equation x n — B x n ~ * + 
Cx n ~~ 2 — T> a ’”~ 3 + &c. -j-dx 3 -t-cx 3 + bx-t-A= 0 
are impoflible: But whatever be the Number of impof- 
fible Roots in the Equation nx n ~ l . — n — 1 B x n ~ 2 ~b 
n — zCx n ~* — &c. + z c x +b = 0, there are at 
lead as many in the Equation x n — B x n ~ I + 
Cx n '~ 7 &c. Hb c x 2 4- b x -bA — 0. Thus all the 
Roots of the Equation 4 x 3 — 3 B x 2 + z C X 
*D = 0 may be real, and yet two or perhaps ail the 
four 
