( 5*8 ) 
always exceed LxN— KxO+IxT— HxQ+ (Sc. 
h Y — .. — which 
n x - - l yJllZLl x &c. x U m ^ 1 
2 3 m 
is always a pofitive Number, when the Roots a , 
^ are real Numbers pofitive or negative. 
L>ct the Equation be of the feventh Degree or 
x? — Bx 6 q- Cx'- c Dx + + Ex i —Fx 2 -£Gx— 
A == o, whofe Roots let be a , b , , r, d, e , ^ g, in 
which Cafe » = 7 . Let A/ — E = ab c d -j- a b c e 
^ ^ c f a b c g -f- b c d e + &c. then m z=z 4 . 
£ = — N = — E, K =z C, O = G, I=—B, 
5P = — ^7, ^ = ab c d—abc e | 2 -J- abc d — ^ £ c /| 2 -}- 
abed — a b c g\ 2 -f" Therefore — ~ x 
1 
X E 2 or 
*7 
E 2 exceeds T> x 
35 * 
/r_CxG + ^x^ by 
Kabcd—abceV -1 x ^ bed — a b cfU 4, 
70 '70 
&e. 
From this Proposition, is deduced the following 
Rule for determining the Number of impoffible Roots 
in any given ./Equation. From each of the Unche 
of the middle Terms of that Power of a Binomial, 
1 whofe 
