( ?<? ) 
/ 
II. A Method for determining the Number of im- 
pojfible <I(oots in adfeSled / Equations . (By Mr . 
George Campbell. 
Lemma I. 
I N every adfeded quadratick ./Equation a x 2 — - B x -J- 
A—O, whofe Roots are real, a fourth Part of the 
Square of the Coefficient of the fecond Term is greater 
than the Redangle under the Coefficient of the firft 
Term and the absolute Number or % B 1 \ ax A; and 
viceverfa if % B 2 \axA , the Roots of the /Equation 
ax 2 — B x -{-A= o, will be real. But if * B 2 \ a xA, 
the Roots will be impoflible. This is evident from the 
Roots of the /Equation being 
5 B-\- y/'iB 2 — axA 
a 1 
\B — V iB 2 — axA 
a 
Lemma II. 
Whatever be the Number of impoflible Roots in the 
-/Equation x* — B x”~ l + C x n ~ 2 _ T) x n ~ i -f ©f. 
±^x i + cx 2 +bxTA=o 9 there are juft as many in 
the /Equation Ax* — b x n ~' x -f- c x n ~ 2 — dx " — * -f“ 
f§c. ± 2) x 3 T Cx 2 ± B x T 1 = o. For the Roots 
of the laft ./Equation are the Reciprocals of thofe of 
the firft, as is evident from corqmon Algebra. Let the 
Roots of the biquadratick /Equation x * — B x * -f. 
Cx' — T) x be a , b , c y d y whereof let c y d 
be impoflible, then the Roots of the /Equation 
Z z z z Ax* — * 
