Roots of Equations . 377 
coefficients are possible, and whose roots are therefore of the 
form specified in the proposition. 
In the same manner the proposition may be proved, when 
y 6 &c. is an odd number ; and thus it appears that it 
is true in all equations. * 
Cor. 1. If v\ or y, be positive, the roots of the quadratic 
factor x x — 2 zx -j- — v*= o, and therefore two roots of the 
proposed equation are possible. If y = o, two roots are equal ; 
and if y be negative, two roots are impossible. 
Cor. 2. If a possible value of % be determined, and substi- 
tuted in b, c, d, &c. the original equation will have as many 
pairs of possible roots as there are changes of signs in the equa- 
tion y rn -f by m ~ l - f- c y m ~~ 2 -f- &c . =0; and as many pairs of 
impossible roots as there are continuations of the same sign. 
# See Dr. Waring’s Med, Alg. 
3c 
MDCCXCVIII. 
