A Proof by Elementary Methods. 221 
B. If the given equation has some of its coefficients imaginary 
{ao-\-ia'^)x'^-\-{a^-{-ia\)x''~'-\- ... = 0 (1) 
. • . {a^x''-^a^x''-' + ...y + {a'X+CL[x''~' + ...)^ = 0 (2) 
an equation of degree 2n with real coefficients. We have proved that this 
has n pairs of factors of the form a? — (a + t/3). 
Since (2) is the product of (1) and its conjugate {a^ - ia'o)x'^ . . . , 
(1) or its conjugate must hold when x = a-{-i(3 = r{Gos 0 + t sin 6). 
And if (ao+t^io)r'*(cos nO-i-i sin 7td)... =0, 
— t(Xo)r"(cos nd — L sinnd)... =0 follows. 
Therefore we may say that of the 2n roots of (2) those of form a + i/B are 
roots of (1) and those of form {a — t(3) are roots of the conjugate. 
