a double sign, spoken of as an ambiguity, being placed wherever there is a doubt as to whether the sign of a term is positive or negative.
Examining the product we see that
(i.) An ambiguity replaces each continuation of sign in the original multinomial;
(ii.) The signs before and after an ambiguity or set of ambiguities are unlike;
(iii.) A change of sign is introduced at the end.
Let us take the most unfavorable case and suppose that all the ambiguities are replaced by continuations; from (ii.) we see that the number of changes of sign will be the same whether we take the upper or the lower signs; let us take the upper; thus the number of changes of sign cannot be less than in
ian ae oD
and this series of signs is the same as in the original multinomial with an additional change of sign at the end.
If then we suppose the factors corresponding to the negative and imaginary roots to be already multiplied together, each factor x-a corresponding to a positive root introduces at least one change of sign; therefore no equation can have more positive roots than it has changes of sign.
To prove the second part of Descartes’ Rule, let us suppose the equation complete and substitute —y for x; then the permanences of sign in the original equation become variations of sign in the transformed equation. Now the transformed equation cannot have more positive roots than it has variations of sign, hence the original equation cannot have more negative roots than it has permanences of sign.
Whether the equation f(x)=0 be complete or incomplete its roots are equal to those of f(-x) but opposite to them in sign; therefore the negative roots of f(x)=0 are the positive roots of f(-x)=0; but the number of these positive roots cannot exceed the number of variations of sign in f(-x); that is, the number of negative roots of f(x)=0 cannot exceed the number of variations of sign in f(-x).
