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466 Rh from the last result we have p_n=\pm 1, and thus we have two classes of reciprocal equations.

(i.) If p_n=1 then

Pi=Pr-v P2= Pn—-2» Ps = Pn-a °°°5

that is, the coefficients of terms equidistant from the beginning and end are equal.

(ii.) If p_n =- 1, then

Pi =— Pra-v P2 = — Pa-» P3 = — Pu-3 °**5

hence if the equation is of 2m dimensions p_m=—pm or p_m=0. In this case the coefficients of terms equidistant from the beginning and end are equal in magnitude and opposite in sign, and if the equation is of an even degree the middle term is wanting.

588. Standard Form of Reciprocal Equations. Suppose that f(x) = 0 is a reciprocal equation.

If f(x) = 0 is of the first class and of an odd degree it has a root —1; so that f(x) is divisible by x+1. If \phi(x) is the quotient, then \phi(x) = 0 is a reciprocal equation of the first class and of an even degree.

If f(x) = 0 is of the second class and of an odd degree, it has a root +1; in this case f(x) is divisible by x — 1, and as before \phi(x) =0 is the reciprocal equation of the first class and of an even degree.

If f(x) =0 is of the second class and of an even degree, it has a root +1 and a root —1; in this case f(x) is divisible by x^2—1, and as before \phi(x)=0 is a reciprocal equation of the first class of an even degree.

Hence any reciprocal equation is of an even degree with its last term positive, or can be reduced to this form; which may therefore be considered as the standard form of reciprocal equations.

589. A reciprocal equation of the standard form can be reduced to an equation of half its dimensions. Let the equation be