A linear reciprocal system with periodic coefficients is stable if the system monodromy matrix has simple structure and eigenvalues all of modulus unity. Under the assumption that the former condition is true, it is proved in this paper that this criterion is equivalent to the condition that the roots, α, of an algebraic equation all lie in the interval −1 ⩽ α ⩽ 1, and an explicit scheme is presented for the derivation of this algebraic equation in terms of the coefficients of the characteristic equation of the system monodromy matrix. The modified criterion provides considerable computational advantage over the usual form of the criterion.