Hypotheses of uniform diagonal dominance are stated for G(iω) and G(iω)−l, where G(s) = Q(s)P(a)−1B is the transfer matrix of the non-linear differential equation P(D)x+ BFQ(D)x = 0, in which F is a diagonal matrix. When G(iω)−1satisfies these hypotheses, sufficient conditions for stability and instability are obtained by applying the circle criterion to the loci of the diagonal elements of G(iω)−1and restricting the corresponding diagonal elements of F to lie within concentric circles having reduced radii. A similar result is also proved for the loci of the diagonal elements of G(iω).