An algebraic result is obtained which enables the authors to establish the equivalence of \(A\)-stability and \(G\)-stability for one-leg methods. They obtain an alternative proof of the second Dahlquist barrier. In the following two of their results are stated: Theorem: If the polynomials \(\rho\) and \(\sigma\) are relatively prime, then \(A\)-stability and \(G\)-stability are equivalent. Hence each of \(\rho\) and \(\sigma\) is of degree \(r\geq 1\) and has real coefficients. They satisfy the property \(| z| \geq 1\Rightarrow \text{Re }\rho(z)\sigma (\bar z)\geq 0.\) Theorem: The order of accuracy of a linear multistep \(A\)-stable method is limited to two. For the methods of order two, the best possible error constant is 1/12.