The second-order tensor virial equation (TVE) method applied to rotating polytropes and degenerated dwarfs has indicated that the Kelvin bar modes becomes secularly unstable when T/Vertical BarWVertical Bar> or approx. =0.14, where T is the rotational kinetic energy and W is the total gravitational energy, for a wide variety of rotation laws and equations of state. However, recent advances in Lagrangian perturbation techniques have shwon that the TVE method provides neither a necessary nor a sufficient condition for secular stability in differentially rotating stars, because the trial eigenfunction has recently been suggested which satisfies the Kelvin circulation theorem. When used a Lagrangian variational formulation this trial eigenfunction yields a sufficient condition for secular instability. We find that the application of this sufficient condition for secular instability of barlike modes gives substantially the same result as the TVE method: differentially rotating polytropes and degenerate dwarfs are secularly unstable for T/Vertical BarWVertical Bar> or approx. =0.14 for a wide range of compressibilities and rotation laws. The difference between our improved secular stability limits and the TVE limits ranges from 1 to 7% and increases with the degree of central concentration of mass and angular momentum of the equilibrium model.