  The naive use of higher order perturbation theory leads the left--hand cut
integrals in $\pi\pi$ dispersion relations~\cite{hjy,Xiao01} divergent. This
problem is discussed and solved. Also we point out that the Adler zero
condition imposes three constraints on the dispersion relations. The $\sigma $
pole position is determined using the improved method, $M_\sigma= 483\pm 13
{\rm MeV}$,$\Gamma_\sigma=
  705 \pm 50{\rm MeV}$. The scattering length parameter is found to be in
excellent agreement with the experimental result.
