  A polarization analysis of the fine-structure intervals for the $n = 17$
Rydberg states of Mg and the $n = 29$ states of Si$^{2+}$ is performed. The
coefficients of all terms in the polarization expansion up to $r^{-8}$ were
computed using a semi-empirical single electron analysis combined with the
relativistic all-order single-double method (MBPT-SD) which includes all
single-double excitations from the Dirac-Fock wave functions to all orders of
perturbation theory. The revised analysis yields dipole polarizabilities of
$\alpha_1 = 35.04(3)$ a.u. for Mg$^+$ and $\alpha_1 = 7.433(25)$ a.u. for
Si$^{3+}$, values only marginally larger than those obtained in a previous
analysis (E. L. Snow and S. R. Lundeen (2007) Phys. Rev. A {\bf 75} 062512,
{\em ibid} (2008) {\bf 77} 052501). The polarizabilities are used to make
estimates of the multiplet strength for the resonant transition for both ions.
The revised analysis did see significant changes in the slopes of the
polarization plots. The dipole polarizabilities from the MBPT-SD calculation,
namely 35.05(12) a.u. and 7.419(16) a.u., are within 0.3% of the revised
experimental values.
