  We revisit secular stability against quasi-radial collapse for rigidly
rotating supermassive stars (SMSs) in general relativity. We suppose that the
SMSs are in a nuclear-burning phase and can be modeled by polytropic equations
of state with the polytropic index $n_p$ slightly smaller than $3$. The
stability is determined in terms of the turning-point method. We find a fitting
formula of the stability condition for the plausible range of $n_p$ ($2.95 \alt
n_p \alt 3$) for SMSs. This condition reconfirms that, while non-rotating SMSs
with mass $\sim 10^5M_\odot$--$10^6M_\odot$ may undergo a
general-relativistically induced quasi-radial collapse, rigidly rotating SMSs
with a ratio of rotational to gravitational potential energy ($\beta$) of $\sim
10^{-2}$ are likely to be stable against collapse unless they are able to
accrete $\sim 5$ times more mass during the (relatively brief) hydrogen-burning
phase of their evolution. We discuss implications of our results.
