  We quantitatively examine the significance of star formation triggered in the
swept-up shell around an expanding HII region. If the swept-up molecular gas is
sufficiently massive, new OB stars massive enough to repeat the triggering
process will form in the shell. We determine the lower limit (M_thr) for the
mass of the star that sweeps up the molecular gas, where at least one new star
with mass M_* > M_thr forms after the shell fragmentation. To calculate the
threshold stellar mass, M_thr, we examine how massive molecular shells can form
around various central stars, by performing detailed numerical radiation
hydrodynamics calculations. The mass of the photodissociated gas is generally
larger than the mass of the photoionized gas. However, the swept-up molecular
mass exceeds the photodissociated mass with a higher-mass star of M_* > 20
Msun. The accumulated molecular mass generally increases with the stellar mass,
and amounts to 10^{4-5} Msun for M_* > 20 Msun with an ambient density of n
\sim 100/cc. The threshold stellar mass is M_thr \sim 18 Msun with the
star-formation efficiency of \epsilon \sim 0.1 and n \sim 100/cc. We examine
the generality of this mode of run-away triggering for different sets of
parameters, and found that M_thr \sim 15-20 Msun in various situations. If the
ambient density is too high or the star-formation efficiency is too low, the
triggering is not run-away, but a single event.
