  We investigate, using 3D hydrodynamic simulations, the fragmentation of
pressure-confined, vertically stratified, self-gravitating gaseous layers. The
confining pressure is either thermal pressure acting on both surfaces, or
thermal pressure acting on one surface and ram-pressure on the other. In the
linear regime of fragmentation, the dispersion relation we obtain agrees well
with that derived by Elmegreen & Elmegreen (1978), and consequently deviates
from the dispersion relations based on the thin shell approximation (Vishniac
1983) or pressure assisted gravitational instability (W\"unsch et al. 2010). In
the non-linear regime, the relative importance of the confining pressure to the
self-gravity is a crucial parameter controlling the qualitative course of
fragmentation. When confinement of the layer is dominated by external pressure,
self- gravitating condensations are delivered by a two-stage process: first the
layer fragments into gravitationally bound but stable clumps, and then these
clumps coalesce until they assemble enough mass to collapse. In contrast, when
external pressure makes a small contribution to confinement of the layer, the
layer fragments monolithically into gravitationally unstable clumps and there
is no coalescence. This dichotomy persists whether the external pressure is
thermal or ram. We apply these results to fragments forming in a shell swept up
by an expanding H II region, and find that, unless the swept up gas is quite
hot or the surrounding medium has low density, the fragments have low-mass ( ~<
3 M_Sun ), and therefore they are unlikely to spawn stars that are sufficiently
massive to promote sequential self-propagating star formation.
