  We present simulations of initially stable isothermal clouds exposed to
ionising radiation from a discrete external source, and identify the conditions
that lead to radiatively driven implosion and star formation. We use the
Smoothed Particle Hydrodynamics code SEREN and an HEALPix-based
photo-ionisation algorithm to simulate the propagation of the ionising
radiation and the resulting dynamical evolution of the cloud. We find that the
incident ionising flux, $\Phi_{_{\rm LyC}}$, is the critical parameter
determining the cloud evolution. At moderate fluxes, a large fraction of the
cloud mass is converted into stars. As the flux is increased, the fraction of
the cloud mass that is converted into stars and the mean masses of the
individual stars both decrease. Very high fluxes simply disperse the cloud.
Newly-formed stars tend to be concentrated along the central axis of the cloud
(i.e. the axis pointing in the direction of the incident flux). For given cloud
parameters, the time, $t_{_\star}$, at which star formation starts is
proportional to $\Phi_{_{\rm LyC}}^{-1/3}$. The pattern of star formation found
in the simulations is similar to that observed in bright-rimmed clouds.
