  We present a new computation of the linear tidal interaction of a
protoplanetary core with a thin gaseous disc in which it is fully embedded. For
the first time a discussion of the orbital evolution of cores with eccentricity
(e) significantly larger than the gas-disc scale height to radius ratio (H/r)
is given. We find that the direction of orbital migration reverses for
e>1.1H/r. This occurs as a result of the orbital crossing of resonances in the
disc that do not overlap the orbit when the eccentricity is very small. Simple
expressions giving approximate fits to the eccentricity damping rate and the
orbital migration rate are presented. We go on to calculate the rate of
increase of the mean eccentricity for a system of protoplanetary cores due to
dynamical relaxation. By equating the eccentricity damping time-scale with the
dynamical relaxation time-scale we deduce that an equilibrium between
eccentricity damping and excitation through scattering is attained on a 10^3 to
10^4 yr time-scale, at 1au. The equilibrium thickness of the protoplanet
distribution is such that it is generally well confined within the gas disc. By
use of a three dimensional N-body code we simulate the evolution of a system of
protoplanetary cores, incorporating our eccentricity damping and migration
rates. Assuming that collisions lead to agglomeration, we find that the
vertical confinement of the protoplanet distribution permits cores to build up
from 0.1 to 1 earth mass in only ~10^4 yr, within 1au. The time-scale required
to achieve this is comparable to the migration time-scale. We deduce that it is
not possible to build up a massive enough core to form a gas giant planet
before orbital migration ultimately results in the preferential delivery of all
such bodies to the neighbourhood of the central star. [Abridged]
