  High-precision astrometric and radial-velocity observations require accurate
modelling of stellar motions in order to extrapolate measurements over long
time intervals, and to detect deviations from uniform motion caused for example
by unseen companions. We aim to explore the simplest possible kinematic model
of stellar motions, namely that of uniform rectilinear motion relative to the
Solar System Barycentre, in terms of observable quantities including error
propagation. The apparent path equation for uniform rectilinear motion is
solved analytically in a classical (special-relativistic) framework, leading to
rigorous expressions which relate the (apparent) astrometric parameters and
radial velocity to the (true) kinematic parameters of the star in the
barycentric reference system. We present rigorous and explicit formulae for the
transformation of stellar positions, parallaxes, proper motions, and radial
velocities from one epoch to another, assuming uniform rectilinear motion and
taking into account light-time effects. The Jacobian matrix of the
transformation is also given, allowing accurate and reversible propagation of
errors over arbitrary time intervals. The light-time effects are generally very
small but exceeds 0.1 mas or 0.1 m/s over 100 yr for at least 33 stars in the
Hipparcos Catalogue. For high-velocity stars within a few tens of pc from the
Sun light-time effects are generally more important than the effects of the
curvature of their orbits in the Galactic potential.
