Evolution of short-period cometary orbits under the influence of close and moderately close approaches to Jupiter is studied. We have restricted the discussion to orbits in Jupiter's orbital plane and have neglected distant perturbations by Jupiter. The general evolution is a diffusion towards orbits of larger semimajor axes and a loss of comets, mainly due to ejection along hyperbolic orbits. We have found that the dynamical evolution of the orbits severely alters the assumed initial distribution of orbits during a time of some hundred years. We also tried to obtain an idea of the importance of close approaches as compared to the effect of moderate approaches to Jupiter for the evolution of the orbits. This was done by calculating the evolution of a group of comets twice, first considering the effect of perturbations up to a distance of 1.0 AU from Jupiter and then to a distance of 2.0 AU. The statistical results of the two calculations are in general agreement, indicating that close approaches to Jupiter mainly determine the evolution of the short-period comets.