  We apply a recently developed perturbative formalism which describes the
evolution under their self-gravity of particles displaced from a perfect
lattice to quantify precisely, up to shell crossing, the effects of
discreteness in dissipationless cosmological N-body simulations. We give simple
expressions, explicitly dependent on the particle density, for the evolution of
power in each mode as a function of red-shift. For typical starting red-shifts
the effect of finite particle number is to {\it slow down} slightly the growth
of power compared to that in the fluid limit (e.g. by about ten percent at half
the Nyquist frequency), and to induce also dispersion in the growth as a
function of direction at a comparable level. In the limit that the initial
red-shift tends to infinity, at fixed particle density, the evolution in fact
diverges from that in the fluid limit (described by the Zeldovich
approximation). Contrary to widely held belief, this means that a simulation
started at a red-shift much higher than the red-shift of shell crossing
actually gives a worse, rather than a better, result. We also study how these
effects are modified when there is a small-scale regularization of the
gravitational force. We show that such a smoothing may reduce the anisotropy of
the discreteness effects, but it then {\it increases} their average effect.
This behaviour illustrates the fact that the discreteness effects described
here are distinct from those usually considered in this context, due to
two-body collisions. Indeed the characteristic time for divergence from the
collisionless limit is proportional to $N^{2/3}$, rather than $N/ \log N$ in
the latter case.
