We measure the scaling properties of the probability distribution of the smoothed density field in $N$-body simulations of expanding universes with scale-free initial power-spectra, with particular attention to the predictions of the stable clustering hypothesis. We concentrate our analysis on the ratios $S_Q(\ell)\equiv {\bar ��}_Q/{\bar ��}_2^{Q-1}$, $Q \leq 5$, where ${\bar ��}_Q$ is the averaged $Q$-body correlation function over a cell of radius $\ell$. The behavior of the higher order correlations is studied through that of the void probability distribution function. As functions of ${\bar ��}_2$, the quantities $S_Q$, $3 \leq Q \leq 5$, exhibit two plateaus separated by a smooth transition around ${\bar ��}_2 \sim 1$. In the weakly nonlinear regime, ${\bar ��}_2 \la 1$, the results are in reasonable agreement with the predictions of perturbation theory. In the nonlinear regime, ${\bar ��}_2 > 1$, the function $S_Q({\bar ��}_2)$ is larger than in the weakly nonlinear regime, and increasingly so with $-n$. It is well-fitted by the expression $S_Q= ({\bar ��}_2/100)^{0.045(Q-2)}\ {\widetilde S}_Q$ for all $n$. This weak dependence on scale proves {\em a small, but significant departure from the stable clustering predictions} at least for $n=0$ and $n=+1$. The analysis of $P_0$ confirms that the expected scale-invariance of the functions $S_Q$ is not exactly attained in the part of the nonlinear regime we probe, except possibly for $n=-2$ and marginally for $n=-1$. In these two cases, our measurements are not accurate enough to be discriminant.

31 pages, postscript file, figure 1 missing. Postscript file including figure 1 available at ftp://ftp-astro-theory.fnal.gov:/pub/Publications/Pub-95-256-A