  Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of
Brownian motion. Our main result is to show that for each fixed $t$ $${\int
(L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2}
\stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}\eta$$
as $h\to 0$, where $\eta$ is a normal random variable with mean zero and
variance one that is independent of $L^{x}_{t}$. This generalizes our previous
result for the second moment. We also explain why our approach will not work
for higher moments
