  Let $\Gamma$ be a lattice of a semisimple Lie group $L$. Suppose that one
parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on
$L/\Gamma$ with respect to the probability Haar measure $\mu$. For certain
proper subgroup $U$ of the unstable horospherical subgroup of $\{g_t\}$ we show
that given $x\in L/\Gamma$ for almost every $u\in U$ the trajectory $\{g_tux:
0\le t\le T\}$ is uniformly distributed with respect to $\mu$ as $T\to \infty$.
