  A new formulation is presented for a variational calculation of $N$-body
systems on a correlated Gaussian basis with arbitrary angular momenta. The
rotational motion of the system is described with a single spherical harmonic
of the total angular momentum $L$, and thereby needs no explicit coupling of
partial waves between particles. A simple generating function for the
correlated Gaussian is exploited to derive the matrix elements. The formulation
is applied to various Coulomb three-body systems such as $e^-e^-e^+, tt\mu,
td\mu$, and $\alpha e^-e^-$ up to $L=4$ in order to show its usefulness and
versatility. A stochastic selection of the basis functions gives good results
for various angular momentum states.
