  The global rotational degrees of freedom in the Schr\"{o}dinger equation for
an $N$-body system are completely separated from the internal ones. After
removing the motion of center of mass, we find a complete set of $(2\ell+1)$
independent base functions with the angular momentum $\ell$. These are
homogeneous polynomials in the components of the coordinate vectors and the
solutions of the Laplace equation, where the Euler angles do not appear
explicitly. Any function with given angular momentum and given parity in the
system can be expanded with respect to the base functions, where the
coefficients are the functions of the internal variables. With the right choice
of the base functions and the internal variables, we explicitly establish the
equations for those functions. Only (3N-6) internal variables are involved both
in the functions and in the equations. The permutation symmetry of the wave
functions for identical particles is discussed.
