  In this paper, we consider the elliptic relative equilibria of the restricted
$N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear
central configuration or a $(1+n)$-gon central configuration. We obtain the
symplectic reduction to the general restricted $N$-body problem. For the first
case, by analyzing the relationship between this restricted $N$-body problems
and the elliptic Lagrangian solutions, we obtain the linear stability of the
restricted $N$-body problem by the $\omega$-Maslov index. Via numerical
computations, we also obtain conditions of the stability on the mass parameters
under $N=4$ and the symmetry of the central configuration. For the second case,
there exist three positions $S_1,S_2$ and $S_3$ of the massless body (up to
rotations of angle $\frac{2\pi}{n}$). For ${m_0\over m}$ sufficiently large, we
show that the elliptic relative equilibria is linearly unstable if the
eccentricity $0\le e<e_0$ and the massless body lies at $S_1$ or $S_2$; while
the elliptic relative equilibria is linear stability if the massless body lies
at $S_3$.
