  In this paper, we consider the elliptic relative equilibria of the restricted
$4$-body problems, where the three primaries form an Euler collinear
configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic
reduction to the general restricted $N$-body problem. By analyzing the
relationship between this restricted $4$-body problems and the elliptic
Lagrangian solutions, we obtain the linear stability of the restricted $4$-body
problem by the $\omega$-Maslov index. Via numerical computations, we also
obtain conditions of the stability on the mass parameters for the symmetric
cases.
