Abstract: We first reduce the linearized Hamiltonian system near the Euler-Moulton solutions of the collinear n-body problem to (n -1) independent Hamiltonian systems, in which the one is the linearized system of the Kepler 2-body problem at Kepler orbits, and the other (n - 2) are the essential part of the linearized Hamiltonian system of some collinear 3-body with di erent mass parameters. Then using the Maslov-type !-index theory of symplectic paths and the theory of linear operators we compute the !-indices, and hence obtain certain properties of linear stability of the Euler elliptic solutions of the n-body problem. As an example, we carry out the detailed derivation of the stability properties for an Euler-Moulton solutions of the 4-body problem with two small masses in the middle.
