Periodic and asymptotic orbits are obtained for the motion of an infinitesimal body subject to the Newtonian attraction of four finite bodies. The finite bodies are equal in mass and remain relatively fixed at the vertices of a square while they revolve about their common centre of gravity with uniform angular velocity. The infinitesimal body moves in the vicinity of the 13 libration points of the finite bodies. Asymptotic orbits of two dimensions and periodic orbits of two and of three dimensions are obtained.