This work concerns an analytic autonomous Hamiltonian system of differential equations with two degrees of freedom which admits and unstable equilibrium point. More specifically, the eigenvalues corresponding to the equilibrium point include one real pair and one imaginary pair and so we know (by a theorem of A. Liapunov) that passing through the equilibrium point there is an invariant two dimensional manifold on which all solutions are periodic. Furthermore, the level surfaces of the Hamiltonian function are invariant three dimensional manifolds which, for appropriate values of that function, contain exactly one of these periodic solutions. Our aim is to study the orbits on such a level surface. The nature of the work is most easily described in terms of the following example: consider two bowls connected by a trough so that, when inverted they look like two mountains with a pass between. The differential equations are taken to be those describing the motion of a point mass sliding without friction on this “double bowl”. The Hamiltonian function is the sum of the potential energy, i.e. the height in the bowl, and the kinetic energy, also obtained as usual. Since the kinetic energy is positive, fixing the value of the Hamiltonian function corresponds to limiting the height to which the mass can go. Our problem concerns the case where the mass can go high enough to get from one bowl to the other with just a little room to spare. Having fixed on an appropriate level surface (of the Hamiltonian function) we first study the behavior of orbits near the equilibrium point, which in the example above, corresponds to the saddle point in the trough connecting