We calculate the flux of non-adsorbed, collisionless molecules which follow linear trajectories and obey the cosine law, through spheres of radius R, linked by windows of radius r, formed by removing two spherical caps from each sphere and arranged in a linear sequence. Two methods of calculation have been used; in the first, the molecular path tracing (MPT) method, no assumptions, other than those mentioned above, were introduced. The expression for the flux is obtained in the form of an integral which has been evaluated numerically. In the second method, a radiating disc theorem was employed which does not take account of flux variation over a cross section, but which leads to a very simple analytical result which shows similar behaviour to that predicted by the MPT method. A simple empirical correction has been calculated to provide an accurate analytical equation. The second method can also be readily extended to a network of linked spheres and, augmented by the empirical correction, leads to a model for the description of the flow of a very dilute, weakly adsorbed gas in a system of this kind. The relationship between geometrical structure and flow is discussed with reference to another similar model and to zeolitic networks.