The proposal to utilize tensioned rectangular membranes as antennae in interspace communication has resulted in the focus of considerable attention on the free vibration of membranes with complicated in-plane loadings. In this paper the problem of analizing the vibration of a rectangular membrane with linear variation in tension in one direction is examined. This tension variation occurs due to gravitational forces when the membranes are tested experimentally in vibration laboratories. The Rayleigh-Ritz energy method is employed to obtain analytical results. Convergence is found to be rapid. Mode shapes are compared with those of known classical solutions when no tension variation is permitted. Generation of the eigenvalue matrix is demonstrated to be a very simple task.