This paper concerns the free vibration of linear systems which are subject to damping. The general theory for a system having n degrees of freedom is recalled and illustrated by means of a simple example. Using an analytical method, the approximate theory for small damping is derived, showing the nature of the entrainment of the principal modes; the results are found to confirm those of Southwell (who uses a physical argument) and they are illustrated by an example. The approximate theory is shown to need modification when there are two nearly-equal natural frequencies and the appropriate corrections are found. The analysis is made somewhat complicated by the existence of two criteria of “ smallness ” instead of one; these are the difference between the close natural frequencies and the amount of the damping.