The rapid progress made in mechanical engineering during the present century has brought many troublesome problems in its wake. One of the most serious of these troubles arises from resonant and such like vibration. It has, therefore, become increasingly necessary for a designer to take account of vibration. A powerful instrument for this purpose was provided by the late Lord Rayleigh, whose method for finding an approximation to the fundamental frequency of vibration of an elastic system is fairly well known. In this paper an attempt is made inter alia to explain Rayleigh's Principle and to clarify, so far as the author is able to do so, the basis of the principle. Other devices for ascertaining approximate frequencies are also given and an appendix has been added on forced vibration experiments.