To explain the object of the following paper, it will be best to relate the circumstance which gave rise to it. Some time ago Professor Willis requested my consideration of a certain differential equation in which he was interested, at the same time explaining its object, and the mode of obtaining it. The equation will be found in the first article of this paper, which contains the substance of what he communicated to me. It relates to some experiments which have been performed by a Royal Commission, of which Professor Willis is a member, appointed on the 27th of August, 1847, “ for the purpose of inquiring into the conditions to be observed by engineers in the application of iron in structures exposed to violent concussions and vibration. ” The object of the experiments was to examine the effect of the velocity of a train in increasing or decreasing the tendency of a girder bridge over which the train is passing to break under its weight. In order to increase the observed effect, the bridge was purposely made as slight as possible: it consisted in fact merely of a pair of cast or wrought iron bars, nine feet long, over which a carriage, variously loaded in different sets of experiments, was made to pass with different velocities.