Thomson devoted great attention from time to time to the science of hydrodynamics. This is perhaps the most abstruse subject in the domain of applied mathematics, and when viscosity (the frictional resistance to the relative motion of particles of the fluid) is taken into account, passes beyond the resources of mathematical science in its present state of development. But leaving viscosity entirely aside, and dealing only with so-called perfect fluids, the difficulties are often overwhelming. For a long time the only kind of fluid motion considered was, with the exception of a few simple cases, that which is called irrotational motion. This motion is characterised by the analytical peculiarity, that the velocity of an element of the fluid in any direction is the rate of variation per unit distance in that direction of a function of the coordinates (the distances which specify the position) of the particle. This condition very much simplifies the analysis; but when it does not hold we have much more serious difficulties to overcome. Then the elements of the fluid have what is generally, but quite improperly, called molecular rotation. For we know little of the molecules of a fluid; even when we deal with infinitesimal elements, in the analysis of fluid motion, we are considering the fluid in mass. But what is meant is elemental rotation, a rotation of the infinitesimal elements as they move. We have an example of such motion in the air when a ring of smoke escapes from the funnel of a locomotive or the lips of a tobacco-smoker, in the motion of part of the liquid when a cup of tea is stirred by drawing the spoon from one side to the other, or when the blade of an oar is moving through the water. In these last two cases the depressions seen in the surface are the ends of a vortex which extends between them and terminates on the surface. In all these examples what have been called vortices are formed, and hence the name vortex motion has been given to all those cases in which the condition of irrotationality is not satisfied. The first great paper on vortex motion was published by von Helmholtz in 1858, and ten years later a memoir on the same subject by Thomson was published in the _Transactions of the Royal Society of Edinburgh_. In that memoir are given very much simpler proofs of von Helmholtz's main theorems, and, moreover, some new theorems of wide application to the motion of fluids. One of these is so comprehensive that it may be said with truth to contain the whole of the dynamics of a perfect fluid. We go on to indicate the contents of the principal papers, as far as that can be done without the introduction of analysis of a difficult description.