Thomson's great paper on vortex-motion was read before the Royal Society of Edinburgh in 1867, and was recast and augmented in the following year. It will be possible to give here only a sketch of its scope and main results. The fluid is supposed contained in a closed fixed vessel which is either simply or multiply continuous (see p. 156), and may contain immersed in it simply or multiply continuous solids. When these solids exist their surfaces are part of the boundary of the liquid; they are surrounded by the liquid unless they are anywhere in contact with the containing vessel, and their density is supposed to be the same as that of the liquid. They may be acted on by forces from without, and they act on the liquid with pressure-forces, and either directly or through the liquid on one another. The first result obtained is fairly obvious. The centre of mass of the whole system must remain at rest whatever external forces act on the solids, since the density is the same everywhere within the vessel, and the vessel is fixed; that is to say, there is no momentum of the contents of the vessel in any direction. For whatever motion of the solids is set up by the external forces, must be accompanied by a motion of the liquid, equal and opposite in the sense here indicated. After a discussion of what he calls the impulse of the motion, which is the system of impulsive forces on the movable solids which would generate the motion from rest, Thomson proceeds to prove the important proposition that the rotational motion of every portion of the liquid mass, if it is zero at any one instant for every portion of the mass, remains always zero. This is done by considering the angular momentum of any small spherical portion of the liquid relatively to an axis through the centre of the sphere, and proving that in order that it may vanish, for every axis, the component velocities of the fluid at the centre must be derivable from a velocity-potential. The angular momentum of a particle about an axis is the product of the component of the particle's momentum, at right angles to the plane through the particle and the axis, by the distance of the particle from the axis. The sum of all such products for the particles making up the body (when proper account is taken of the signs according to the direction of turning round the axis) is the angular momentum. The proof of this result adopted is due to Stokes. The angular velocities of an element of fluid at a point x, y, z, about the axes of x, y, z are shown to be ½(∂w⧸∂y － ∂v⧸∂z), etc.