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 of electrical images, whereupon the theory was extended by Stokes, Hicks, and Lewis. In 1849, Thomson gave the maximum and minimum theorem peculiar to hydrodynamics, which was afterwards extended to dynamical problems in general.

A new epoch in the progress of hydrodynamics was created, in 1856, by Helmholtz, who worked out remarkable properties of rotational motion in a homogeneous, incompressible fluid, devoid of viscosity. He showed that the vortex filaments in such a medium may possess any number of knottings and twistings, but are either endless or the ends are in the free surface of the medium; they are indivisible. These results suggested to Sir William Thomson the possibility of founding on them a new form of the atomic theory, according to which every atom is a vortex ring in a non-frictional ether, and as such must be absolutely permanent in substance and duration. The vortex-atom theory is discussed by J. J. Thomson of Cambridge (born 1856) in his classical treatise on the Motion of Vortex Rings, to which the Adams Prize was awarded in 1882. Papers on vortex motion have been published also by Horace Lamb, Thomas Craig, Henry A. Rowland, and Charles Chree.

The subject of jets was investigated by Helmholtz, Kirchhoff. Plateau, and Rayleigh; the motion of fluids in a fluid by Stokes, Sir W. Thomson, Köpcke, Greenhill, and Lamb; the theory of viscous fluids by Navier, Poisson, Saint-Venant, Stokes, O. E. Meyer, Stefano, Maxwell, Lipschitz, Craig, Helmholtz, and A. B. Basset. Viscous fluids present great difficulties, because the equations of motion have not the same degree of certainty as in perfect fluids, on account of a deficient theory of friction, and of the difficulty of connecting oblique pressures on a small area with the differentials of the velocities.

Waves in liquids have been a favourite subject with