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 English
mathematicians. The early inquiries of Poisson and
Cauchy were directed to the investigation of waves produced
by disturbing causes acting arbitrarily on a small portion of the fluid. The velocity of the long wave was given
approximately by Lagrange in 1786 in case of a channel of
rectangular cross-section, by Green in 1839 for a channel of
triangular section, and by P. Kelland for a channel of any
uniform section. Sir George B. Airy, in his treatise on Tides
and Waves, discarded mere approximations, and gave the exact equation on which the theory of the long wave in a channel of uniform rectangular section depends. But he gave no general solutions. J. McCowan of University College at Dundee
discusses this topic more fully, and arrives at exact and complete solutions for certain cases. The most important application of the theory of the long wave is to the explanation of tidal phenomena in rivers and estuaries.
The mathematical treatment of solitary waves was first taken up by S. Earnshaw in 1845, then by Stokes; but the first