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 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. McGowan 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 sound approximate theory was given by J. Boussinesq in 1871, who obtained an equation for their form, and a value for the velocity in agreement with experiment. Other methods of approximation were given by Rayleigh and J. McCowan. In connection with deep-water waves, Osborne Reynolds gave in 1877 the dynamical explanation for the fact that a group of such waves advances with only half the rapidity of the individual waves.

The solution of the problem of the general motion of an ellipsoid in a fluid is due to the successive labours of Green (1833), Clebsch (1856), and Bjerknes (1873). The free motion of a solid in a liquid has been investigated by W. Thomson, Kirchhoff, and Horace Lamb. By these labours, the motion of a single solid in a fluid has come to be pretty well understood, but the case of two solids in a fluid is not