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1. This paper treats of the propagation of vibrations over the surface of a “semi-infinite” isotropic elastic solid, i.e., a solid bounded only by a plane. For purposes of description this plane may be conceived as horizontal, and the solid as lying below it, although gravity is not specially taken into account.

The vibrations are supposed due to an arbitrary application of force at a point. In the problem most fully discussed this force consists of an impulse applied vertically to the surface; but some other cases, including that of an internal source of disturbance, are also (more briefly) considered. Owing to the complexity of the problem, it has been thought best to concentrate attention on the vibrations as they manifest themselves at the free surface. The modifications which the latter introduces into the character of the waves propagated into the interior of the solid are accordingly not examined minutely.

The investigation may perhaps claim some interest on theoretical grounds, and also in relation to the phenomena of earthquakes. Writers on seismology have naturally endeavoured from time to time to interpret the phenomena, at all events in their broader features, by the light of elastic theory. Most of these attempts have been based on the laws of wave-propagation in an unlimited medium, as developed by Green and Stokes; but Lord Rayleigh’S discovery of a special type of surface-waves has made it evident that the influence of the free surface in modifying the character of the vibrations is more definite and

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