Page:Great Neapolitan Earthquake of 1857 Vol 2.djvu/382

Rh for the material, we should be enabled, to calculate, the volume and amplitude of the wave, upon various hypotheses which might be framed, as to the mode of its origination. For, supposing the compression to originate at a single point $$o$$, the column of least resistance of the surrounding medium, is the vertical line $$o\ d$$, and if the pressure upon the unit of surface, at the focal point, be unknown, its greatest limit for the production of a wave, is that, when the elasticity at the base, balances the whole weight of this column, for any increase of pressure beyond that, will lift it bodily. If the pressure at $$o$$, be known or assumed, then, if $$\mathrm$$ = the unit of length, of a compressed prism of the material, $$l$$ = the compression due to the pressure, on the length unit; $$\frac $$ measures the compressing force, and $$o\ t$$, is the total compression from $$o$$, of the entire column $$o\ d$$, or $$l\, :\, o\, t\, :\,:\, \mathrm\, :\, o\, d$$.

And as the compression is supposed equal all round, from the initial point $$o$$—

the whole compressed volume, which is equal to the volume of the wave at a given moment near the instant of starting.

If we suppose the volume of the wave constant, up to the time it reaches the surface at $$d$$, or a point at some small distance as $$s$$; then, the volume of the spherical shell, whose exterior radius is $$o\ d$$, and interior $$o\ c$$, must equal that of the wave, or

whence, $$c\ d$$ = the wave amplitude on reaching the surface may be obtained; or if this be observed, the depth of the focus ascertained by the methods we have employed, and