Page:Great Neapolitan Earthquake of 1857.djvu/192

6th. In the case of hollow right cylinder, fractured from its base.

Here

$k^{2}={\frac {5\beta ^{2}+\beta '^{2}}{16}}$,

and

${\begin{cases}V={\frac {g}{8}}g\times {\frac {(5\beta ^{2}+\beta '^{2})L}{\beta \alpha ^{2}}}\\or\\V=4.02\times {\frac {(5D^{2}+D'^{2})L}{D\alpha ^{2}}}\end{cases}}$

(XXVIII.)

In the 5th and 6th cases, and generally in cases of solid columns or minarets, &c., or hollow prismatic or cylindriical towers, the fracture at the base, is never perfectly, and all through, horizontal. When the breadth or diameter is small, however, in proportion to the height, the irregularity of the fracture is not great, and the slope from horizontal also small; and no serious error is introduced by considering the plane of fracture as horizontal.

Secondly. The wave-path subnormal (force emergent and oblique to the horizon).

When the wave-path is subnormal, in any prismatic structure, the first and second semiphases of the wave, act upon it as already explained; the former to produce inceptive overturn upon the axis (arris of the base) $\mathrm {A}$, and the latter upon the axis $\mathrm {B}$.

Both cases are included in Eq. XXI.

$V=g\times {\frac {\mathrm {L} }{\alpha }}\times {\frac {k^{2}}{f\beta }}$

Writing for $f$ = the perpendicular height of the centre of