Page:Great Neapolitan Earthquake of 1857.djvu/193

gravity above the base of fracture, as in the preceding

equations, $f$ or $f'$ of Fig. 104, and Equations XII., XIII., &c., &c., in which

$f=a\cos {(\phi +e)}$

$f'=a\cos {(\phi -e)}$

and $a$, as before, the distance of the centre of gravity from the axes of inceptive rotation, we have

$V=g\times {\frac {\mathrm {L} k^{2}}{\beta a\alpha }}\times \sec {(\phi \pm e)}$

(XXIX.)

The sign $+$ applying to the first semiphase, and that $-$ to the second semiphase of the wave.

7th. In the case of a solid cube, fractured from its horizontal base, by subnormal wave.

Substituting for $k^{2}$ its value, and also for $a$,

$k^{2}={\frac {\alpha ^{2}}{3}}={\frac {\beta ^{2}}{3}}$

$a={\frac {\alpha }{\sqrt {2}}}$

we obtain

${\begin{cases}V={\frac {\sqrt {2}}{3}}g\times {\frac {\mathrm {L} }{\alpha }}\times \sec {(45^{\circ }\pm e)}\\or\\V=15.17\times {\frac {\mathrm {L} }{\alpha }}\times \sec {(45^{\circ }\pm e)}\end{cases}}$

(XXX.)