Page:Great Neapolitan Earthquake of 1857.djvu/193

gravity above the base of fracture, as in the preceding

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

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

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

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

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

(XXIX.)

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

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

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

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

we obtain

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

(XXX.)