Page:Great Neapolitan Earthquake of 1857.djvu/184

Substituting the expression in Eq. I., we find

$\mathrm {V} ^{2}={\frac {2g\alpha k^{2}}{({\frac {\alpha \cos e-\beta \sin e}{2}})}}\times (1-\cos \phi )$

but

${\frac {1}{2}}(\alpha \cos e-\beta \sin e)=f=a\cos(\phi +e)$

or

$\mathrm {V} ^{2}=2gl\times {\frac {1-\cos \phi }{\cos ^{2}(\phi +e)}}$

(XIII.)

Which, when $e=0$, reduces itself to Eq. II. for the normal wave, or $V^{2}=\mathrm {V} ^{2}$.

But if hte structure be overturned by the second semiphase, inertia acts in the direction of transit $\mathrm {OC}$, and tends to make it overturn round the axis $\mathrm {B}$.

The overturning couple is

${\frac {\mathrm {MV} (\alpha \cos e+\beta \sin e)}{2}}$

and

$f'={\frac {\alpha \cos e+\beta \sin e}{2}}=a\cos(\phi -e)$

$f$ in the former, and $f'$ in the latter cases, denoting the perpendiculars let fall upon the wave-path from the axes $\mathrm {A}$ and $\mathrm {B}$ respectively.

Substituting in Eq. I., as before, we find

$\mathrm {V} ^{2}=2gl\times {\frac {1-\cos \phi }{\cos ^{2}(\phi -e)}}$

(XIV.)

When $e=0$ or the wave normal, this also reduces to Eq. II., as must necessarily result from the fact that both semiphases of the wave (assuming the velocity practically the same in both) are equally effective in producing over-