Page:Great Neapolitan Earthquake of 1857.djvu/185

throw by horizontal shock; the structure presenting similar aspects to the wave-path, in both semiphases.

When $\phi +e=90^{\circ }$, wave-path passes through the centre of gravity and axis of overturning, and $\mathrm {V}$ becomes infinite, so that the structure cannot be overturned by any velocity of shock during the first semiphase of the wave. When $e=90^{\circ }$, and $\phi =0$, the wave-path is vertical, and the structure cannot be overthrown in the wave-path by any velocity, but may be conceived lifted, in the second semiphase of the wave, by its own inertia of motion first impressed.

And when $\phi -e=0$, the wave-path is perpendicular to the diagonal, or $f'=a$, and the wave in its second semiphase, produces its maximum effect, that maximum in the first semiphase, of course occurring when the wave is normal.

Proceeding to the consideration of the special problems:—

8th. In the case of a solid cube overturned by subnormal wave.

Preserving the foregoing notation, the structure shall overturn round $\mathrm {A}$ (Fig. 102), in the first semiphase, or round $\mathrm {B}$ in the second semiphase, of the wave.

$\alpha$ being the side of the cube,

$l={\frac {\sqrt {2}}{3}}\alpha$

and since $\phi =45^{\circ }$ (Eq. XIII. and XIV.) become

$\mathrm {V} ^{2}={\frac {4}{3}}g\alpha \times {\frac {{\sqrt {2}}-1}{\cos ^{2}(45^{\circ }\pm e)}}$

(XV.)

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