Page:Great Neapolitan Earthquake of 1857.djvu/177

substituting in (Eq. II.) and since

$\cos ^{2}\phi ={\frac {\alpha ^{2}}{\alpha ^{2}+\beta ^{2}}}$

(VII.)

$V^{2}={\frac {15\beta ^{2}+16\alpha ^{2}}{12\alpha ^{2}}}\times g{\sqrt {\alpha ^{2}+\beta ^{2}}}\left(1-\cos \phi \right)$ (VII.)

4th. In the case of a hollow rectangular parallelopiped overturned (Fig. 96).

Let the edges of the parallelopiped be $\alpha ,\beta ,\gamma$ the thickness of its walls being small in relation to their lengths, and suppose it overturned round the edge or axis $\gamma$.

It is easily demonstrable that

$k^{2}={\frac {2\beta \left(\alpha ^{2}+\beta ^{2}\right)+\gamma \ \left(2\alpha ^{2}+3\beta ^{2}\right)}{6\left(\beta +\gamma \right)}}$

and

$a={\tfrac {1}{2}}{\sqrt {\alpha ^{2}+\beta ^{2}}}$

and therefore

${\frac {k^{2}}{\alpha }}={\frac {2\beta \left(\alpha ^{2}+\beta ^{2}\right)+\gamma \ \left(2\alpha ^{2}+3\beta ^{2}\right)}{3\left(\beta +\gamma \right){\sqrt {\alpha ^{2}+\beta ^{2}}}}}$

(VIII.)

from which, substituting the value of $l$ in Eq. II., and remembering that $\tan \phi ={\frac {\beta }{\alpha }}$ we obtain $V$, the horizontal velocity.