Page:Great Neapolitan Earthquake of 1857.djvu/186

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138

PARTICULAR PROBLEMS.

9th. In the case of a solid parallelolpiped overturned (subnormnal wave).

Here

l={\frac {2}{3}}{\sqrt {\alpha ^{2}+\beta ^{2}}}

therefore

\mathrm {V} ^{2}={\frac {4}{3}}g\times {\sqrt {\alpha ^{2}+\beta ^{2}}}\times {\frac {1-\cos \phi }{\cos ^{2}(\phi \pm e)}}

(XVI.)

the signs $+$ and $-$ being attended to as before.^[1]

10th. In the case of a solid right cylinder overturned (subnormal wave)

.

In this case

l={\frac {15\beta ^{2}+16\alpha ^{2}}{24{\sqrt {\alpha ^{2}+\beta ^{2}}}}}

and

\mathrm {V} ^{2}={\frac {g}{12}}\times {\frac {15\beta ^{2}+16\alpha ^{2}}{\sqrt {\alpha ^{2}+\beta ^{2}}}}\times {\frac {1-\cos \phi }{\cos ^{2}(\phi \pm e)}}

(XVII.)

$+$ and $-$ applying as before.

11th. In the case of a hollow parallelopiped overturned (subnormal wave)

.

Here, from Eq. VIII., XIII., and XIV., we have

\mathrm {V} ^{2}={\frac {2g}{3}}\times {\frac {2\beta (\alpha ^{2}+\beta ^{2})+\gamma (2\alpha ^{2}+3\beta ^{2})}{(\beta +\gamma ){\sqrt {\alpha ^{2}+\beta ^{2}}}}}\times {\frac {1-\cos \phi }{\cos ^{2}(\phi \pm e)}}

(XVIII.)

↑ Eq. XVI. has been applied in the text of Part II. under the form

\mathrm {V} ^{2}={\frac {4}{3}}g\times {\frac {(\alpha ^{2}+\beta ^{2})^{\frac {3}{2}}}{(\alpha \cos e\pm \beta \sin e)^{2}}}\times (1-\cos \phi )

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