# Page:Great Neapolitan Earthquake of 1857.djvu/186

138
PARTICULAR PROBLEMS.

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

Here

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

therefore

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

the signs ${\displaystyle +}$ and ${\displaystyle -}$ being attended to as before.[1]

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

.

In this case

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

and

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

${\displaystyle +}$ and ${\displaystyle -}$ applying as before.

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

.

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

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

1. Eq. XVI. has been applied in the text of Part II. under the form
 ${\displaystyle \mathrm {V} ^{2}={\frac {4}{3}}g\times {\frac {(\alpha ^{2}+\beta ^{2})^{\frac {3}{2}}}{(\alpha \cos e\pm \beta \sin e)^{2}}}\times (1-\cos \phi )}$