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

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

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

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

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

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