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 \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)