Page:Great Neapolitan Earthquake of 1857.djvu/195

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TOTAL VELOCITY—FRACTURE AND OVERTURNING.

147

we have

{\begin{cases}\mathrm {V} ={\frac {1}{2}}g\times {\frac {\mathrm {L} \beta (\beta +2\gamma )}{\alpha ^{2}(\beta +\gamma )}}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\\or\\\mathrm {V} =16.10\times {\frac {\mathrm {L} \beta (\beta +2\gamma )}{\alpha ^{2}(\beta +\gamma )}}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\end{cases}}

(XXXIII.)

11th. In the case of a hollow square prismatic tower, fractured from its horizontal base, by subnormal wave.

As before, $\beta =\gamma$ , and

{\begin{cases}\mathrm {V} ={\frac {5}{4}}g\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\\or\\\mathrm {V} =24.15\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\end{cases}}

(XXXIV.)

12th. In the case of a hollow right cylinder, fractured from its horizontal base, by subnormal wave.

Here

k^{2}={\frac {5\beta ^{2}+\beta '^{2}}{16}}

a={\frac {1}{2}}\alpha \sec {\phi }

;

therefore

{\begin{cases}\mathrm {V} ={\frac {1}{8}}g\times {\frac {\mathrm {L} (5\beta ^{2}+\beta '^{2})}{\alpha ^{2}\beta }}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\\or\\\mathrm {V} =4.02\times {\frac {\mathrm {L} (5D^{2}+D'^{2})}{\alpha ^{2}D}}\times {\frac {\cos {\phi }}{\cos {(\phi \pm e)}}}\end{cases}}

(XXXV.)

Thirdly. Let the structure be fractured, from its horizontal base, and also overturned, whether by a normal or a subnormal wave.

If the structure be observed fractured only, at the base,

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