Page:Great Neapolitan Earthquake of 1857.djvu/191

Here ${\displaystyle \beta =D}$, the diameter of the cylinder,

${\displaystyle k^{2}={\frac {5}{16}}\beta ^{2}}$;

therefore

${\displaystyle {\begin{cases}V={\frac {5}{8}}g\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\\or\\V=20.12{\frac {\mathrm {LD} }{\alpha ^{2}}}\end{cases}}}$

(XXV.)

Assuming the thickness of the walls small, as before, and ${\displaystyle \gamma }$ being that side of the parallelopiped, which is the axis of inceptive rotation,

${\displaystyle k^{2}={\frac {\beta ^{2}(\beta +2\gamma )}{4(\beta +\gamma )}}}$;

therefore

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

(XXVI.)

Then, ${\displaystyle \beta =\gamma }$, and (Eq. XXVI.) becomes

${\displaystyle {\begin{cases}V={\frac {3}{4}}g\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\\or\\V=24.15\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\end{cases}}}$

(XXVII.)