Page:Great Neapolitan Earthquake of 1857.djvu/186

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]

.

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.

.

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.)