Page:Great Neapolitan Earthquake of 1857.djvu/178

5th. In the case of a hollow cylinder overturned (Fig. 97).

Let the altitude of the right circular cylinder be $\alpha$, the exterior and interior diameters of its base $\beta ,\beta '$ so that the thickness of the wall $={\frac {\beta -\beta ^{2}}{2}}$

then

$k^{2}={\frac {15\beta ^{2}+16\alpha ^{2}+3\beta '^{2}}{48}}$

which is the case of the solid cylinder when $\beta '=0$, and when the thickness of the wall is so small in relation to the other dimensions that it may be neglected

$k^{2}={\frac {9\beta ^{2}+8\alpha ^{2}}{24}}$

In either case, the distance of the centre of gravity from the axis or point of rotation at the base is

$a={\tfrac {1}{2}}{\sqrt {\alpha ^{2}+\beta ^{2}}}$

Hence, in the first case,

${\frac {k^{2}}{\alpha }}={\frac {15\beta ^{2}+16\alpha ^{2}+3\beta '^{2}}{24{\sqrt {\alpha ^{2}+\beta ^{2}}}}}$

(IX.)

and in the second case,

${\frac {k^{2}}{\alpha }}=l={\frac {9\beta ^{2}+8\alpha ^{2}}{12{\sqrt {\alpha ^{2}+\beta ^{2}}}}}$

(X.)

and

$V^{2}=2g\times {\frac {9\beta ^{2}+8\alpha ^{2}}{12{\sqrt {\alpha ^{2}+\beta ^{2}}}}}\times {\frac {1-\cos \phi }{\cos ^{2}\phi }}$.