Page:Great Neapolitan Earthquake of 1857.djvu/187

12th. In the case of a hollow right cylinder overturned (subnormal wave).

Here, from what has preceded, we have

$\mathrm {V} ^{2}={\frac {g}{12}}\times {\frac {15\beta ^{2}+16\alpha ^{2}+3\beta ^{2}}{\sqrt {\alpha ^{2}+\beta ^{2}}}}\times {\frac {1-\cos \phi }{\cos ^{2}(\phi \pm e)}}$

(XIX.)

and

$\mathrm {V} ^{2}={\frac {g}{6}}\times {\frac {9\beta ^{2}+8\alpha ^{2}}{\sqrt {\alpha ^{2}+\beta ^{2}}}}\times {\frac {1-\cos \phi }{\cos ^{2}(\phi \pm e)}}$

(XX.)

Proceeding now to

Cap. B.—Bodies or Structures Fractured.

If the fracturing force, or $\mathrm {MV}$, in the direction of the wave-path, $\mathrm {M}$ being the mass broken off, act transversely to the plane of fracture, the case is one simply of cohesion destroyed by an impulsive force, in which $2\mathrm {MV}$ is equal to the statical strain that would have produced the same fracture; and if the direction of the force be such as to produce rotation in the mass fractured off, there will be a dynamic couple to be taken into account; and lastly, if the plane of fracture occur so, that it is not transverse to the line of force, the latter may be resolved into one that shall be so, which is all that need be said as to direct fractures, such as those passing down vertically or diagonally, as fissures through walls, &c.; and the rather because, precious as these become as indices of direction, they should never be adopted as measures of wave-velocity, from the uncertainty that must always attend the knowledge of the coefficient of force necessary to produce fracture through the joints across the beds of masonry, &c.

Proceeding, therefore, to the determination of fracture occurring at the base, or in horizontal planes, or in those of