Page:Great Neapolitan Earthquake of 1857.djvu/189

In accordance with the theory of Leibnitz, we therefore have

$\mathrm {M} Vf=\mathrm {FA} {\frac {k^{2}}{\beta }};$

$\mathrm {M} =$ being the mass of the detached portion;

$V=$ the velocity of the wave in its path (normal), $f=$ the perpendicular height of the centre of gravity above the base of fracture; $\mathrm {F} =$ the coefficient of dynamic cohesion, or the force upon the unit of surface of the material fractured, which, when suddenly applied, is sufficient to produce fracture; $\mathrm {A} =$ the area of the base of fracture in such units; $k=$ the radius of gyration of the plane of fracture with respect to the axis $\mathrm {A}$ or $\mathrm {B}$. $\beta =$ the width of $\mathrm {AB}$.

If $\mathrm {W} =$ the weight of the mass broken off, $g=$ the velocity due to gravity in one second, then

$V=g{\frac {\mathrm {FA} }{\mathrm {W} }}\times {\frac {k^{2}}{f\beta }};$

and if the detached mass be any regular prism or right

extension. If the compressions and extensions follow the same law, and have the same coefficient, this axis, or neutral line, will divide the base into two equal parts, which would eutirely change the necessary amount of the fracturing force." I should admit the correctness of the conclusion thus expressed, if I could altogether, the premises, and their applicability to the matter in hand may be disposed of in a few words. The actual extensibility of all building materials, and still more their compressibility, are so extremely snmll, that for our present purposes beth may be regarded as in the text, without sensible error, the compressibility for the small intensity of pressures we are dealing with is insensible, and therefore the position of the axis of rotation is practically that assigned to it. If the objector will point out any good ground for adopting any oth axis of rotation, I shall be ready to employ it.