Page:Great Neapolitan Earthquake of 1857.djvu/205

If $b$ denote the height through which the centre of gravity has descended, to reach the ground, or the horizontal plane pessing through the centre of gravity, when so deposited, and $a$ the horizontal distance, traversed by the same centre, on striking the ground, then

$b=a\tan {e}+{\frac {a^{2}}{4\mathrm {H} \cos ^{2}{e}}}$

from which the following expressions are easily deduced for the angle of emergence (which is alternate and equal, to the angle of projection) and for the velocity:

$\mathrm {Tan\,} {e}={\frac {-2\mathrm {H} \pm {\sqrt {4\mathrm {H} (\mathrm {H} +b)-a^{2}}}}{a}}$

(XXXVIII.)

$\mathrm {V} ^{2}={\frac {a^{2}g}{2\cos ^{2}{e}(b-a\tan {e})}}$

(XXXIX.)

In the second semiphase of the wave (or return shock) the displaced body is thrown, not downwards, but more or less upwards, if projected by the inertia of motion, acquired from a subnormal wave. If the wave were perfectly normal, the projection of course would be horizontal, and $e=0$ for both semiphases.

In the case of projection by subnormal wave, observing that the axis of $y$ is measured vertically upwards, and that of $x$ horizontally, from the origin, in the centre of gravity of the body as before, we have for the trajectory (Fig. 106)

$y=x\tan {e}-{\frac {x^{2}}{4\mathrm {H} \cos ^{2}{e}}}$

(XL.)

and substituting in this as before

$y=-b$

$x=+a$