Page:Great Neapolitan Earthquake of 1857.djvu/206

we find

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

whence the angle of emergence or of projection is

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

(XLI.)

and the velocity of projection

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

(XLII.)

As the velocity of projection by earthquake-shock has been proved, by the examination of this shock of December, 1857, to be small, and therefore $\mathrm {H}$, the height due to it, also small; we can find either $\mathrm {V}$ or $e$, geometrically, by the application of Prof. Galbraith's very beautiful problem, for determining graphically, either of these quantities for a projectile; and as this method may be applied by any unmathematical observer, who measures on the ground, the vertical and horizontal heights of a body thrown, and can use a pair of compasses, it will be well to transcribe it.

Let $\mathrm {A}$ (Fig. 107) be the top of any tower or other elevation from which a body has been projected. From $\mathrm {A}$ draw $\mathrm {AB}$ vertical and $=4\mathrm {H}$ ($\mathrm {H}$ being the height due to the velocity, supposed given). Through $\mathrm {A}$ draw $\mathrm {AX}$ horizontal. Bisect $\mathrm {CB}$ in $\mathrm {Y}$, and on $\mathrm {BC}$ describe the semicircle $\mathrm {BXC}$. Bisect $\mathrm {BA}$ in $\mathrm {O}$, and with $\mathrm {O}$ as centre and $\mathrm {OX}$ as radius