# Page:Great Neapolitan Earthquake of 1857.djvu/208

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EMERGENCE AND VELOCITY BOTH OBTAINED.

If in the same locality, we are enabled to observe two different bodies, both projected, and to measure the vertical and horizontal distances to the point of fall, we can determine both the angle of emergence of the wave-path (${\displaystyle e}$) and the maximum velocity of the wave. Thus, for example, let both the bodies, be projected by the second semiphase of the wave, and let ${\displaystyle ab}$ and a ${\displaystyle a'b'}$ denote the coordinates in ${\displaystyle x}$ and ${\displaystyle y}$, of the two trajectories; then by Eq. XL. we have

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

from which we find

 ${\displaystyle \mathrm {Tan\,} {e}={\frac {a^{2}b'-a'^{2}b}{aa'(a'-a)}}}$ (XLIII.)
 ${\displaystyle \mathrm {H} \cos ^{2}{e}={\frac {aa'(a'-a)}{4(ab'-a'b)}}}$ (XLIV.)

and substituting for ${\displaystyle \mathrm {H} }$ its value ${\displaystyle {\frac {\mathrm {V} ^{2}}{2g}}}$ we find

 ${\displaystyle \mathrm {V} ^{2}=g\times {\frac {aa'(a'-a)}{2\cos ^{2}{e}(ab'-a'b)}}}$ (XLV.)

In the case, of the upper portion of a wall, thrown off from the lower which remains standing, which is a very frequent one, the equations to apply, are the same as for a body, projected and overturned from the summit; the upper portion turning over first, upon one arris, and then being thrown more or less from the base of the wall, in a trajectory. The preceding equations embrace, probably, every case likely to occur to observation.