# Page:Great Neapolitan Earthquake of 1857.djvu/395

311
VELOCITY DETERMINED.

If we apply this same method, to the jamb stones alone, or, as above, to the lintel alone, or to the whole viewed as a single mass, we arrive at the same value for ${\displaystyle \mathrm {V} }$, within extremely narrow limits.

The assumption upon which this method depends (as respects the horizontal velocity impressed) is open to the objection of being slightly arbitrary; whether by compensation of errors however, or not, the result arrived at is extremely near to the truth, as will appear further on, and is controlled by the following calculation, which is open to no such objection.

Taking the overthrow of this "camine," in connection with that of the wall ${\displaystyle \mathrm {B} }$, at the south end of the palazzo from which the upper part was thrown off. We have here two different bodies at the same spot, projected by the same shock, and by the same phase (the second) of the wave; and we can apply the method developed in Part I. (Eq. XL. to XLVI.) to determine both the emergence of the wave-path, and the velocity of the wave particle, in that path.

${\displaystyle a}$ and ${\displaystyle a^{\prime }}$, ${\displaystyle b}$ and ${\displaystyle b^{\prime }}$ being the respective horizontal and vertical distances of projection, we have

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

and

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

whence

 ${\displaystyle \mathrm {Tan\,} {e}={\frac {a^{2}b^{\prime }-a^{\prime 2}b}{aa^{\prime }(a^{\prime }-a)}}}$
 ${\displaystyle {\text{and }}\mathrm {H} \cos ^{2}{e}={\frac {aa^{\prime }(a^{\prime }-a)}{4(ab^{\prime }-a^{\prime }b)}}}$