Page:Great Neapolitan Earthquake of 1857.djvu/207

Rh describe the circle $$\mathrm{XT}$$. This is the locus circle (i.e. that in which the line of aim—in our case the wave-path, shall cut the vertical drawn through the point $$\mathrm{D}$$, at which the projectile falls—whether the angle $$e$$, be above or below, the horizontal line through the point of projection). Let $$\mathrm{D}$$ be the point at which the projectile falls to the ground; draw $$\mathrm{DFE}$$ vertical through its centre of gravity. The directions $$\mathrm{AE}$$ and $$\mathrm{AF}$$, formed by its intersections with the vertical, give the superior, and inferior angles of elevation, for the given horizontal range and elevation, and coincide in result with Eq. XXXVIII. and XLI.

The wave-path must always, be either horizontal or emergent. Hence in the first semiphase of the wave, although the motion of the projectile is contrary to that of the wave transit, the angle $$e$$, given by the above construction, will be the superior one, and also in the second semiphase of the wave, in which the motion of the projectile is in the same direction with the wave transit, the angle $$e$$ will be still the superior one.

The values of $$\mathrm{V}$$ given by Eq. XXXIX. and XLII. are those of the projectile itself, but are less than the maximum velocity of the earth-wave by the velocity destroyed by adhesion, &c. The latter produces rotation in the body, and we generally find it overturned, as well as projected. The velocity, therefore, destroyed by adhesion is equal to that which has produced the rotation, $$v$$, and may be arrived at by the Eq. I. to XX. inclusive, and that velocity so found reduced to the direction of the wavepath, and added to the velocity of projection, will give the total velocity, or $$\mathrm{V}$$ = the maximum velocity of the wave.