Let the straight line m, m1, m2, m3 represent the surface of the earth shaken by an earthquake. For small earthquakes, to consider the surface of the earth as a plane will not lead to serious errors.
If an earthquake originates at c, then to reach the surface at m it traverses a distance h in the time t. To reach the surface at m1 it traverses a distance h + x1 in a time t2. If v equals the velocity of propagation,
then t = h/v, t1 = h + x1/v,
t2 = h + x2/v, &c.
Seebach now says that if we have given the position of m or epicentrum of the shock, and draw through it rectangular axes like m m3 and m t3, and lay down on m m3 in miles the distances from M of the various stations which have been shaken, and in equal divisions for minutes lay down on m t3 the differences of time at which m, m1, m2, &c. were shaken, then m1 t1, m2 t2, &c. are the co-ordinates of points on an hyperbola. The degree of exactness with which this hyperbola is in any given case constructed is a check upon the accuracy of the time observations and the position of the epicentrum. The apex of the hyperbola is the epicentrum.
The intersection of the asymptote with the ordinate axis is the time point of the first shock, which, because the scale for time and for space were taken as equal, gives the absolute position of the centrum. This intersection is shown by dotted lines. Knowing the position of the centrum, we can directly read from our diagram how far the disturbance has been propagated in a given time.
CHAPTER XI.
THE DEPTH OF AN EARTHQUAKE CENTRUM.
The depth of an earthquake centrum—Greatest possible depth of an earthquake—Form of the focal cavity.
Depth of centrum.—The first calculations of the depth at which an earthquake originated were those made by Mallet for the Neapolitan earthquake of 1857. These were made on the assumption that the earth wave radiated in straight lines from the origin, and, therefore, at points at different distances from the epicentrum it had different angles of emergence. These angles of emergence were chiefly calculated from the inclination of fissures produced in certain buildings, which were assumed to be at right angles to the direction of the normal motion. If we have determined the epicentrum of an earthquake and the muzoseismal circle, and make either the assumption that the angle of emergence in this circle has been 45° or 54° 44′ 9″ (see page 54), it is evidently an easy matter by geometrical construction to determine the depth of the centrum. Höfer followed this method when investigating the earthquake of Belluno.