When a, b, and c are three stations at which a shock is observed at the times t1, t2, and t3; a, b, and c are the distances between a, b, and c, and ϕ is the angle made by the coseismal lines x a x, y b y, and the line a b, which are assumed to be parallel.
This I applied in the case of the Iquique earthquake, but owing to the smallness of the angles between the three stations a, b, and c, the result was unsatisfactory. The problem ought to be restricted, first, to places which are a long distance away from a centre, and, secondly, to places which are not nearly in a straight line. This problem may be solved more readily by geometrical methods. Plot the three stations a, b, and c on a map, join the two stations between which there was the greatest difference in the time observation. Let these, for example, be a and c. Divide the line a c at point d, so that a d : d c as the interval between the shock felt at a and b is to the interval between the shock felt at b and c. The line b d will be parallel to the direction in which the wave advanced.
The difference in time of the arrival of two disturbances.—In the various calculations which have been made to determine an origin based on the assumption of a known or of a constant velocity, we have only dealt with a single wave, which may have been a disturbance in the earth or in the water. A factor which has not yet been employed in this investigation is the difference in time between the arrival of two disturbances; one propagated, for instance, through the earth, and the other, for example, through the ocean. The difference in the times of the arrival of two waves of this description is a quantity which is so often recorded that it is well not to pass it by unnoticed. To the waves mentioned we might also add sound waves, which so frequently accompany destructive earthquakes, and, in some localities, as, for instance, in Kameishi, in North Japan, are also commonly associated with earthquakes of but small intensity. It was by observing the difference in time between the shaking and the sound in different localities that Signor Abella was enabled to come to definite conclusions regarding the origin of the disturbances which affected the province of Neuva Viscoya in the Philippines, in 1881; the places where the interval of time was short, or the places where the two phenomena were almost simultaneous, being, in all probability, nearer to the origin than when the intervals were comparatively large. I myself applied the method with considerable success when seeking for the origin of the Iquique earthquake of 1877. The assumptions made in that particular instance were, first, that the velocity of the disturbance through the earth was known, and, secondly, that the velocity with which a sea wave was propagated was also known.
A method similar to the above was first suggested by Hopkins. It depended on the differences of velocity with which normal and transversal waves are propagated.[86]
Seebach’s method.—To determine the true velocity of an earthquake, the time of the first shock, and the depth of the centre.
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,
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.