If the propagation of the wave is considered as a horizontal one, as would be done when calculating the position of the epicentrum or point above the origin, by means of the times of arrival of a sea wave, the ordinate z of the first five equations would be omitted. Working in this way the resulting four equations, viz.
2a1 x + 2b1 y + um2 - wm2 = a12 + b12
&c. &c. &c.
remained unchanged.
Applying this method to the same example as that used as illustration for the two previous methods, we obtain for the co-ordinates of Mejillones, Iquique, Cobija, Tocopilla, and Huanillos, measured in geographical miles, and the times in Iquique time at which the wave reached each, as given in the following table; ox and oy being, drawn through Mejillones.
| Co-ordinates | Time of arrival | ||||
|---|---|---|---|---|---|
| OX | OY | h. | m. | ||
| Mejillones | a or 0 | b or 0 | 8 | 46 | p. m. |
| Iquique | a1 or 150 | b1 or 96 | 8 | 40 | „ |
| Cobija | a2 or 36 | b2 or 14 | 8 | 38 | „ |
| Tocopilla | a3 or 66 | b3 or 31 | 8 | 32 | „ |
| Huanillos | a4 or 102 | b4 or 58 | 8 | 30 | „ |
From this data we find the co-ordinates x and y of this origin to be 85·8 and 56·7; whilst the velocity of propagation = 45 feet per second.
Measuring these ordinates upon the map, we obtain a centre lying very near Long. 71° 5′ W. and Lat. 21° 22′ S., a position which is very near to that which has already been obtained by other methods.
If instead of Huanillos we substitute the ordinates and time of arrival of the sea wave for Pabalon de Pica, another point for the origin will be obtained lying farther out at sea. To obtain the best result, the method to be taken will evidently be, first to reject those places at which it seems likely that some mistake has been made with the time observations, and then with the remaining places to form as many equations as possible, and from these to obtain a mean value. This is a long and tedious process, and as the time observations of this particular earthquake are probably one and all more or less inaccurate, it is hardly worth while to follow the investigation farther.
In this example, as in the preceding ones, it will be observed that it has been sea waves that have been dealt with, rather than earth vibrations. It is evident, however, that these latter vibrations may be dealt with in a similar manner.
In these determinations it will also have been observed that it is assumed that the disturbance has radiated from a centre, and, therefore, approached the various stations in different directions. If we assume that we have three stations very near to each other as compared with their distances from the origin, so that we can assume that the wave fronts at the various stations were parallel, the determination of the direction in which the wave advanced appears to be much simplified. The determination of the direction in which a wave has passed across three stations was first given by Professor Haughton.