_x_ and _y_ determine the origin of the shock. The absolute velocity _v_ equals √ _u_. From _v_ and _w_ we obtain _t_. Substituting _x_, _y_, _v_, and _t_ in the first equation we obtain _z_. We have here assumed that the points of observation have all about the same elevation above sea level. If it is thought necessary to take these elevations into account, a sixth equation may be introduced. 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. 2_a__{1} _x_ + 2_b__{1} _y_ + _u__m_^2 - _w__m_^2 = _a__{1}^2 + _b__{1}^2 &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 | _a_{1}_ or 150 | _b_{1}_ or 96 | 8 40 „ | | Cobija | _a_{2}_ or 36 | _b_{2}_ or 14 | 8 38 „ | | Tocopilla | _a_{3}_ or 66 | _b_{3}_ or 31 | 8 32 „ | | Huanillos | _a_{4}_ or 102 | _b_{4}_ 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.