II. _The method of circles._—Given the times _t__{0}, _t__{1}, _t__{2}, &c., at which a shock arrived at a number of places A_{0}, A_{1}, A_{2}, &c., to determine the position from which the shock originated. Suppose A_{0} to be the place which the shock reached first, and that it reached A_{1}, A_{2}, A_{3}, &c., successively afterwards. Let _t__{1} - _t__{0} = _a_ _t__{2} - _t__{0} = _b_ _t__{3} - _t__{0} = _c_, &c. With A_{1}, A_{2}, A_{3}, &c. as centres, describe circles with radii proportional to the known qualities _a_, _b_, _c_, &c., and also a circle which passes through A_{0} and touches these circles. The centre of the last circle will be the _epicentrum_. The radii proportional to _a_, _b_, _c_, &c. may be represented by the quantities _ax_, _bx_, _cx_, &c., where _x_ is the velocity of propagation of the shock. It will be observed that the times at which the shock arrived at three places might alone be sufficient. If, instead of taking the times of arrival of a shock, the arrival of a sea wave be taken, the result would be a closer approximate to the absolute truth. It will be observed that this method is not a direct one, but is one of trial. If, however, an imaginary case be taken, and three given points of observation, A_{0}, A_{1}, A_{2}, be plotted on a piece of paper, it will be found that it is not a difficult matter to determine two numbers proportional to _a_ and _b_ which will allow you to draw two circles so that they may be touched by a third circle drawn through A_{0}. This problem has practically been applied in the case of the arrival of a sea wave at a number of places on the South American coast, at the time of the earthquake of May 9, 1877. This is illustrated as follows. The places which were chosen were Huanillos, Tocopilla, Cobija, Iquique, Mejillones. In the following table the first column gives the times at which the sea wave arrived at each of these places in Iquique time; in the second column the difference between these times and the time at which it reached Huanillos is given; in the third column the distances through which a sea wave, propagated at the rate of 350 feet per second, could travel during the intervals noted in the second column is given. +------------+----------+----------+------------+ | | Arrival |Time after| Distance at| | | of sea |arrival at|350 feet per| | | wave | Huanillos| second | +------------+----------+----------+------------+ | | h. m. | minutes | miles | | Huanillos | 8 30 | 0 | 0 | | Tocopilla | 8 32 | 2 | 8 | | Cobija | 8 38 | 8 | 32 | | Iquique | 8 40 | 10 | 40 | | Mejillones | 8 46 | 16 | 64 | +------------+----------+----------+------------+