II. The method of circles.—Given the times t0, t1, t2, &c., at which a shock arrived at a number of places a0, a1, a2, &c., to determine the position from which the shock originated.
Suppose a0 to be the place which the shock reached first, and that it reached a1, a2, a3, &c., successively afterwards.
Let t1 - t0 = a
t2 - t0 = b
t3 - t0 = c, &c.
With a1, a2, a3, &c. as centres, describe circles with radii proportional to the known qualities a, b, c, &c., and also a circle which passes through a0 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, a0, a1, a2, 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 a0. 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 of sea wave | Time after arrival at Huanillos | Distance at 350 feet per 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 |
The distances marked in the third column are used as radii of the circles drawn round the places to which they respectively refer.
The centre of the circle drawn to touch the circles of the first column, and at the same time to pass through Huanillos, is marked c.