The position from which the shock originated appears therefore to have occurred very near to a place lying in Long. 7° 15′ W. and Lat. 21° 22′ S.
Fig. 31.
The actual operations which were gone through in making the accompanying map were as follows. First, the places with which we had to deal were represented on a map in orthographical projection, the centre of projection being near to the centre of the map. This was done so that the measurements which were made upon the map might be more correct than those we should obtain from an ordinary chart where this portion of the world was not the centre of projection. Next, a number was taken as equal to the velocity with which the sea wave had travelled. The first velocity taken was about 400 feet per second—this being about the velocity with which, theoretically, it must have travelled in an ocean having a depth equal to that indicated upon the charts—also it seemed to have travelled at this rate from the various times of arrival as recorded at places along the coast. Circles were then drawn round Tocopilla, Cobija, Iquique, and Mejillones with radii equal to 2, 8, 10, and 15, each multiplied by (60 × 400). It was then seen by trial that it was impossible to draw a single circle which should touch four circles and also pass through Huanillos. These four circles were, in fact, too large. Four new but smaller circles, which are shown in the map, were next drawn, their radii being respectively equal to the numbers 2, 8, 10, and 16, each multiplied by (60 × 350), and it was found that a circle, with a centre c, could be drawn which would practically touch the four circles, and at the same time would pass through Huanillos.
III. The method of hyperbolas.—The method which I call that of hyperbolas is only another form of the method of circles. It is, however, useful in special cases, as, for instance, where we have the times of arrival of earthquakes at only two stations. Between Tokio and Yokohama, at which places I frequently obtain tolerably accurate time records, the method has been applied on several occasions with advantage. In the preceding example let us suppose that the only time records which we had were for Huanillos and Mejillones, and that the wave was felt at the latter place sixteen minutes or 960 seconds after it was experienced at the former. Calling these places h and m respectively, round m draw a circle equal to the 960 multiplied by the velocity with which the wave was propagated. It is then evident that the origin of this disturbance must be the centre of a circle which passes through h and touches the circle drawn round m. Join h m, cutting the circle round m in y. Bisect y h in v. It is evident that v is one possible origin for the disturbance. Next, from m, in the direction of h, draw any line m z p; join z h; bisect z h at right angles by the line o p n. Because ph = pz, it is evident that p is a second possible origin. Proceeding in this way a series of points lying to the right and left of v on the curve r v t may be found, and we may therefore say that the origin lies somewhere in the curve r v t. By increasing or decreasing our velocity we vary the position of the curve r v t, and, instead of a line on which our origin may be, we obtain a band. As the assumed velocity increases, the circle round m becomes larger, and has its limit when it passes through h, where the two arms of the curve r v t will close together and form a prolongation of the line m y h as the assumed velocity diminishes. The circle round m becomes smaller until it coincides with the point m. At such a moment the curve r v t opens out to form a straight line bisecting m h at right angles. The curve r v t is a hyperbola with a vertex v and foci h and m. Inasmuch as pm - ph = a constant quantity. If we have the time given at which the shock or wave arrived at a third station as at Iquique, it is evident that a second hyperbola r′ v′ t′ might be drawn with Iquique and Huanillos as foci, and that the mutual intersection of these two hyperbolas with a third hyperbola, having for its foci Iquique and Mejillones, would give the origin of the wave. The obtaining of a mutual intersection would depend on the assumed velocity, and the accuracy of the result, like that of the method of circles, would depend upon the trials we made. The method here enunciated may be carried farther by describing hyperboloids instead of hyperbolas, the mutual intersection of which surfaces would, in the case of an earth wave, give the actual origin or centrum rather than the point above the origin or epicentrum.
IV. The method of co-ordinates.—Given the times at which a shock arrived at five or more places, the position of which we have marked upon a map, or chart, to determine the position on the map of the centre of the shock, its depth, and the velocity of propagation.
Commencing with the place which was last reached by the shock, call these places p, p1, p2, p3, and p4, and let the times taken to reach these places from the origin be respectively t, t1, t2, t3, and t4.
Through p draw rectangular co-ordinates, and with a scale measure the co-ordinates of p1, p2, p3, and p4, and let these respectively be a1, b1; a2, b2; a3, b3; a4, b4. Then if x, y, and z be the co-ordinates of the origin of the shock, d, d1, d2, d3, and d4, the respective distances of p, p1, p2, p3, and p4 from this origin, and v the velocity of the shock, we have
- x2 + y2 + z2 = d2 = v2 t2
- (a1 - x)2 + (b1 - y)2 + z2 = v2 t12
- (a2 - x)2 + (b2 - y)2 + z2 = v2 t22
- (a3 - x)2 + (b3 - y)2 + z2 = v2 t32
- (a4 - x)2 + (b4 - y)2 + z2 = v2 t42
Because we know the actual times at which the waves arrived at the places p, p1, p2, p3, p4, we know the values t—t1, t—t2, t—t3, t—t4. Call these respectively m, p, q, and r. Suppose t known, then