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_.