1. We know from observations on artificial earthquakes that the velocity of propagation is greater between stations near to the origin of the shock than it is between more remote stations; and also the velocity of propagation varies with the initial force which produced the disturbance. If our points of observation are sufficiently close together as compared with their distance from the origin of the disturbance, it is probable that errors of this description are small and will not make material differences in the general results. 2. We have reasons for believing that the transit velocity of an earthquake is dependent on the nature of the rocks through which it is propagated. Errors which arise from causes of this description will, however, be practically eliminated if our observation points are situated on an area sufficiently large, so that the distribution of the causes tending to alter the velocity of a shock balance each other. It must be remarked, that causes of this description may also produce an alteration in the direction of our shock. Other errors which may sometimes enter into our results, when determining the origin of shocks by means of observations on velocities, are the assumptions that the disturbance has travelled along the surface from the _epicentrum_ and not in a direct line from the _centrum_. Again, it is assumed that the origin is a point, whereas it may possibly be a cavity or a fissure. Lastly, if we desire extreme accuracy, we must make due allowance for the sphericity of the earth and the differences of elevation of the observing stations. I. _The method of straight lines._—Given a number of pairs of points A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}, &c., at each of which the shock was felt simultaneously, to determine the origin. Theoretically if we bisect the line which joins A_{0} and A_{1} by a line at right angles to A_{0}, A_{1}, and similarly bisect the lines B_{0}, B_{1}, C_{0}, C_{1}, all these bisecting lines _a__{0}, _a__{1}, _b__{0}, _b__{1}, _c__{0}, _c__{1}, &c., ought to intersect in a point, which point will be the _epicentrum_ or the point above the origin. This method will fail, first, if A_{0}, A_{1}, B_{0}, B_{0}, C_{0}, C_{1} form a continuous straight line, or if they form a series of parallel lines. Hopkins gives a method based on a principle similar to the one which is here employed—namely, given that a shock arrives simultaneously at _three_ points to determine, the centre. In this case, the relative positions of the three points, where the time of arrival was simultaneous, must be accurately known, and these three points must not lie in a straight line, or the method will fail. For practical application the problem must be restricted to the case of three points which do not lie nearly in the same straight line.