[40] In order to simplify these figures, the rivers, most of the inlets, and other details are omitted. Small figures are added along the railway lines to denote the distance in miles from the stations in Charleston. [41] If this were so, the decrease in intensity would be only apparent; but it may have been real, and a possible explanation on this supposition is given later on (p. 135). [42] If _c_ be the depth of the focus, _a_ the intensity at unit distance from the focus, and _y_ the intensity on the surface at distance _x_ from the epicentre, then y=a/(c^2+x^2) The inclination of the curve at any point is given by dy/dx=-2*a*x/(c^2+x^2)^2, and this is a maximum when d^2y/dx^2 or (3*x^2-c^2)/(c^2+x^2)^3 is zero, which is satisfied when c=x*sqrt(3) [43] _British Association Report_, 1858, pp. 101-103. [44] The above time would have to be increased by one second if the depth of the focus were very small, and diminished by one second if it were as great as 23 miles; the difference in either case being less than the probable error. [45] The method employed is as follows: Let t_0 be the computed time (9h. 51m. 6s.) at the focus, _x_ seconds the error in this estimate, _t_ the reported time at a given place, _D_ its distance from the focus in miles, and _y_ the number of seconds required to travel one mile; then, assuming that _y_ does not vary with the distance, we have x+Dy=t+t_0. An equation of this form is obtained from each observation, and the method of least squares is then applied to determine the most probable values of _x_ and _y_. [46] This seems to me the more probable course. It is possible, however, that the fault-line may pass from Mount Holly Station to the east of the Woodstock epicentre as shown in Fig. 28, and then to the west of the Rantowles epicentre, the fault changing its direction of hade in the intermediate district. CHAPTER VI. THE RIVIERA EARTHQUAKE OF FEBRUARY 23RD, 1887.