170 PROCEEDINGS OF THE [July, 1887. 



This radius KG depends only upon the depth KO of the focus and not upon the 

 intendty of disturbance at this focus O. 



5. But we may also suppose that the intensity of the disturbance varies in- 

 versely as the square of the distance ; that is that the product of the intensity 

 by the square of the distance is a constant quantity, and for the propagation 

 of a disturbance in a homogeneous medium, mathematical and physical consid- 

 erations indicate this as the law of the diminution of the intensity. 



6. In the preceding case, it was shown that when OK was taken to repre- 

 sent the intensity at K, then at any other point on the surface as A, 0<x will 

 represent the intensity at that point A, and Oa was taken as much smaller 

 than OK, as OK is smaller than OA, In the present case, where the intensity 

 multiplied by the square of the distance is constant, we must on the line OA, 

 again cut off a portion as much smaller than Oa, as OK is smaller than OA ; in 

 the figure, this portion extends from O to the dot marked on the line Oa, 

 which dot is not lettered in the figure, to avoid complexity. In like manner, the 

 proper lengths reckoned from O, are marked off by dots on the lines radiating 

 from O. If a curve ( not drawn in the figure ) be conceived to pass through 

 all the dots thus placed, it will in this case take the place of the semi-circle in 

 the preceding case. 



7. Now, if from each dot in this curve, a line be drawn perpendicular to OK, 

 these will show the direction and intensity of the horizontal disturbance at A, 

 B, &c., to K, and from the form of the curve it will be evident, that one of 

 these lines must be greater than any other. The position of this longest line 

 can be found, by a process to be given presently, and the line has been drawn 

 in its proper place in the figure, indicated by Q^. The radiating line from O 

 through q, cuts the semi-circle in e, and the surface of the earth in E, which 

 point is therefore the place of maximum horizontal disturbance at the surface, 

 in the present case, and is nearer to K, than is C in the preceding case. The 

 position of this point E is such that if on KE a square be described, KO will 

 equal the diagonal of that square, and so if KE be known, then KO, the depth 

 of the focus also becomes known. Now the side of a square is to its diagonal 

 as 1 is to-,/2, or as 1 is to 1.414 ; hence if KE, the radius of circle of greatest 

 horizontal disturbance, be estimated at 10 or 11 miles, as in the preceding dis- 

 cussion, then KO will be 14 or 15 miles as there stated. The angle of emer- 

 gence of the wave of disturbance at C, in preceding case is 45° ; at E in pres- 

 ent case it is 54° 14'. 



8. The position of the line Qg was found thus. Draw the line eR parallel 



to A K, then it may be shown ( by the calculus ) that KK equals -j KO. Now 



reversing our steps, first take KE= jKO, draw Ke parallel to KA, cutting 

 the circle in e ; then draw OeE, cutting the surface KA in the required point 

 E in the circle of greatest horizontal disturbance. The line OE, cuts the curve 

 passing through the dots, as above mentioned at q, and the line Qg was drawn 



parallel to KA. Since KR is \ of KO, it f oUows that Ee is j EO, then Oe 

 is double Ee, and by a well known theorem in geometry, therefore the square 



