Earthquake- Wave in the Interior of the Earth. 583 



The value of S may be calculated most conveniently by 

 eliminating 6 from the equations 





This gives 





The time of describing the whole of a path is just twice 

 the time up to the vertex, therefore 



f 



Mdm 



v \/x 2 — aV 2 



where z is the value of x for which ,r 2 =aV 2 (.r). It S is 

 regarded as a known function of the solution of this 

 integral equation may be made to depend on that of the 

 preceding equation. 



The number of paths which reach a given point on the 

 surface may be determined geometrically by drawing the 

 curve which is the locus of the vertices of the paths*. If a 

 radius making an angle \& with the radius to the epicentre 

 is cut by this curve in n points, n distinct rays of the type 

 under consideration will reach a point on the surface specified 

 by the angle 6. 



Since the equation 



pip*** 



is linear as regards /(a) and --'-- (log as), it is clear that if the 



values of -- (log as) corresponding to particular forms of f(u) 



are known, the value corresponding to a function f (a) 

 which is a linear combination of these particular forms may 

 be written down at once. In the following table is given 

 the solution of the integral equation in a number of cases. 



* This method was used by P. G. Tait in the theory of mirage to 

 determine the number of visible images. Transactions of the Koyal 

 Society of Edinburgh, vol. xxx. (1881) ; Scientific Papers, i.'p. 427. 



