Earthquake- Wave in the Interior of the Earth. 579 



to an arbitrary constant. Thus the equation o£ the first half 

 of the path is 



f« dv 



+ a 1 / 2 = const. 



Since = when #=1 the constant is zero *. 



Let x De the value of for the line of symmetry and z the 

 corresponding value of x, then (z, %) are the polar coordinates 

 of the vertex and 2% is the angle which the radius through 

 the point of emergence makes with the radius through the 

 origin. These quantities are connected by the equation 



*~£-7r=, < 5 » 



Let ty be the angle at which the radius cuts the path at any 

 point, then 



or . ccv /^N 



T X 



Thus ^= — when x = av; the value of z is therefore given 



by the equation 



*-«<*) (?) 



If e is the angle which the ray makes with the surface 

 when it emerges and U the value of v at the surface of the 

 sphere, i. e. when a=l, we have 



cose = aU (8) 



The value of a for any particular ray may be determined 

 from the equation 



(9) 



a = 



d<9" 



where S' and 6 f now refer to the point on the surface at 

 which the ray emerges. Now S' may be supposed to have 

 been determined as a function of 6' by observation: hence 

 the equations 



«§>• >-* 



* We assume here that the disturbances radiate from the hypocentre, 

 i. e. the point = 0, a?=l. 



2P2 



