582 Mr. H. Bateman on the Velocity of an 



The result indicates that the solution of the integral equation 



J a VV—^ 



is given by 



1 dx 

 This formula enables us to determine as a function 



of v . Let xd V 



\ d i=^ ( i5 > 



Then by eliminating rj from this equation and 



(16) 



' "GO* 



we obtain u as a function of .a?. 



It is important to notice that although there may be more 

 than one value of S for a given value of 6, there is only one 

 value of x or /( a ) f° r a given value of a. ior a is deter- 

 mined by the equation 



cose = aU, 



and the angle of emergence e is equal to the angle which the 

 path makes with the surface at the epicentre, thus a deter- 

 mines the ray uniquely and therefore determines f(u) 

 uniquely. 



we get 



Now 



therefore 



1 dr 9 c » 



= ir sin (^)^j\(£a)/(«Kfl 

 o o 



__ _ 2 d poo ^P 



7T ^J C08 & • ^J J o(£«)/c«y«. 



00 



J COS (f,)J (£a)^= -^~ z « 2 >^| 

 = a 2 ^]' 



1 dx _ __ 2 tf_ f A a _l^L 

 This was the method by which tlie result was first obtained. 



