AN INTERRUPTED MEDIUM. 533 



If we substitute these approximations in V we obtain, to two dimensions 



i -TIT y /\ 



2 sin <p a i (0 4) s + 2 sin (p { (0 4) (ir) } = an identical equation : 

 and in (2') 



S< P( 7-y j fr - " ~2~ 



also an identical equation. 



If we proceed one dimension higher we get 



or 



2(0-4) - 



(z-r) (0 2 + ^)-2 / a 2 +2 r a 

 * - ^ - z-J- 



/-a 2 4- - 

 -0-4 + 



which (omitting higher dimensions) is again an identical equation. 

 Also (2') has no term of the next dimension. 



We proceed to make the substitutions in I. II. III. IV. 



fl2 > 



I. hr$ina + e(t rcosa)= (Q $) r a ~- (ir) 



Hence the equation to two dimensions is 



i-r s p- (i-rf-r (i-r} (a?-a~6~+^>) 

 - B + V 



Dividing, we get 



n / \ /i N z'7 r(a 7) 



z -l-.v --- - 



i r + r a (a + 4) + - 



L ^^_- = 



or 



-n 4* . , 4; ' r 



But t= -- -sm0 .-. -, --n 7 approx. 



a + b a + b 6 -4* rl 



2 at 2 a 2-a 



ami r = , cos 



a+b a+b 7 y 



26 * 26 2y-a 

 cos<p.\ = 



