AN INTERRUPTED MEDIUM. 529 



For (a.) we get r=i, a=0, /?= ^-, 7=0 



4 2 6 



For (b.) we get *=*, a=7r, (3=0, 7 = - 



2z 2 ft* 



<r = -, - T\ - 3 P = r sin 

 (a + 6) cos <p * a + 6 



2? 2 a i . 



For (c.) we get J=r, 7 = , /3 = ^ + | 



which includes both the other cases. 



The problem which we have now solved may be considered as the simplest 

 form in which reflection can be produced without refraction. Perhaps, in the 

 application of this theory to light, no instance might be found exactly to corres- 

 pond with this, but it has the advantage of serving as a first approximation, by 

 means of which the more complete solution of the problem can be arrived at by 

 successive steps. 14 will appear in the sequel that the solution of the problem is 

 too laborious to invite us to any thing beyond a second approximation. The 

 results of that approximation will, however, we think, be found sufficiently im- 

 portant in themselves to warrant us in proceeding thus far with the solution of 

 our problem. 



We now proceed to the discussion of our equations. Let 



cos /3 = cos j3' + h, sin /3=sin /3' 



cos 7=cos y+, sin7=si 



(ir cos a) sin (fi = e cos /3' > cos /3' > h . . by (1) 



r,, (i r cos a) sin d) o h 



and eos/3'=L _jL + _ 



Similarly sin/3' = - "^"* + L ..... by (2) 



(i r cos a) sin (i) th 

 cos / 3 = -7^- - + t=j- 



r sin a sin d> te 



