348 Messrs. Edser and Senior on the Diffraction of 



Let AC (fig. 1) be a plane surface o£ separation between 

 two media, of which the lower is the denser ; and let AB be 

 the trace of one of a series of plane waves, incident in the 

 direction BC on the surface. In order that there should be 

 a resultant disturbance at P, a point in the upper medium, 



Fie-. 1. 



wavelets originating in the immediate neighbourhood of 

 some point A must reach P in the same phase. Let NAN 7 

 be the normal to the surface at A. Then if i is the angle of 

 incidence, this will be equal to the angle BAC which the 

 incident wave makes with the surface AC. Let the angle 

 PAN be equal to 6, and let the upper medium be the free 

 aether, while the refractive index of the lower medium is 

 equal to fi. Then in order that wavelets from A and E. two 

 neighbouring points on the surface AC, should reach P in 

 the same phase, we must have 



AP = /*.DE + EP, (1) 



where DE is the perpendicular distance of the point E from 

 the plane wave AB. 



Let AP = d, while AE = 8. Then 



(EP) 2 = d 2 -f-S 2 -2^sin0 



DE = Ssin/, 



d = (d 2 + S 2 - 2dS sin 6)^ + ^8 sin i, 



and from (1), 



.". sin 6 = /j, sin i — ^-, (fx 2 sin 2 i — 1) ', 



(2) 



If S is of very small magnitude, and the point P is at an 

 appreciable distance from A, d will be very great in com- 

 parison with 8, and the second term to the right of (2) will 

 be negligibly small. In these circumstances 



sin6=fismi, (3) 



