518 The Electromagnetic Theory of Light [CH. xvm 



where, in order that the equations of propagation may be satisfied, we must 

 have 



sin 3 cos 



s / fo 



VZ, 



(554). 



The boundary conditions must be satisfied for all values of y and t. Since 

 y and t enter only through exponentials in the different waves, this requires 



that we have 



KJ sin #j = tf 2 sin 2 = tt 3 sin 3 ..................... (555), 



*iK =& =*K ........................... (556). 



From (556) we must have K^ = /c s , and hence from (555), sm# 1 

 Since lt and O s must not be identical, we must have 0, = TT 6%. Thus 

 The angle of incidence is equal to the angle of reflection. 

 We further have, from equations (555) and (556), 



sin d 



where n is the index of refraction on passing from medium 1 to medium 2, 

 so that the sine of the angle of incidence is equal to n times the sine of the 

 angle of refraction. 



Thus the geometrical laws of reflection and refraction can be deduced at 

 once from the electromagnetic theory. These laws can, however, be deduced 

 from practically any undulatory theory of light. A more severe test of a 

 theory is its ability to predict rightly the relative intensities of the incident, 

 reflected and refracted waves. 



595. The first boundary condition to be satisfied is the continuity, at 

 the boundary, of the ^-components of electric force. Thus we must have 



Z' + Z'" = Z" .............................. (558). 



Similarly, the magnetic boundary conditions lead to the relations 



) = /*>", 



On substituting from equations (552), (553) and (554), these become 

 respectively, 



2 Z'' ............ (559), 



A / l cos 0, (Z' - Z'") = A /^ cos 2 Z" ............ (560). 



/*! V /^ 2 



From equation (557), equation (559) is at once seen to be identical with 

 equation (558), so that all the boundary conditions are satisfied if 





