% 2o8 Prof. A. McAulay on Reflexion and 



(taken as the point called above) at any instant be, in order, 



HXT I XT I XT // XT If 

 , -til , -H-2 j -"-I 5 -U-2 ? 



and the corresponding vectors of ray-velocity 



P, Pi', P2', pi" P 2 " 



and similarly for the other vectors involved. 



The theorem is true only for the usual form of electro- 

 magnetic theory, i. e. for the case when both media are 

 assumed n on- magnetic. It is to be remembered that this form 

 is apparently as satisfactory as the more general form in 

 explaining all known optical phenomena. The following is 

 the theorem : — 



If we suppose mechanical forces 



HXT I XT f XT II XT If 

 , — ill J ™*8>% 5 -n-2 



to act at points ivhose vector coordinates are 



p, Pi, P-2, Pi", p2 f , 



they will reduce to a couple whose plane is parallel to the face. 

 [See figures 6 and 7 below.] 



Before giving the proof some remarks are desirable. If 

 the incident wave-front and the two wave-surfaces (centre at 

 point of incidence) be given, by Huyghens's construction 

 we know the five wave-fronts. These fronts, supposed drawn 

 touching the wave-surfaces at the appropriate points, all 

 contain the line of intersection of the incident front with the 

 face, or as we shall call it the trace of that front on the face. 

 Since the fronts are known the directions of the M.M.F.'s 

 are known, and it only remains to find their magnitudes. 

 These M.M.F.'s, drawn as described in the enunciation, 

 are in the corresponding fronts, and therefore all intersect 

 one line, the trace. 



Thus when the circumstances of incidence are given the 

 four magnitudes of the M.M.F.'s of the refracted and reflected 

 waves are required. The theorem gives five conditions, but 

 one of them is always satisfied by reason of the five vectors 

 all intersecting one line. The theorem is in every case not 

 only necessary, but sufficient to determine the unknown 

 quantities. 



In the proof about to be given, whenever 2± ( ) is 

 applied to a sum, referring to the five rays, the sign + is to 

 be understood to refer to one medium, and the sign — to the 

 other. 



Let k be the unit normal of the face. The boundary con- 

 ditions are that the tangential components of H and E, and 



