Refraction at the Surface of Crystals. 239 



the normal components of B and D, are continuous. Thus 



0=2 + V*H, 0=Z±YkE = Z±YkYB P 

 [Prop. III.], 



0=2 + S£B, = 2 + S£D. 



[Here we have apparently six conditions to determine four 

 unknowns. That the two last conditions are contained in the 

 others may be thus shown : — or, cr/ . . . are perpendicular to 

 the fronts and therefore to the trace. Their components 

 parallel to the face are equal ; for expressing that the foot of 

 the perpendicular from on the trace is in each front 



— l = Sao"=Sa(7 1 '= . . ., 



where a is the vector perpendicular from on the trace. 

 Hence 



Yk<r=Yko- 1 1 =...= v , 



say, where r\ is parallel to the trace. But 



2±SAB = 2 + S*<rE = 2±Si;E = 



by the condition 2 + V&E = 0. Similarly the condition 

 2 + S£D = can be deduced from 2±V*H = 0. We shall, 

 however, use the condition 2 + SH$ = 0.] 



Make now the assumption that the media are non-magnetic, 

 i. e. that B is parallel and proportional to H. In this case 

 both normal (2 + S£B = 0) and tangential (2 + VAH=0) com- 

 ponents of H are continuous, i. e. 



2 + H = 0. 



Also from the condition 2 + V/;E = 0, 



t±YkY P K = Q. 



The first of these expresses that the mechanical system of 

 the enunciation reduces to a couple. The second expresses 

 that the vector moment of the forces about is parallel to 

 the normal of the face, i. e. that the plane of the couple is 

 parallel to the face. This is Hamilton's theorem. 



It will be noticed also that if the conditions of the enuncia- 

 tion are satisfied, the boundary conditions are satisfied. 

 Thus the theorem is always sufficient to give the four 

 unknowns. 



To exhibit the ease of application in particular cases to 

 useful interpretations take the case of two isotropic media. 

 The theorem, it will be noticed, was not arrived at in considering 

 this comparatively simple case, but it seems to me to serve as 

 a much better resume of our knowledge, even in this case, 

 than any other I have come across. 



