ON THE ACTION OF MAGNETISM ON LIGHT. 349 



The following expression must then be continuous across the interface in 

 order that the surface integral may be null : 



lK\dz dxj dddt ^dt\dy dzj J ^ 



Now we must have 5^ and Et] continuous to avoid a breach in the medium 

 at the interface, therefore the coefficients of these quantities must also be 

 continuous across the interface ; and as regards the third term either ^^ 

 is continuous, or else its coefficient mast vanish.' The other conditions 

 of continuity do not allow ^C to be continuous ; therefore the third term 

 gives simply the surface condition as to X in the form 



This very slight pressure X is by the previous analysis continuous across 

 the interface ; it is important because it appears in a rotationally active 

 form in the equations ; the formula shows that, at the interface it is 

 proportional to the normal component of the magnetic force. 



It appeai-s therefore that we have here a consistent scheme of equa- 

 tions of reflexion and refraction, without the necessity of condoning any 

 dynamical difficulties in the process, the result being in all respects 

 implicitly involved in the expression for the energy function of the 

 medium. 



The introduction of the circumstance of conduction, or absorption of 

 the energy of vibration, can hardly afiect the analytical form of the 

 boundary conditions as to displacement and traction across an interface. 

 If this be allowed, the problem of reflexion at a magnet will involve the 

 same equations of propagation in the magnet as the above, with the 

 exception that the velocity constant is complex, both the magneto-optic 

 terms and the boundary conditions being otherwise unaltered. How far 

 this theory can compete with others in giving a full explanation of our 

 experimental knowledge would take too long time at present to inquire ; 

 but the considerations to be explained in the latter part of this paper 

 will, I think, give it strong claims to being a correct formulation of the 

 phenomena. 



' The difficulty has been raised that this procedure leaves 5f discontinuous, and 

 so apparently leads to rupture of the media at the interface. The reply to this point 

 would be that if the necessity of the continuity of S( is admitted, the very formula- 

 tion of the problem will involve innate inconsistency, as no other equation of condi- 

 tion can be introduced into the variational equation ; while on the other hand the 

 vanishing of the coefficient of S^, as above, shows that there is no resistance offered 

 to stretching along the normal of the layer of the medium at the interface, and 

 therefore the continuity of 5f will be actually adjusted by a stretching of the inter- 

 facial layer which involves no dynamical consequences. The part of S^ to be thus 

 adjusted is very small, depending on C ; the mode of adjustment would probably be 

 more fully in evidence, if we passed to the hmit through a medium of slight com- 

 pressibility. Precisely the converse mode of adjustment is in fact required in Lord 

 Kelvin's labile aether. In any case we can hold to the axiom (§ 24) that the variation 

 of the Action introduces all the conditions that are really essential. 



