ON THE ACTION OF MAGNETISM ON LIGHT. 345 



for variations of (^,»/,<r) subject to the imposed condition 



dx dy dz 



which expresses that the magnetic flux is constrained to be circuital. 

 This condition is included, in the Lagrangian manner, by adding on to 

 the above variation, -which is equated to zero, a term 



l*t(l4;4D*' 



and determining \ afterwards as a function of position, so that the im- 

 posed condition shall be satisfied. Thus we have 



-IKf- ■■■)*}=<'• 



Changing from the differential coefficients of ?^ to ^l itself by integra- 

 tion by parts, and similarly for Ir) and 11^ in the usual manner, we obtain 

 finally 



L-[c^^ / [f, ^4 f-;^ ^-J^ + 4.^W+ ... + .. . 



+ [('^ ^'» - 5^ dL,-^-K\ll \ld^^ ...+... j = 0. 



This is to be true for all forms of ^|, Sjj, S^ which necessitates equations 

 of the type 



dH , d dXS d dX5 , , d\ ^ 



It \- — — |-47r — = U 



dt^ dy dh dz dg dx 



throughout the medium ; while at an interface, supposed for an instant 

 to be normal to the axis of x, so that (Z, m, }i):=(l, 0, 0), we must have 



dh dg 



continuous. 



Now from the bodily equations we deduce at once 



V2X=0 ; 



therefore X is mathematically the potential function of a mass-distribution 

 on the interlaces only, and so is continuous across them. It follows that 

 the only way of securing the required continuity at an interface is (i) to 

 postulate that r] and lif are continuous across it, owing to the continuous 



structure of the svstem, and that therefore —- and -— are also continuous 

 '' dg dh 



across it ; and (ii) either to postulate that H is constrained to be nuU or 

 else that \ is null all over it. The alternative taken in Maxwell's electro- 

 dynamics is that \ is null everywhere in the field, as in fact representing 

 no observed physical phenomenon ; then the boundary conditions are 

 four, that the tangential components of the magnetic force are continu- 

 ous, as also the tangential components of the electric force. 



