342 



EEPOET 1893. 



B, can involve only products of the operators -t-j -tj -^5 and we must 



ax ay dz 



have identically 



dR. 



dx dy 





dz 



These conditions necessitate that R^., R^, R. shall be each null. 



The equations of the magnetically modified medium are therefore 

 restricted to a definite form by the hypothesis that the wave remains 

 strictly transversal. The equations so obtained, of the type 



d'^u . 2 .■ry.d^dudvdw\ 



df^ dx\dx dy dz J 



+ 



d_ 



dt 



dx\dx 

 d , d' 



/ d , ^ d d\ fdv_dw\ 



\ "dx "dy 'dzj \dz dyj' 



contain only terms that are invariantive for transformation of the co- 

 ordinates ; they thus retain the same form when referred to new axes. 

 They therefore satisfy Verdet's ]aw, that the rotatory coefficient for any 

 other direction, which may be taken as the new axes of z, is proportional 

 to the component of the magnetic field in that direction, as they ought 

 to do. 



Not only so, but Verdet's law requires that the equations shall be 

 expressible in terms of invariants of the three vectors 



independently of particular axes of coordinates. Hence P^., Pj,, P. must 

 be so expressible ; and they must be the components of a vector, of the 

 first degree in the first and third of the above vectors, of the second 

 degree in the remaining one. The invariants which can enter are simply 

 the geometrical relations of the figure formed by the above three vectors 

 drawn as rays from an origin. The only possible forms are the scalars 



d"^ I d} d"^ ^ 4- • ^ 4- - ^ du dv dw 

 dx^ dy^ dz^^ " dx "dy ' dz^ dx dy dz' 



and the vectors of which the x components are 



dv dw 

 dz dy 



These combine to give the most general form for P^., represented by the 

 equation 



^'=^(£-^+|-^+£) i^y^-^'^) 



\ dx " dy ' dz) \dz dy ) 



i 



