ON THE ACTION OF MAGNETISM ON LIGHT. 341 



6. The considerations just given bring together evidence of various 

 kinds, that for ordinary media the ^2 rotatory term is to be taken as very 

 subordinate to the k^ term. Using the k^ term alone, the equations of 

 propagation of a wave travelling along the lines of magnetic force (now 

 leaving out dispersion) will be of the form 



dt^ dz"^ ^ dzHt' 



d^v 2 '^^^ _ '^^'^ 



^d^~°' dz^ "'d^dt 



Let us now attempt to deduce general equations of propagation along 

 any direction. These clearly must involve three constants k^, k^, k^ pre- 

 portional to the components of the magnetic field along the axes of co- 

 ordinates. For they must lead to the experimental law that the rotation 

 for any direction of the wave is proportional to the component in that 

 direction of the intensity of the magnetic field; in particular this law 

 must be satisfied for the directions of the axes of coordinates. Further, 

 the vibrations may be assumed to remain purely transverse, so that we 

 must have no compression of the medium ; and therefore the condition 



du dv ,dw ^ 



dx dy dz 



is to remain satisfied after the rotational terms are added to the equations. 

 The equations, then, must for an isotropic medium conform to the 

 general type 



d'^u A_2 1 "R d fdu dv dw\,d-p 



dt^ dx\dx dy dz J dt " 



Xi 



in which P^., P^, P. are linear functions of the second spacial difiFerential 

 coeflBcients of the displacements ; and transversality of the unmodified 



wave requires A+B=0. Further when w, ,- and — are all null, these 



oiB dy 



functions must reduce to the forms 



d^v _ d^u f^ 



Hence 



d^v d'^w 



'dz'' "' % 



^x ='-"^^79 ~ S 37:2 "^ ^'^ 



in which Q^ involves only products of — , -— , — . 



dx dy dz 



Transversality of the disturbed wave requires 



dx dy dz ' 

 hence changing the expression for P^ to 



P.= 



f . d , . (i , . <i\ fdv _dw \ I -p 



\ ^ dx ^ dy ' dz) \dz dy J 



