Light Vibrations in a Magnetic Field. 33 



for the data, it may be interesting to write down shortly the equa- 

 tions which, upon Lorentz's theory as to the cause of the Zeeman 

 effect and Larmor's theory as to dispersion, we may expect to 

 hold.* 



Assuming, then, that the electric displacement in the medium is 

 partly due to the electric force and partly to the displacement oC 

 electrons, we may write for its components in the wave face sup- 

 posed plane and perpendicular to z = 0, 



/=KP+B, = KQ + ey, 



where P, Q are electric force, e the electron, and a', y are coordinates 

 that measure the displacement of this latter. From these and the 

 usual equations connecting magnetic and electric force we get for the 

 medium 



For the motion of the matter, if m be the mass of whatever moves 

 with the electron, matter or effective inertia of ether, and H be the 

 component of the magnetic force normal to the wave, and k the 

 coefficient of restitution of the matter displacement which controls 

 its free period, 



mx + kx = eP + eHy, my + ky = eQ eHx. 



It is to be observed that if we assume the motion periodic these 

 equations can be reduced to the form that Drudef and Leathern^ 

 have shown to lead to results that agree with the observations on 

 the effects of magnetised media on the transmission and reflection of 

 light. 



If we substitute in these equations what obviously solves them, 

 the equations of a right- or left-handed circularly polarised wave of 

 frequency n = pJ2ir t and wave-length X = 2r/j, and whose amplitude 

 in x and y is a, and in P and Q is A, we get 



a(mp 2 eH .p Jc)+eA. = 0. 



If we substitute for k mp Q z , where p is a measure of the fre- 

 quency of the free period corresponding to the forces kx, ky which 

 have been assumed above to control the motion of the electron, we 

 get for the velocity of propagation p/q = V, 



and 



A in (p* p ~)eILp 



* Larmor, * Phil. Trans.,' A, 1897. 

 f ' Wied. Ann.,' 1896. 

 j ' Phil. Trans.,' A, 1897. 



