Vortex Theory of Electromagnetic Action. 405 



■d/dZ d?\ -d/dr, d^\ I 



+r, dd{dl—d7v) + K deKdx~Ty) h ■ (17) 



where 



d d , n cl d 



and from this he has obtained an expression for the rotation 

 of the plane of polarization of light in the field. 



I wish to put this in a somewhat more general form, in order 

 to" develop the connexion between it and the electromotive 

 force discovered experimentally by Mr. E. H. Hall (Phil. Mag. 

 March 1880), which leads, as Prof. Rowland has shown, to 

 the same expression for the rotation of the plane of polariza- 

 tion of the light. Let us suppose, then, that forces X, Y, Z 

 are applied per unit mass at each point of our medium, and 

 also that we may consider it as incompressible. Then, if W 

 be the potential energy and B the coefficient of rigidity, 



To obtain the equations of motion we follow the method 

 adopted by Fitzgerald (" On the Electromagnetic Theory of 

 the Reflexion and Refraction of Light," Phil. Trans. 1880).; 

 and from the condition that 



SJ(T+W)<ft=0 ; 



we arrive finally at the equations 



^c|(f-$)=pX-B W c. . . (19) 



Now we have seen already that, on the molecular vortex 

 theory, if «', /3', y f denote the magnetic force, /, g } It the elec- 

 tric displacement, and F, G, H the components of vector po- 

 tential due to the displacements considered, 



