Vortex Theory of Electromagnetic Action. 411 



small. Let us suppose that in the aether,©!^? 6 ^ become known 

 to us as magnetic force ; then this last term expresses the 

 magnetic energy in the field. And if we suppose, further, 

 that 



&7Tp 



since (£> Y — a &c, the magnetic energy of the whole field may 

 be written 



and this may be transformed to 



ifejlf** 7 '* + vvn ' v + ^ * dxdydz 



by an application of Green's theorem whenever f, rj, £ are 

 functions of the same function of a, y, z. Hence, remember- 

 ing that 



V 2 ?=8tt/, *j=F,/=u&c., 



the electrokinetic energy becomes 



$$$ ( Fw + Gv + Hm dx dl J dz ' 

 Since in the kinetic energy of the field we have, on the mole- 

 cular vortex theory, a term involving (oy\ + a>\ + col), we see 

 that if we put for a^ 



a being the rotation which constitutes the magnetic force, and 

 coi the rotation due to the wave of displacement travelling 

 through, we shall introduce a term ao) 1 + ^(o 2 + r yo) 5 , which is 

 the hypothesis from which we started. 



If we start from Mr. Fitzgerald's standpoint, and assume 

 terms in the electrokinetic energy of the form 



to explain the phenomena of the rotation of the plane of pola- 

 rization without reference to theories of molecular vortices, 

 we may show that this assumption also leads to Hall's effect. 

 We wish to transform this term into one in which the kinetic 

 energy is expressed in terms of /, g, h, f, g, h and constants. 

 Then, if T be the kinetic energy, and P, Q, R the components 



