268 Mr. W. Williams on the Relation of Dimensions 



7. Flux through vortex = Potential energy f magnetic 

 shell. 



8. The action of a vortex filament upon the surrounding 

 liquid is determined by the quantities L, M, 1ST. These cor- 

 respond to the components F, G, H of electromagnetic 

 momentum (A). 



Thus, when fi is of the dimensions of density and magnetic 

 energy kinetic, the interpretations of the dimensional formulae 

 of electromagnetic quantities are identical with those of the 

 corresponding quantities in the case of vortex motion. 



In the Phil. Mag. vol. xxxi. p. 149, Prof. J. J. Thomson 

 develops a method of representing electromagnetic effects by 

 means of tubes of electrostatic induction distributed throughout 

 the field. The axes of these tubes coincide with the axes of 

 electric displacement, and the tubes terminate normally upon 

 the surfaces of charged bodies. By means of these tubes it is 

 possible to picture the changes going on when electricity 

 passes through electrolytes and conductors, and when changes 

 are produced in the electromagnetic field. If, now, we suppose 

 these tubes of electrostatic induction to be vortex filaments, 

 this method of representing electromagnetic phenomena 

 becomes identical with that indicated by the interpretations 

 above given to the dimensional formulae. 



The two rotational theories of electromagnetism which have 

 been hitherto put forward are discussed by Mr. Oliver Heavi- 

 side (Elec. Jan. 23, 1891). The first of these is that already 

 deduced above from dimensional considerations, by supposing 

 fjb to be of the dimensions of density. Here the axis of ro- 

 tation at any point is the axis of the electric displacement ; 

 the angular velocity is the current-density ; the velocity is 

 magnetic force ; the torque called into play when the angular 

 velocity of a vortex changes is the electric force ; the terminal 

 torque, the source of the disturbance, is the voltage ; the 

 relative angular displacements of the vortices when their 

 velocities change is the electric displacement ; the linear 

 momentum of the irrotationally moving liquid is magnetic in- 

 duction or intensity of magnetization ; the angular momentum 

 is the vector potential ; the strength of a vortex is the strength 

 of the current ; specific resistance is a quantity of the nature 

 of viscosity, and k~ l of the nature of rigidity. 



In the other system things are inverted. Here k is the 

 density of the medium, electrical energy is kinetic, and the axes 

 of the vortices coincide with the directions of the magnetic dis- 

 placements. The interpretations of the formula? of the various 

 quantities are as below. 



