that, in fact, "the physical lines of magnetic force are currents."
The comparison with the lines of flow of a liquid is applicable to electric as well as to magnetic lines of force. In this case the vector which corresponds to the velocity of the fluid is, in free aether, the electric force E. But when different dielectrics are present in the field, the electric force is not a circuital vector, and therefore cannot be represented by lines of force; in fact, the equation
,
is now replaced by the equation
,
where ε denotes the specific inductive capacity or dielectric constant at the place (x, y, z). It is, however, evident from this equation that the vector εE is circuital; this vector, which will be denoted by D, bears to E a relation similar to that which the magnetic induction B bears to the magnetic force H. It is the vector D which is represented by Faraday's lines of electric force, and which in the hydrodynamical analogy corresponds to the velocity of the incompressible fluid.
In comparing fluid motion with electric fields it is necessary to introduce sources and sinks into the fluid to correspond to the electric charges; for D is not circuital at places where there. is free charge. The magnetic analogy is therefore somewhat the simpler.
In the latter half of his memoir Maxwell discussed how Faraday's "electrotonic state" might be represented in mathematical symbols. This problem be solved by borrowing from Thomson's investigation of 1847 the vector a, which is defined in terms of the magnetic induction by the equation
;
if, with Maxwell, we call a the electrotonic intensity, the. equation is equivalent to the statement that "the entire electrotonic intensity round the boundary of any surface measures the number of lines of magnetic force which pass
