Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/333

each other if they were traversed by electric currents proportional to the circulations.

There is, however, an important difference between the two cases, which was subsequently discussed by W. Thomson, who pursued the analogy in several memoirs.^[1] In order to represent the magnetic field by a conservative dynamical system, we shall suppose that it is produced by a number of rings of perfectly conducting material, in which electric currents are circulating; the surrounding medium being free aether. Now any perfectly conducting body acts as an impenetrable barrier to lines of magnetic force; for, as Maxwell showed,^[2] when a perfect conductor is placed in a magnetic field, electric currents are induced on its surface in such a way as to make the total magnetic force zero throughout the interior of the conductor.^[3] Lines of force are thus deflected by the body in the same way as the lines of flow of an incompressible fluid would be deflected by an obstacle of the same form, or as the lines of flow of electric current in a uniform conducting mass would be deflected by the introduction of a body of this form and of infinite resistance. If, then, for simplicity we consider two perfectly conducting rings carrying currents, those lines of force which are initially linked with a ring cannot escape from their entanglement, and new lines cannot become involved in it. This implies that the total number of lines of magnetic force which pass through the aperture of each ring is invariable. If the coefficients of self and mutual induction of the rings are denoted by L₁, L₂, L₁₂, the electrokinetic energy of the system may be represented by

$T={\tfrac {1}{2}}(L_{1}i_{1}^{2}+2L_{12}i_{1}i_{2}+L_{2}i_{2}^{2})$,

where i₁, i₂, in denote the strengths of the currents; and the condition that the number of lines of force linked with each circuit is to be invariable gives the equations

$L_{1}i_{1}+L_{12}i_{2}={\text{ constant}}$,

$L_{12}i_{1}+L_{2}i_{2}={\text{ constant}}$.

1. Thomson's Reprint of Papers in Elect. and Mag., §§ 573, 733, 751 (1870-1872).
2. Maxwell's Treatise on Elect. and Mag., § 664.
3. For this reason W. Thomson called a perfect conductor an ideal extreme diamagnetic.