Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/244

Hence, the mutual potential energy of the two currents is

$ii^{\prime }\int \limits _{s^{\prime }}\iint \limits _{S}\left(\mathrm {curl} \ {\frac {\mathbf {ds^{\prime }} }{r}}\mathbf {.dS} \right)$,

which by Stokes's transformation may be written in the form

$ii^{\prime }\int _{s}\int _{s^{\prime }}{\frac {(\mathbf {ds.ds^{\prime }} )}{r}}$.

This expression represents the amount of mechanical work which must be performed against the electro-dynamic ponderomotive forces, in order to separate the two circuits to an infinite distance apart, when the current-strengths are maintained unaltered.

The above potential function has been obtained by considering the ponderomotive forces; but it can now be connected with Faraday's theory of induction of currents. interpreting the expression

$\iint \limits _{S}(\mathbf {B.dS} )$

in terms of lines of force, we see that the potential function represents the product of i into the number of unit-lines of magnetic force due to s′, which pass through the gap formed by the circuit s; and since by Faraday's law the currents induced in s depend entirely on the variation in the number of these lines, it is evident that the potential function supplies all that is needed for the analytical treatment of the induced currents. This was Neumann's discovery.

The electromotive force induced in a circuit s by the motion of other circuits s′, carrying currents i′, is thus proportional to the time-rate of variation of the potential

$i^{\prime }\int _{s}\int _{s^{\prime }}{\frac {\mathbf {ds.ds^{\prime }} }{r}}$;

so that if we denote by a the vector

$i^{\prime }\int _{s^{\prime }}{\frac {\mathbf {ds^{\prime }} }{r}}$,