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and the electromotive intensity, by Ohm's law, which states that ${\mathfrak {K}}=C{\mathfrak {E}}$, becomes ${\mathfrak {K}}^{2}/C$, where C is the specific conductivity. But this is the energy appearing as heat in the circuit per unit volume according to Joule's law. If we sum up the quantity in (3) thus transformed, for the whole space within a closed surface the integral of the first term can be integrated by parts, and we find that it consists of two terms — one an expression depending on the surface alone to which each part of the surface contributes a share depending on the values of the electromotive and magnetic intensities at that part, the other term being the change per second in the magnetic energy (that is, the second term of (2)) with a negative sign. The integral of the second term of (3) is the total amount of heat developed in the conductors within the surface per second. We have then the following result.

The change per second in the electric energy within a surface is equal to a quantity depending on the surface — the change per second in the magnetic energy — the heat developed in the circuit.

Or rearranging.

The change per second in the sum of the electric and magnetic energies within a surface together with the heat developed by currents is equal to a quantity to which each element of the surface contributes a share depending on the values of the electric and magnetic intensities at the element. That is, the total change in the energy is accounted for by supposing that the energy passes in through the surface according to the law given by this expression.

On interpreting the expression it is found that it implies that the energy flows as stated before, that is, perpendicularly to the plane containing the lines of electric and magnetic force, that the amount crossing unit area per second of this plane is equal to the product

${\frac {\mathrm {electromotive\ intensity\times magnetic\ intensity\times sine\ included\ angle} }{4\pi }}$

while the direction of flow is given by the three quantities, electromotive intensity, magnetic intensity, flow of energy, being in right-handed order.

It follows at once that the energy flows perpendicularly to the fines of electric force, and so along the equipotential surfaces where these exist. It also flows perpendicularly to the lines of magnetic force, and so along the magnetic equipotential surfaces where these exist. If both sets of surfaces exist their lines of intersection are the lines of flow of energy.

The following is the full mathematical proof of the law:—

The energy of the field may be expressed in the form (Maxwell's 'Electricity', vol. ii, 2nd ed., p. 253)

${\frac {1}{2}}\iiint (Pf+Qg+Rh)dx\ dy\ dz+{\frac {1}{8\pi }}\iiint (a\alpha +b\beta +c\gamma )dx\ dy\ dz$

the first term the electrostatic, the second the electromagnetic energy.