Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/148

every part of the field, instead of confining our attention to the electrified bodies.

The fact that $Q$ attains a minimum value when the components of the electric force are expressed in terms of the first derivatives of a potential, shews that, if it were possible for the electric force to be distributed in any other manner, a mechanical force would be brought into play tending to bring the distribution of force into its actual state. The actual state of the electric field is therefore a state of stable equilibrium, considered with reference to all variations of that state consistent with the actual distribution of free electricity.

Green's Theorem.

100.] The following remarkable theorem was given by George Green in his essay ‘On the Application of Mathematics to Electricity and Magnetism.’

I have made use of the coefficient $K$, introduced by Thomson, to give greater generality to the statement, and we shall find as we proceed that the theorem may be modified so as to apply to the most general constitution of crystallized media.

We shall suppose that $U$ and $V$ are two functions of $x,y,z$, which, with their first derivatives, are finite and continuous within the space bounded by the closed surface $S$.

We shall also put for conciseness

${\frac {d}{dx}}K{\frac {dU}{dx}}+{\frac {d}{dy}}K{\frac {dU}{dy}}+{\frac {d}{dz}}K{\frac {dU}{dz}}=-4\pi \rho$,

(1)

and

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}=-4\pi \rho '$,

(2)

where $K$ is a real quantity, given for each point of space, which may be positive or zero but not negative. The quantities $\rho$ and $\rho '$ correspond to volume-densities in the theory of potentials, but in this investigation they are to be considered simply as abbreviations for the functions of $U$ and $V$ to which they are here equated.

In the same way we may put

$lK{\frac {dU}{dx}}+mK{\frac {dU}{dy}}+nK{\frac {dU}{dz}}=4\pi \sigma$,

(3)

and

$lK{\frac {dV}{dx}}+mK{\frac {dV}{dy}}+nK{\frac {dV}{dz}}=4\pi \sigma '$,

(4)

where $l,m,n$ are the direction-cosines of the normal drawn inwards