Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/156

Rh Let us confine our attention to the last of these three groups of terms, merely observing that the other groups are essentially positive. By Green’s theorem

the first integral of the second member being extended over the surface of the region and the second throughout the enclosed space. But on the surfaces $$S_{1},S_{2}$$ &c. $$U=0$$, so that these contribute nothing to the surface-integral.

Again, on the surface $$S_0$$, $$\tfrac{dV_{0}}{d\nu}=0$$, so that this surface contributes nothing to the integral. Hence the surface-integral is zero.

The quantity within brackets in the volume-integral also disappears by equation (3), so that the volume-integral is also zero. Hence $$Q$$ is reduced to

Both these quantities are essentially positive, and therefore the minimum value of $$Q$$ is when

or when $$U$$ is a constant. But at the surfaces $$S$$, &c. $$U=0$$. Hence $$U=0$$ everywhere, and $$V_0$$ gives the unique minimum value of $$Q$$.

The quantity $$Q$$ in its minimum form can be expressed by means of Green’s theorem in terms of $$V_{1},V_{2}$$, &c., the potentials of $$S_{1},S_{2}$$, and $$E_{1},E_{2}$$, &c., the charges of these surfaces,

or, making use of the coefficients of capacity and induction as defined in Article 87,

The accurate determination of the coefficients $$q$$ is in general difficult, involving the solution of the general equation of statical electricity, but we make use of the theorem we have proved to determine a superior limit to the value of any of these coefficients.