Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/146

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with the condition (1)

then $$V, u, v, w$$ can be found without ambiguity from these four equations.

Corollary II. The general characteristic equation

where $$V$$ a finite quantity of single value whose first derivatives are finite and continuous except at the surface $$S$$, and at that surface fulfil the superficial characteristic

can be satisfied by one value of $$V$$, and by one only, in the following cases.

Case 1. When the equations apply to the space within any closed surface at every point of which $$V = C$$.

For we have proved that in this case $$a, b, c$$ have real and unique values which determine the first derivatives of $$V$$, and hence, if different values of $$V$$ exist, they can only differ by a constant. But at the surface $$V$$ is given equal to $$C$$, and therefore $$V$$ is determinate throughout the space.

As a particular case, let us suppose a space within which $$\rho = 0$$ bounded by a closed surface at which $$V=C$$. The characteristic equations are satisfied by making $$V=C$$ for every point within the space, and therefore $$V=C$$ is the only solution of the equations.

Case 2. When the equations apply to the space within any closed surface at every point of which $$V$$ is given.

For if in this case the characteristic equations could be satisfied by two different values of $$V$$, say $$V$$ and $$V'$$, put $$U = V-V'$$, then subtracting the characteristic equation in $$V'$$ from that in $$V$$, we find a characteristic equation in $$U$$. At the closed surface $$U=0$$ because at the surface $$V = V'$$, and within the surface the density is zero because $$\rho=\rho'$$. Hence, by Case 1, $$U=0$$ throughout the enclosed space, and therefore $$V = V'$$ throughout this space.