Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/393

${\begin{array}{rl}W=&\iiint \left\{r_{1}\left.{\frac {\overline {dV}}{dx}}\right|^{2}+r_{2}\left.{\frac {\overline {dV}}{dy}}\right|^{2}+r_{3}\left.{\frac {\overline {dV}}{dz}}\right|^{2}\right.\\&\left.+(p_{1}+q_{1}){\frac {dV}{dy}}{\frac {dV}{dz}}+(p_{2}+q_{2}){\frac {dV}{dz}}{\frac {dV}{dx}}+(p_{3}+q_{3}){\frac {dV}{dx}}{\frac {dV}{dy}}\right\}\;dx\,dy\,dz\\&+\iiint \left\{R_{1}^{2}u^{2}+R_{2}^{2}v^{2}+R_{3}^{2}w^{2}\right.\\&\quad \quad \quad \quad +\left.(P_{1}+Q_{1})vw+(P_{2}+Q_{2})wu+(P_{3}+Q_{3}uv\right\}\;dx\,dy\,dz\\&+\iiint \left(u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}}\right)\;dx\,dy\,dz.\end{array}}$

(28)

Since

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$

(29)

the third term of $W$ vanishes within the limits.

The second term, being the rate of conversion of electrical energy into heat, is also essentially positive. Its minimum value is zero, and this is attained only when $u,\,v,$ and $w$ are everywhere zero.

The value of $W$ is in this case reduced to the first term, and is then a minimum and a unique minimum.

305.] As this proposition is of great importance in the theory of electricity, it may be useful to present the following proof of the most general case in a form free from analytical operations.

Let us consider the propagation of electricity through a conductor of any form, homogeneous or heterogeneous.

Then we know that

(1) If we draw a line along the path and in the direction of the electric current, the line must pass from places of high potential to places of low potential.

(2) If the potential at every point of the system be altered in a given uniform ratio, the currents will be altered in the same ratio, according to Ohm's Law.

(3) If a certain distribution of potential gives rise to a certain distribution of currents, and a second distribution of potential gives rise to a second distribution of currents, then a third distribution in which the potential is the sum or difference of those in the first and second will give rise to a third distribution of currents, such that the total current passing through a given finite surface in the third case is the sum or difference of the currents passing through it in the first and second cases. For, by Ohm's Law, the additional current due to an alteration of potentials is independent of the original current due to the original distribution of potentials.

(4) If the potential is constant over the whole of a closed surface,