Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/392

have reason to believe that it does not exist in any known substance. It should be found, if anywhere, in magnets, which have a polarization in one direction, probably due to a rotational phenomenon in the substance.

304.] Let us next consider the general characteristic equation of $V,$

${\begin{aligned}{\frac {d}{dx}}\left(r_{1}{\frac {dV}{dx}}+p_{3}{\frac {dV}{dy}}+q_{2}{\frac {dV}{dz}}\right)+{\frac {d}{dy}}\left(q_{3}{\frac {dV}{dx}}+r_{2}{\frac {dV}{dy}}+p_{1}{\frac {dV}{dz}}\right)\qquad \\+{\frac {d}{dz}}\left(p_{2}{\frac {dV}{dx}}+q_{1}{\frac {dV}{dy}}+r_{3}{\frac {dV}{dz}}\right)+4\pi \rho =0,\end{aligned}}$

(24)

where the coefficients of conductivity $p,\,q,\,r$ may have any positive values, continuous or discontinuous, at any point of space, and $V$ vanishes at infinity.

Also, let $a,\,b,\,c$ be three functions of $x,\,y,\,z$ satisfying the condition

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}+4\pi \rho =0;$

(25)

and let

$\left.{\begin{aligned}a&=&r_{1}{\frac {dV}{dx}}+p_{3}{\frac {dV}{dy}}+q_{2}{\frac {dV}{dz}}+u,\\b&=&q_{3}{\frac {dV}{dx}}+r_{2}{\frac {dV}{dy}}+p_{1}{\frac {dV}{dz}}+v,\\c&=&p_{2}{\frac {dV}{dx}}+q_{1}{\frac {dV}{dy}}+r_{3}{\frac {dV}{dz}}+w.\end{aligned}}\right\}$

(26)

Finally, let the triple-integral

${\begin{aligned}&W=\iiint \,\{R_{1}a^{2}+R_{2}b^{2}+R_{3}c^{2}\\&\qquad \qquad +(P_{1}+Q_{1})bc+(P_{2}+Q_{2})ca+(P_{3}+Q_{3})ab\}\;\;dx\,dy\,dz\end{aligned}}$

(27)

be extended over spaces bounded as in the enunciation of Art. 97, where the coefficients $P,Q,R$ are the coefficients of resistance.

Then $W$ will have a unique minimum value when $a,\,b,\,c$ are such that $u,\,v,\,w$ are each everywhere zero, and the characteristic equation (24) will therefore, as shewn in Art. 98, have one and only one solution.

In this case $W$ represents the mechanical equivalent of the heat generated by the current in the system in unit of time, and we have to prove that there is one way, and one only, of making this heat a minimum, and that the distribution of currents $(abc)$ in that case is that which arises from the solution of the characteristic equation of the potential $V.$

The quantity $W$ may be written in terms of equations (25) and (26),