Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/390

Equation of Continuity in a Homogeneous Medium.

301.] If we express the components of the electromotive force as the derivatives of the potential $V,$ the equation of continuity

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

(15)

becomes in a homogeneous medium

$r_{1}{\frac {d^{2}V}{dx^{2}}}+r_{2}{\frac {d^{2}V}{dy^{2}}}+r_{3}{\frac {d^{2}V}{dz^{2}}}+2s_{1}{\frac {d^{2}V}{dydz}}+2s_{2}{\frac {d^{2}V}{dzdx}}+2s_{3}{\frac {d^{2}V}{dxdy}}=0.$

(16)

If the medium is not homogeneous there will be terms arising from the variation of the coefficients of conductivity in passing from one point to another.

This equation corresponds to Laplace's equation in an isotropic medium.

302.] If we put

$[rs]=r_{1}r_{2}r_{3}+2s_{1}s_{2}s_{3}-r_{1}s_{1}^{2}-r_{2}s_{2}^{2}-r_{3}s_{3}^{2},$

(17)

and

$[AB]=A_{1}A_{2}A_{3}+2B_{1}B_{2}B_{3}-A_{1}B_{1}^{2}-A_{2}B_{2}^{2}-A_{3}B_{3}^{2},$

(18)

where

$\left.{\begin{array}{rcl}[rs]A_{1}&=&r_{2}r_{2}-s_{1}^{2},\\{}[rs]B_{1}&=&s_{2}s_{3}-r_{1}s_{1},\\--&-&---\end{array}}\right\}$

(19)

and so on, the system $A,B$ will be inverse to the system $r,s,$ and if we make

$A_{1}x^{2}+A_{2}y^{2}+A_{3}z^{2}+2B_{1}yz+2B_{2}zx+2B_{3}xxy=[AB]\rho ^{2},$

(20)

we shall find that

$V={\frac {c}{4\pi }}{\frac {1}{\rho }}$

(21)

is a solution of the equation.

In the case in which the coefficients $T$ are zero, the coefficients $A$ and $B$ become identical with $R$ and $S.$ When $T$ exists this is not the case.

In the case therefore of electricity flowing out from a centre in an infinite homogeneous, but not isotropic, medium, the equipotential surfaces are ellipsoids, for each of which $\rho$ is constant. The axes of these ellipsoids are in the directions of the principal axes of conductivity, and these do not coincide with the principal axes of resistance unless the system is symmetrical.

By a transformation of this equation we may take for the axes of $x,y,z$ the principal axes of conductivity. The coefficients of the forms $s$ and $B$ will then be reduced to zero, and each coefficient