Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/403

functions of the derivatives of $V,$ the forms of which are given in the equations

$\left.{\begin{aligned}u&=r_{1}X+p_{3}Y+q_{2}Z,\\v&=q_{3}X+r_{2}Y+p_{1}Z,\\w&=p_{2}X+q_{1}Y+r_{3}Z,\end{aligned}}\right\}$

(3)

where $X,Y,Z$ are the derivatives of $V$ with respect to $x,y,z$ respectively.

Let us take the case of the surface which separates a medium having these coefficients of conduction from an isotropic medium having a coefficient of conduction equal to $r.$

Let $X',Y',Z'$ be the values of $X,Y,Z$ in the isotropic medium, then we have at the surface

$V=V',$

(4)

or

$X\,dx+Y\,dy+Z\,dz=X'\,dx+Y'\,dy+Z'\,dz,$

(5)

when

$l\,dx+m\,dy+n\,dz=0.$

(6)

This condition gives

$X'=X+4\pi \sigma l,\quad \quad Y'=Y+4\pi \sigma m,\quad \quad Z'=Z+4\pi \sigma n,$

(7)

where $\sigma$ is the surface-density.

We have also in the isotropic medium

$u'=rX',\quad \quad v'=rY',\quad \quad w'=rZ',$

(8)

and at the boundary the condition of flow is

$u'l+v'm+w'n=ul+vm+wn,$

(9)

or

${\begin{aligned}&\qquad r(lX+mY+nZ+4\pi \sigma )\\&l(r_{1}X+p_{3}Y+q_{2}Z)+m(q_{3}X+r_{2}Y+p_{1}Z)+n(p_{2}X+q_{1}Y+r_{3}Z),\end{aligned}}$

(10)

whence

${\begin{aligned}4\pi \sigma r=(l(r_{1}-r)+mq_{3}+np_{2})X+(lp_{3}+m(r_{2}-r)+nq_{1})Y\qquad \\+(lq_{2}+mp_{1}+n(r_{3}-r))Z.\end{aligned}}$

(11)

The quantity $\sigma$ represents the surface-density of the charge on the surface of separation. In crystallized and organized substances it depends on the direction of the surface as well as on the force perpendicular to it. In isotropic substances the coefficients $p$ and $q$ are zero, and the coefficients $r$ are all equal, so that

$4\pi \sigma =({\frac {r_{1}}{r}}-1)(lX+mY+nZ),$

(12)

where $r_{1}$ is the conductivity of the substance, $r$ that of the external medium, and $l,m,n$ the direction-cosines of the normal drawn towards the medium whose conductivity is $r$.

When both media are isotropic the conditions may be greatly