Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/144

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}+4\pi \rho =0$;

(7)

and the superficial characteristic at the surfaces $S$,

$l(K_{1}{\frac {dV_{1}}{dx}}-K_{2}{\frac {dV_{2}}{dx}})+m(K_{1}{\frac {dV_{1}}{dy}}-K_{2}{\frac {dV_{2}}{dy}})+n(K_{1}{\frac {dV_{1}}{dz}}-K_{2}{\frac {dV_{2}}{dz}})+4\pi \rho =0$

(8)

$K$ being a quantity which may be positive or zero but not negative, given at every point of space.

Finally, let $8\pi Q$ represent the triple integral

$8\pi Q\iiint {\frac {1}{K}}(a^{2}+b^{2}+c^{2})\,dx\,dy\,dz$,

(9)

extended over a space bounded by surfaces, for each of which either

$V$ = constant,

or

$la+mb+nc=Kl{\frac {dV}{dx}}+Km{\frac {dV}{dy}}+Kn{\frac {dV}{dz}}=q$,

(10)

where the value of $q$ is given at every point of the surface; then, if $a,b,c$ be supposed to vary in any manner, subject to the above conditions, the value of $Q$ will be a unique minimum, when

$a=K{\frac {dV}{dx}},{\color {White}xxxx}b=K{\frac {dV}{dy}},{\color {White}xxxx},c=K{\frac {dV}{dz}}$.

(11)

Proof.

If we put for the general values of $a,b,c,$

$a=K{\frac {dV}{dx}}+u,{\color {White}xxxx}b=K{\frac {dV}{dy}}+v,{\color {White}xxxx},c=K{\frac {dV}{dz}}+w$;

(12)

then, by substituting these values in equations (5) and (7), we find that $u,v,w$ satisfy the general solenoidal condition

(1) ${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$.

We also find, by equations (6) and (8), that at the surfaces of discontinuity the values of $u_{1},v_{1},w_{1}$ and $u_{2},v_{2},w_{2}$ satisfy the superficial solenoidal condition

(2) $l(u_{l}-u_{2})+m(v_{1}-v_{2})+n(w_{1}-w_{2})=0\,\!$

.

The quantities $u,v,w$, therefore, satisfy at every point the solenoidal conditions as stated in the preceding lemma.