Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/145

We may now express $Q$ in terms of $u,v,w$ and $V$,

$Q=\iiint K(\left.{\frac {\overline {dV}}{dx}}\right|^{2}+\left.{\frac {\overline {dV}}{dy}}\right|^{2}+\left.{\frac {\overline {dV}}{dz}}\right|^{2})dx\,dy\,dz+\iiint {\frac {1}{K}}(u^{2}+v^{2}+w^{2})dx\,dy\,dz$

$+2\iiint (u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}})dx\,dy\,dz$. (13)

The last term of $Q$ may be written $2M$, where $M$ is the quantity considered in the lemma, and which we proved to be zero when the space is bounded by surfaces, each of which is either equipotential or satisfies the condition of equation (10), which may be written

(4) $lu+mv+nw=0\,\!$.

$Q$ is therefore reduced to the sum of the first and second terms.

In each of these terms the quantity under the sign of integration consists of the sum of three squares, and is therefore essentially positive or zero. Hence the result of integration can only be positive or zero.

Let us suppose the function $V$ known, and let us find what values of $u,v,w$ will make $Q$ a minimum.

If we assume that at every point $u=0$, $v=0$, and $w=0$, these values fulfil the solenoidal conditions, and the second term of $Q$ is zero, and $Q$ is then a minimum as regards the variation of $u,v,w$.

For if any of these quantities had at any point values differing from zero, the second term of $Q$ would have a positive value, and $Q$ would be greater than in the case which we have assumed.

But if $u=0$, $v=0$, and $w=0$, then

(11) ${\color {White}xxx}a=K{\frac {dV}{dx}},{\color {White}xxx}b=K{\frac {dV}{dy}},{\color {White}xxx}c=K{\frac {dV}{dz}}$.

Hence these values of $a,b,c$ make $Q$ a minimum.

But the values of $a,b,c$, as expressed in equations (12), are perfectly general, and include all values of these quantities con sistent with the conditions of the theorem. Hence, no other values of $a,b,c$ can make $Q$ a minimum.

Again, $Q$ is a quantity essentially positive, and therefore $Q$ is always capable of a minimum value by the variation of $a,b,c$. Hence the values of $a,b,c$ which make $Q$ a minimum must have a real existence. It does not follow that our mathematical methods are sufficiently powerful to determine them.

Corollary I. If $a,b,c$ and $K$ are given at every point of space, and if we write