Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/157

To determine a superior limit to the coefficient of capacity $q_{11}$, make $V_{1}=1$, and $V_{2},V_{3}$, &c. each equal to zero, and then take any function $V$ which shall have the value 1 at $S_{1}$, and the value 0 at the other surfaces.

From this trial value of $V$ calculate $Q$ by direct integration, and let the value thus found be $Q'$. We know that $Q'$ is not less than the absolute minimum value $Q$, which in this case is ${\tfrac {1}{2}}q_{11}$.

Hence

$q_{11}\ \mathrm {is\ not\ greater\ than} \ 2Q'$

(11)

If we happen to have chosen the right value of the function $V$, then $q_{11}=2Q'$, but if the function we have chosen differs slightly from the true form, then, since $Q$ is a minimum, $Q'$ will still be a close approximation to the true value.

Superior Limit of the Coefficients of Potential.

We may also determine a superior limit to the coefficients of potential defined in Article 86 by means of the minimum value of the quantity $Q$ in Article 98, expressed in terms of $a,b,c$.

By Thomson’s theorem, if within a certain region bounded by the surfaces $S_{0},S_{1}$ &c. the quantities $a,b,c$ are subject to the condition

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

(12)

and if

$la+mb+nc=q\,$

(13)

be given all over the surface, where $l,m,n$ are the direction-cosines of the normal, then the integral

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

(14)

is an absolute and unique minimum when

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

(15)

When the minimum is attained $Q$ is evidently the same quantity which we had before.

If therefore we can find any form for $a,b,c$ which satisfies the condition (12) and at the same time makes

$\iint q\ dS_{1}=E_{1},\ \iint q\ dS_{2}=E_{2}$ &c.;

(16)

and if $Q''$ be the value of $Q$ calculated by (14) from these values of $a,b,c$, then $Q''$ is not less than

${\frac {1}{2}}\left(E_{1}^{2}p_{11}+E_{2}^{2}p_{22}\right)+E_{1}E_{2}p_{12}.$

(17)