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

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.

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

and if

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

is an absolute and unique minimum when

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

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