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

104 and the superficial characteristic at the surfaces $$S$$,

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

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

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

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

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