Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/273

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186.] THEOREM III. If is any function of and , and if and are conjugate functions of and , then

,

integration being between the same limits.

For

;

and

.

Adding the last two equations, and remembering the conditions of conjugate functions (1), we find

Hence

If is a potential, then, by Poisson's equation

and we may write the result

,

or the quantity of electricity in corresponding portions of two systems is the same if the coordinates of one system are conjugate functions of those of the other.