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

 186.] THEOREM III. If $$V$$ is any function of $$x'$$ and $$y'$$, and if $$x'$$ and $$y'$$ are conjugate functions of $$x$$ and $$y$$, then

$\iint\left(\frac{d^{2}V}{dx^{2}}+\frac{d^{2}V}{dy^{2}}\right)dx\ dy=\iint\left(\frac{d^{2}V}{dx'^{2}}+\frac{d^{2}V}{dy'^{2}}\right)dx'\ dy' $,|undefined

integration being between the same limits.

For

$\begin{array}{c} \frac{dV}{dx}=\frac{dV}{dx'}\frac{dx'}{dx}+\frac{dV}{dy'}\frac{dy'}{dx},\\ \\ \frac{d^{2}V}{dx^{2}}=\frac{d^{2}V}{dx'^{2}}\left(\frac{dx'}{dx}\right)^{2}+2\frac{d^{2}V}{dx'dy'}\frac{dx'}{dx}\frac{dy'}{dx}+\frac{d^{2}V}{dy'^{2}}\left(\frac{dy'}{dx}\right)^{2}+\frac{dV}{dx'}\frac{d^{2}x'}{dx^{2}}+\frac{dV}{dy'}\frac{d^{2}y'}{dx^{2}}; \end{array} $;|undefined

and

$\frac{d^{2}V}{dy^{2}}=\frac{d^{2}V}{dx'^{2}}\left(\frac{dx'}{dy}\right)^{2}+2\frac{d^{2}V}{dx'dy'}\frac{dx'}{dy}\frac{dy'}{dy}+\frac{d^{2}V}{dy'^{2}}\left(\frac{dy'}{dy}\right)^{2}+\frac{dV}{dx'}\frac{d^{2}x'}{dy^{2}}+\frac{dV}{dy'}\frac{d^{2}y'}{dy^{2}} $.|undefined

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

$\begin{array}{ll} \frac{d^{2}V}{dx^{2}}+\frac{d^{2}V}{dy^{2}} & =\frac{d^{2}V}{dx'^{2}}\left(\left(\frac{dx'}{dx}\right)^{2}+\left(\frac{dx'}{dy}\right)^{2}\right)+\frac{dV}{dy'^{2}}\left(\left(\frac{dy'}{dx}\right)^{2}+\left(\frac{dy'}{dy}\right)^{2}\right)\\ \\ & =\left(\frac{d^{2}V}{dx'^{2}}+\frac{d^{2}V}{dy'^{2}}\right)\left(\frac{dx'}{dx}\frac{dy'}{dy}-\frac{dx'}{dy}\frac{dy'}{dx}\right) \end{array} $|undefined

Hence

$\begin{array}{ll} \iint\left(\frac{d^{2}V}{dx^{2}}+\frac{d^{2}V}{dy^{2}}\right)dx\ dy & =\iint\left(\frac{d^{2}V}{dx'^{2}}+\frac{d^{2}V}{dy'^{2}}\right)\left(\frac{dx'}{dx}\frac{dy'}{dy}-\frac{dx'}{dy}\frac{dy'}{dx}\right)dx'\ dy'\\ \\ & =\iint\left(\frac{d^{2}V}{dx'^{2}}+\frac{d^{2}V}{dy'^{2}}\right)dx'\ dy' \end{array} $|undefined

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

$\frac{d^{2}V}{dx^{2}}+\frac{d^{2}V}{dy^{2}}+4\pi\rho=0 $|undefined

and we may write the result

$\iint\rho\ dx\ dy=\iint\rho'\ dx'\ dy' $,

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.