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

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

integration being between the same limits.

For

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

and

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

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

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

Hence

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

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

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

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.