Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/149

from the surface $S$. The quantities $\sigma$ and $\sigma '$ correspond to superficial densities, but at present we must consider them as defined by the above equations.

Green's Theorem is obtained by integrating by parts the expression

$4\pi M=\iiint K({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}})dx\,dy\,dz$

(5)

throughout the space within the surface $S$.

If we consider ${\tfrac {dV}{dx}}$ as a component of a force whose potential is $V$, and $K$ ${\tfrac {dU}{dx}}$ as a component of a flux, the expression will give the work done by the force on the flux.

If we apply the method of integration by parts, we find

$4\pi M=\iint VK(l{\frac {dU}{dx}}+m{\frac {dU}{dy}}+n{\frac {dU}{dz}})dS$

$-\iiint V({\frac {d}{dx}}K{\frac {dU}{dx}}+{\frac {d}{dy}}K{\frac {dU}{dy}}+{\frac {d}{dz}}K{\frac {dU}{dz}})dx\,dy\,dz$;

(6)

or

$4\pi M=\iint 4\pi \sigma 'VdS+\iiint 4\pi \rho 'V\,dx\,dy\,dz$.

(7)

In precisely the same manner by exchanging $V$ and $T$, we should find

$4\pi M=+\iint 4\pi \sigma \,U\,dS+\iiint 4\pi \rho \,U\,dx\,dy\,dz.$

(8)

The statement of Green's Theorem is that these three expressions for $M$ are identical, or that

$M=\iint \sigma '\,V\,dS+\iiint \rho '\,V\,dx\,dy\,dz=\iint \sigma \,U\,dS+\iiint \rho \,U\,dx\,dy\,dz$

$={\frac {1}{4\pi }}\iiint K({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}})dx\,dy\,dz.$

(9)

Correction of Green's Theorem for Cyclosis.

There are cases in which the resultant force at any point of a certain region fulfils the ordinary condition of having a potential, while the potential itself is a many-valued function of the coordinates. For instance, if

$X={\frac {y}{x^{2}+y^{2}}},Y=-{\frac {x}{x^{2}+y^{2}}},Z=0$

we find $V=\tan ^{-1}{\tfrac {y}{x}}$, a many-valued function of $x$ and $y$, the values of $V$ forming an arithmetical series whose common difference