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

100.] 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

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

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

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

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