Page:On Faraday's Lines of Force.pdf/42

The proofs of these theorems may be found in any work on attractions or electricity, and in particular in Green's Essay on the Application of Mathematics to Electricity. See Arts. 18, 19 of this paper. See also Gauss, on Attractions, translated in Taylor’s Scientific Memoirs.

THEOREM III.

Let $U$ and $V$ be two functions of $x,y,z$, then

$\scriptstyle \iiint U\left({\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}\right)dxdydz=-\iiint \left({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}}\right)dxdydz$

$\scriptstyle =\iiint U\left({\frac {d^{2}U}{dx^{2}}}+{\frac {d^{2}U}{dy^{2}}}+{\frac {d^{2}U}{dz^{2}}}\right)Vdxdydz$;

where the integrations are supposed to extend over all the space in which $U$ and $V$ have values differing from 0. —(Green, p. 10.)

This theorem shews that if there be two attracting systems the actions between them are equal and opposite. And by making $U=V$ we find that the potential of a system on itself is proportional to the integral of the square of the resultant attraction through all space; a result deducible from Art. (30), since the volume of each cell is inversely as the square of the velocity (Arts. 12, 13), and therefore the number of cells in a given space is directly as the square of the velocity.

THEOREM IV.

Let $\alpha ,\beta ,\gamma ,\rho$ be quantities finite through a certain space and vanishing in the space beyond, and let $k$ be given for all parts of space as a continuous or discontinuous function of $xy,z,$ then the equation in $p$

${\frac {d}{dx}}{\frac {1}{k}}\left(\alpha -{\frac {dp}{dx}}\right)+{\frac {d}{dy}}{\frac {1}{k}}\left(\beta -{\frac {dp}{dy}}\right)+{\frac {d}{dz}}{\frac {1}{k}}\left(\gamma -{\frac {dp}{dz}}\right)+4\pi \rho =0$,

has one, and only one solution, in which p is always finite and vanishes at an infinite distance.

The proof of this theorem, by Prof. W. Thomson, may be found in the Cambridge and Dublin Mathematical Journal, Jan. 1848.