Page:On Faraday's Lines of Force.pdf/41

Rh These equations enable us to deduce the distribution of the current of electricity whenever we know the values of $$\alpha, \beta, \gamma,$$ the magnetic intensities. If $$\alpha, \beta, \gamma$$ be exact differentials of a function of $$x, y, z$$ with respect to $$x, y$$ and $$z$$ respectively, then the values of $$a_2, b_2, c_2,$$ disappear; and we know that the magnetism is not produced by electric currents in that part of the ﬁeld which we are investigating. It is due either to the presence of permanent magnetism within the ﬁeld, or to magnetizing forces due to external causes.

We may observe that the above equations give by differentiation

$\frac{da_2}{dx}+\frac{db_2}{dy}+\frac{dc_2}{dz}=0$

which is the equation of continuity for closed currents. Our investigations are therefore for the present limited to closed currents; and we know little of the magnetic effects of any currents which are not closed.

Before entering on the calculation of these electric and magnetic states it may be advantageous to state certain general theorems, the truth of which may he established analytically.

THEOREM I.

The equation

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

(where $$V$$ and $$\rho$$ are functions of $$x,y,z$$ never inﬁnite, and vanishing for all points at an inﬁnite distance), can be satisﬁed by one, and only one, value of $$V$$. See Art. (17) above.

THEOREM II.

The value of $$V$$ which will satisfy the above conditions is found by integrating the expression

$\iiint \frac{\rho dx dy dz}{(\overline{x-x'}\vert^2+\overline{y-y'}\vert^2+\overline{z-z'}\vert^2)^{\frac{1}{2}}}$,|undefined

where the limits of $$x, y, z$$ are such as to include every point of space where $$\rho$$ is ﬁnite.