Page:On Faraday's Lines of Force.pdf/48

202 For, if we put $$a_1$$ in the form

$\frac{d\beta_0}{dz}-\frac{d\gamma_0}{dy}+\frac{dV}{dx}$,

and treat $$b_1$$ and $$c_1$$ similarly, then we have by integration by parts through inﬁnity, remembering that all the functions vanish at the limits,

$Q=-\iiint \left \{V\left(\frac{d\alpha_1}{dx}+\frac{d\beta_1}{dy}+\frac{d\gamma_1}{dz}\right)+\alpha_0\left(\frac{d\beta_1}{dz}-\frac{d\gamma_1}{dy}\right) +\beta_0\left(\frac{d\gamma_1}{dx}-\frac{d\alpha_1}{dz}\right) +\gamma_0\left(\frac{d\alpha_1}{dy}-\frac{d\beta_1}{dx}\right) \right \} dxdydz ,$

or $Q=+\iiint \{ (4\pi V\rho')-(\alpha_0 a_2+\beta_0 b_2 +\gamma_0 c_2)\} dxdydz $,

and by Theorem III.

$\iiint V\rho' dxdydz = \iint p\rho dxdydz$,

so that ﬁnally

$Q=\iiint{4\pi p\rho-(\alpha_0 a_2+\beta_0 b_2 +\gamma_0 c_2)}dxdydz$.

If $$a_1 b_1 c_1 $$ represent the components of magnetic quantity, and $$\alpha_1 \beta_1 \gamma_1$$ those of magnetic intensity, then $$\rho$$ will represent the real magnetic density, and p the magnetic potential or tension. $$a_2 b_2 c_2 $$ will be the components of quantity of electric currents, and $$\alpha_0 \beta_0 \gamma_0$$ will be three functions deduced from $$a_1 b_1 c_1 $$, which will be found to be the mathematical expression for Faraday's Electrotonic state.

Let us now consider the bearing of these analytical theorems on the theory of magnetism. Whenever we deal with quantities relating to magnetism, we shall distinguish them by the suffix (1). Thus $$a_1 b_1 c_1 $$ are the components resolved in the directions of $$x, y, z$$ of the quantity of magnetic induction acting through a given point, and $$\alpha_1 \beta_1 \gamma_1$$ are the resolved intensities of magnetization at the same point, or, what is the same thing, the components of the force which would be exerted on a unit south pole of a magnet placed at that point without disturbing the distribution of magnetism.

The electric currents are found from the magnetic intensities by the equations

$a_2=\frac{d\beta_1}{dz}-\frac{d\gamma_1}{dy}$ &c.

When there are no electric currents, then

$\alpha_1 dx+\beta_1 dy+\gamma_1 dz = dp_1$,