Page:On Faraday's Lines of Force.pdf/49

Rh a perfect differential of a function of $$x, y, z.$$ On the principle of analogy we may call $$p_1$$ the magnetic tension.

The forces which act on a mass $$m$$ of south magnetism at any point are

$-m\frac{dp_1}{dx},-m\frac{dp_1}{dy}, and -m\frac{dp_1}{dz}, $

in the direction of the axes, and therefore the whole work done during any displacement of a magnetic system is equal to the decrement of the integral

$Q=\iiint \rho_1p_1dx dy dz$

throughout the system.

Let us now call $$Q$$ the total potential of the system on itself. The increase or decrease of $$Q$$ will measure the work lost or gained by any displacement of any part of the system, and will therefore enable us to determine the forces acting on that part of the system.

By Theorem III. $$Q$$ may be put under the form

$Q=+\frac{1}{4\pi}\iiint (a_1\alpha_1+b_1\beta_1+c_1\gamma_1)dx dy dz$

in which $$\alpha_1\beta_1\gamma_1$$ are the differential coefficients of $$p_1$$ with respect to $$x, y, z$$ respectively.

If we now assume that this expression for $$Q$$ is true whatever be the values of $$\alpha_1, \beta_1, \gamma_1$$ we pass from the consideration of the magnetism of permanent magnets to that of the magnetic effects of electric currents, and we have then by Theorem VII.

$Q=\iiint \left \{ p_1\rho_1-\frac{1}{4\pi}(\alpha_0a_2+\beta_0b_2+\gamma_0c_2)\right \}dx dy dz$

So that in the case of electric currents, the components of the currents have to be multiplied by the functions $$\alpha_0, \beta_0, \gamma_0$$ respectively, and the summations of all such products throughout the system gives us the part of $$Q$$ due to those currents.

We have now obtained in the functions $$\alpha_0, \beta_0, \gamma_0$$, the means of avoiding the consideration of the quantity of magnetic induction which passes through the circuit. Instead of this artificial method we have the natural one of considering the current with reference to quantities existing in the same space with the current itself. To these I give the name of Electro-tonic functions, or components of the Electro-tonic intensity.