Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/168

136 in the language of Faraday, the number of lines of magnetic induction, reckoned algebraically, which pass through the shell from the negative to the positive side, lines which pass through the shell in the opposite direction being reckoned negative.

Remembering that the shell does not belong to the magnetic system to which the potential $$V$$ is due, and that the magnetic force is therefore equal to the magnetic induction, we have

$$a=-\frac{dV}{dx}, \quad b=-\frac{dV}{dy}, \quad c=-\frac{dV}{dz},$$

and we may write the value of $$M$$,

$$M = -\phi N$$.

If $$\delta x_1$$ represents any displacement of the shell, and $$X_1$$ the force acting on the shell so as to aid the displacement, then by the principle of conservation of energy,

$$X_1 \delta x_1 + \delta M = 0$$,

or $$X = \phi \frac{\delta N}{\delta x}$$.

We have now determined the nature of the force which corresponds to any given displacement of the shell. It aids or resists that displacement accordingly as the displacement increases or diminishes $$N$$, the number of lines of induction which pass through the shell.

The same is true of the equivalent electric circuit. Any displacement of the circuit will be aided or resisted accordingly as it increases or diminishes the number of lines of induction which pass through the circuit in the positive direction.

We must remember that the positive direction of a line of magnetic induction is the direction in which the pole of a magnet which points north tends to move along the line, and that a line of induction passes through the circuit in the positive direction, when the direction of the line of induction is related to the direction of the current of vitreous electricity in the circuit as the longitudinal to the rotational motion of a right-handed screw. See Art. 23.

490.] It is manifest that the force corresponding to any displacement of the circuit as a whole may be deduced at once from the theory of the magnetic shell. But this is not all. If a portion of the circuit is flexible, so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into