Page:Philosophical magazine 21 series 4.djvu/193

The second term to the action on bodies capable of magnetism by induction.

The third and fourth terms to the force acting on electric currents.

And the fifth to the effect of simple pressure.

Before going further in the general investigation, we shall consider equations (12, 13, 14,) in particular cases, corresponding to those simplified cases of the actual phenomena which we seek to obtain in order to determine their laws by experiment.

We have found that the quantities p, q, and $r$ represent the resolved parts of an electric current in the three coordinate directions. Let us suppose in the first instance that there is no electric current, or that p, q, and $r$ vanish. We have then by (9),

${\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}=0,\ {\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}=0,\ {\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}=0.$

(15)

whence we learn that

$\alpha dx+\beta dy+\gamma dz=d\phi$

(16)

is an exact differential of $\phi$, so that

$\alpha ={\frac {d\phi }{dx}},\ \beta ={\frac {d\phi }{dy}},\ \gamma ={\frac {d\phi }{dz}}:$

(17)

$\mu$ is proportional to the density of the vortices, and represents the "capacity for magnetic induction" in the medium. It is equal to 1 in air, or in whatever medium the experiments were made which determined the powers of the magnets, the strengths of the electric currents, &c.

Let us suppose $\mu$ constant, then

${\begin{array}{c}m={\frac {1}{4\pi }}\left({\frac {d}{dx}}(\mu \alpha )+{\frac {d}{dy}}(\mu \beta )+{\frac {d}{dz}}(\mu \gamma )\right)\\\\={\frac {1}{4\pi }}\mu \left({\frac {d^{2}\phi }{dx^{2}}}+{\frac {d^{2}\phi }{dy^{2}}}+{\frac {d^{2}\phi }{dz^{2}}}\right)\end{array}}$

(18)

represents the amount of imaginary magnetic matter in unit of volume. That there may be no resultant force on that unit of volume arising from the action represented by the first term of equations (12, 13, 14), we must have $m=0$, or

${\frac {d^{2}\phi }{dx^{2}}}+{\frac {d^{2}\phi }{dy^{2}}}+{\frac {d^{2}\phi }{dz^{2}}}=0$

(19)

Now it may be shown that equation (19), if true within a given space, implies that the forces acting within that space are such as would result from a distribution of centres of force beyond that space, attracting or repelling inversely as the square of the distance.