Page:Philosophical magazine 23 series 4.djvu/114

 vibration whose periodic time is $$\tfrac{2\pi}{n}$$, and wave-length $$\tfrac{2\pi}{p}=\lambda$$, propagated in the direction of $$z$$ with a velocity $$\tfrac{n}{p}=v$$, while the plane of the vibration revolves about the axis of $$z$$ in the positive direction so as to complete a revolution when $$z=\tfrac{2\pi}{q}$$.

Now let us suppose $$c^2$$ small, then we may write

and remembering that $$c^{2}=\tfrac{1}{4\pi}\tfrac{r}{\rho}\mu\gamma$$, we find

Here $$r$$ is the radius of the vortices, an unknown quantity. $$\rho$$ is the density of the luminiferous medium in the body, which is also unknown; but if we adopt the theory of Fresnel, and make $$s$$ the density in space devoid of gross matter, then

where $$i$$ is the index of refraction.

On the theory of MacCullagh and Neumann,

in all bodies.

$$\mu$$ is the coefficient of magnetic induction, which is unity in empty space or in air.

$$\gamma$$ is the velocity of the vortices at their circumference estimated in the ordinary units. Its value is unknown, but it is proportional to the intensity of the magnetic force.

Let Z be the magnetic intensity of the field, measured as in the case of terrestrial magnetism, then the intrinsic energy in air per unit of volume is

where $$s$$ is the density of the magnetic medium in air, which we have reason to believe the same as that of the luminiferous medium. We therefore put

$$\lambda$$ is the wave-length of the undulation in the substance. Now if $$\Lambda$$ be the wavelength for the same ray in air, and $$i$$ the index