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

$p={\frac {n}{a}}\ \mathrm {and} \ q={\frac {n^{2}c^{2}}{2a^{3}}};$

(157)

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

$q={\frac {\pi }{2}}{\frac {r}{\rho }}{\frac {\mu \gamma }{\lambda ^{2}v}}.$

(158)

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

$\rho =si^{2},\,$

(159)

where $i$ is the index of refraction.

On the theory of MacCullagh and Neumann,

$\rho =s\,$

(160)

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

$Z^{2}={\frac {1}{8\pi }},\ {\frac {\pi s\gamma ^{2}}{8\pi }},$

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

$\gamma ={\frac {1}{\sqrt {\pi s}}}Z.$

(161)

$\lambda$ 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