Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/260

and dielectric capacity may be expressed in a form which will famish a more rigorous test, as not involving extrapolation.

We have seen on page 242 how we may determine numerically the ratio of the two first terms of equation (25). We thus easily get the ratio of the first and last term, which gives

${\frac {{\text{G}}h^{2}}{4}}={\frac {d\log l}{d\log \lambda }}={\frac {\pi {\text{F}}l^{2}h^{2}}{p^{2}}}\cdot$

(29)

In the corresponding equation for a train of waves of the same amplitude and period in vacuo, $l$ becomes $\lambda ,$ ${\text{F}}$ remains the same, and for ${\text{G}}$ we may write ${\text{G}}'.$ This gives

${\frac {{\text{G}}'h^{2}}{4}}={\frac {\pi {\text{F}}\lambda ^{2}h^{2}}{p^{2}}}\cdot$

(30)

Dividing, we get

${\frac {\text{G}}{{\text{G}}'}}={\frac {d\log l}{d\log \lambda }}{\frac {l^{2}}{\lambda ^{2}}}={\frac {d(l^{2})}{d(\lambda ^{2})}}\cdot$

(31)

Now ${\text{G}}'$ is the dielectric elasticity of pure ether. If ${\text{K}}$ is the specific dielectric capacity of the body which we are considering, ${\text{G}}'/{\text{K}}$ is the dielectric elasticity of the body and ${\text{G}}'/2{\text{K}}$ is the potential energy of the body (per unit of volume), due to a unit of ordinary electrostatic displacement. But ${\text{G}}h^{2}/4$ is the potential energy in a train of waves of amplitude $h.$ Since the average square of the displacement is $h^{2}/2,$ the potential energy of a unit displacement such as occurs in a train of waves is ${\text{G}}/2.$ Now in the electrostatic experiment the displacement distributes itself among the molecules so as to make the energy a minimum. But in the case of light the distribution of the displacement is not determined entirely by statical considerations. Hence

${\frac {\text{G}}{2}}\geqq {\frac {{\text{G}}'}{2{\text{K}}}},$

(32)

${\text{K}}\geqq {\frac {{\text{G}}'}{\text{G}}},$

and

${\text{K}}\geqq {\frac {d(\lambda ^{2})}{d(l^{2})}}\cdot$

(33)

It is to be observed that if we should assume for a dispersion-formula

$n^{-2}=a-b\lambda ^{2}{-2},$

(34)

$1/a,$ which is the square of the index of refraction for an infinite wave-length, would be identical with the second member of (33).

Another similarity between the electrical and optical properties of bodies consists in the relation between conductivity and opacity. Bodies in which electrical fluxes are attended with absorption of energy absorb likewise the energy of the motions which constitute