Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/206

to derive from the same equations the laws of the dispersion of colors, we shall not be able to obtain an equally definite result, since the quantities ${\text{A, B,}}$ etc., and ${\text{A}}',{\text{B}}',$ etc., are unknown functions of the period. If, however, we make the assumption, which is hardly likely to be strictly accurate, but which may quite conceivably be not far removed from the truth, that the manner in which the general or average flux in any small part of the medium distributes itself among the molecules and intermolecular spaces is independent of the period, the quantities ${\text{A, B,}}$ etc., and ${\text{A}}',{\text{B}}',$ etc., will be constant, and we obtain a very simple relation between V and p, which appears to agree tolerably well with the results of experiment.

If we set

${\text{H}}={\frac {{\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta }{\rho ^{2}}},$

(16)

and

${\text{H}}'={\frac {{\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta }{\rho ^{2}}},$

(17)

our general equation (11) becomes

${\text{V}}^{2}={\frac {\text{H}}{2\pi }}-{\frac {2\pi {\text{H}}'}{p^{2}}},$

(18)

where ${\text{H}}$ and ${\text{H}}'$ will be constant for any given direction of oscillation, when ${\text{A, B,}}$ etc., and ${\text{A}}',{\text{B}}',$ etc., are constant. If we wish to introduce into the equation the absolute index of refraction ($n$) and the wavelength in vacuo ($\lambda$) in place of ${\text{V}}$ and $\rho ,$ we may divide both sides of the equation by the square of the constant ($k$) representing the velocity of light in vacuo. Then, since

${\frac {\text{V}}{k}}={\frac {1}{n}},$and$kp=\lambda ,$

our equation reduces to

${\frac {1}{n^{2}}}={\frac {\text{H}}{2\pi k^{2}}}-{\frac {2\pi {\text{H}}'}{\lambda ^{2}}}\cdot$

(19)

It is well known that the relation between $n$ and $\lambda$ may be tolerably well but by no means perfectly represented by an equation of this form.

13. If we now give up the presumably inaccurate supposition that ${\text{A, B,}}$ etc., and ${\text{A}}',{\text{B}}',$ etc., are constant, equation (19) will still subsist, but ${\text{H}}$ and ${\text{H}}'$ will not be constant for a given direction of oscillation, but will be functions of $p,$ or, what amounts to the same, of $\lambda .$ Although we cannot therefore use the equation to derive a priori the relation between $n$ and $\lambda ,$ we may use it to derive the values of ${\text{H}}$ and ${\text{H}}'$ from the empirically determined relation between $n$ and $\lambda .$ To do this, we must make use again of the general principle that an