Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/221

equations is very simple. We have necessarily either $\rho _{2}=0$ and ${\text{V}}^{2}={\text{U}}_{1}^{2},$ or $\rho _{1}=0$ and ${\text{V}}^{2}={\text{U}}_{1}^{2}.$ In this case, the light is linearly polarized, and the directions of oscillation and the velocities of propagation are given by FresneFs law. Experiment has shown that this is the usual case. We wish, however, to investigate the case in which $\phi$ does not vanish. Since the term containing $\phi$ arises from the consideration of those quantities which it was allowable to neglect in the first approximation, we may assume that $\phi$ is always very small in comparison with ${\text{V}}^{3},{\text{U}}_{1}^{3},$ or ${\text{U}}_{2}^{3}.$

17. Equations (28) may be written

${\text{V}}^{2}-{\text{U}}_{1}^{2}=\pm {\frac {\phi }{\text{V}}}{\frac {\rho _{2}}{\rho _{1}}},$${\text{V}}^{2}-{\text{U}}_{2}^{2}=\pm {\frac {\phi }{\text{V}}}{\frac {\rho _{1}}{\rho _{2}}}\cdot$

(29)

By multiplication we obtain

${\text{V}}^{2}({\text{V}}^{2}-{\text{U}}_{1}^{2})({\text{V}}^{2}-{\text{U}}_{2}^{2})=\phi ^{2}.$

(30)

Since $\phi$ is a very small quantity, it is evident from inspection of this equation that it will admit three values of ${\text{V}}^{2},$ of which one will be a very little greater than the greater of the two quantities ${\text{U}}_{1}^{2}$ and ${\text{U}}_{2}^{2}$ another will be a very little less than the less of the same two quantities, and the third will be a very small quantity. It is evident that the values of ${\text{V}}^{2}$ with which we have to do are those which differ but little from ${\text{U}}_{1}^{2}$ and ${\text{U}}_{2}^{2}.$^[1]

For the numerical computation of ${\text{V}},$ when ${\text{U}}_{1},{\text{U}}_{2},$ and $\phi$ are known numerically, we may divide the equation by ${\text{V}}^{2},$ and then solve it as if the second member were known. This will give

${\text{V}}^{2}={\frac {{\text{U}}_{1}^{2}+{\text{U}}_{2}^{2}}{2}}\pm {\sqrt {{\frac {\phi ^{2}}{{\text{V}}^{2}}}+{\frac {({\text{U}}_{1}^{2}-{\text{U}}_{2}^{2})}{4}}}}\cdot$

(31)

By substituting ${\text{U}}_{1}{\text{U}}_{2}$ for ${\text{V}}^{2}$ in the second member, we may obtain a close approximation to the two values of ${\text{V}}^{2}.$ Each of the values obtained may be improved by substitution of that value for ${\text{V}}^{2}$ in the second member of the equation.

For either value of ${\text{V}}^{2}$ we may easily find the ratio of $\rho _{1}$ to $\rho _{2},$ that is, the ratio of the axes of the displacement-ellipse, from one of equations (29), or from the equation

${\frac {\rho _{2}^{2}}{\rho _{1}^{2}}}={\frac {{\text{V}}^{2}-{\text{U}}_{1}^{2}}{{\text{V}}^{2}-{\text{U}}_{2}^{2}}}$

(32)

obtained by combining the two.

1. We should not attribute any physical significance to the third value of ${\text{V}}.$ For this value would imply a wave-length very small in comparison with the length of ordinary waves of light, and with respeet to which our fundamental assumption that the wave-length is very great in comparison with the distances of contiguous molecules would be entirely false. Our analysis, therefore, furnishes no reason for supposing that any such velocities are possible for the propagation of electrical disturbances.