Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/236

12. From the general equation given above (8, 12, or 15), in connection with the solenoidal hypothesis, we may easily derive the laws of the propagation of plane waves in the interior of a sensibly homogeneous medium, and the laws of reflection and refraction at surfaces between such media. This has been done by Maxwell,^[1] Lorentz,^[2] and others,^[3] with fundamental equations more or less similar.

The method, however, by which the fundamental equation has been established in this paper seems free from certain objections which have been brought against the ordinary form of the theory. As ordinarily treated, the phenomena are made to depend entirely on the inductive capacity and the conductivity of the medium, in a manner which may be expressed by the equation

$[{\mathsf {U}}]_{\text{Ave}}=\left({\frac {\text{K}}{4\pi }}-{\frac {p^{2}{\text{C}}}{4\pi ^{2}}}{\frac {d}{dt}}\right)\left({\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}[{\mathsf {U}}]_{\text{Ave}}-\nabla [q]_{\text{Ave}}\right),$

(16)

which will be equivalent to (12), if

${\mathsf {W}}=\left({\frac {\text{K}}{4\pi }}-\iota {\frac {p{\text{C}}}{2\pi }}\right)\left({\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}{\mathsf {W}}-\nabla {\text{Q}}\right),$

(17)

where ${\text{K}}$ and ${\text{C}}$ denote in the most general case the linear vector functions, but in isotropic bodies the numerical coefficients, which represent inductive capacity and conductivity. By a simple transformation {see (9) and (10)}, this equation becomes

$\Theta ^{-1}={\frac {\text{K}}{4\pi }}-\iota {\frac {p{\text{C}}}{2\pi }},$

(18)

where $\Theta ^{-1}$ represents the function inverse to $\Theta .$

Now, while experiment appears to verify the existence of such a law as is expressed by equation (12), it does not show that $\Theta$ has the precise form indicated by equation (16). In other words, experiment does not satisfactorily verify the relations expressed by (16) and (17), if ${\text{K}}$ and ${\text{C}}$ are understood to be the operators (or, in isotropic bodies, the numbers) which represent inductive capacity and conductivity in the ordinary sense of the terms.

The discrepancy is most easily shown in the most simple case, when the medium is isotropic and perfectly transparent, and $\Theta$ reduces to a numerical quantity. The square of the velocity of plane waves is then equal to ${\frac {\Theta }{4\pi }},$ and equation (18) would make it independent of the

1. Phil. Trans., vol. clv (1866), p. 459, or Treatise on Eletricity and Magnetism, chap. xx.
2. Schlömilch's Zeitschrift, vol. xxii, pp. 1–30 and 205–219; xxiii, pp. 197–210.
3. See Fitzgerald, Phil. Trans., vol. clxxi, p. 691; J. J. Thomson, Phil. Mag. (5), vol. ix, p. 284; Rayleigh, Phil. Mag. (5), vol. xii, p. 81.
   That the electromagnetic theory of light gives the conditions relative to the boundary of different media, which are required by the phenomena of reflection and refraction, was first shown by Helmholtz. See Crelle's Journal, vol. lxxii (1870), p. 57.